graphical representation examples

Graphical Representation of Data

Graphical representation of data is an attractive method of showcasing numerical data that help in analyzing and representing quantitative data visually. A graph is a kind of a chart where data are plotted as variables across the coordinate. It became easy to analyze the extent of change of one variable based on the change of other variables. Graphical representation of data is done through different mediums such as lines, plots, diagrams, etc. Let us learn more about this interesting concept of graphical representation of data, the different types, and solve a few examples.

Definition of Graphical Representation of Data

A graphical representation is a visual representation of data statistics-based results using graphs, plots, and charts. This kind of representation is more effective in understanding and comparing data than seen in a tabular form. Graphical representation helps to qualify, sort, and present data in a method that is simple to understand for a larger audience. Graphs enable in studying the cause and effect relationship between two variables through both time series and frequency distribution. The data that is obtained from different surveying is infused into a graphical representation by the use of some symbols, such as lines on a line graph, bars on a bar chart, or slices of a pie chart. This visual representation helps in clarity, comparison, and understanding of numerical data.

Representation of Data

The word data is from the Latin word Datum, which means something given. The numerical figures collected through a survey are called data and can be represented in two forms - tabular form and visual form through graphs. Once the data is collected through constant observations, it is arranged, summarized, and classified to finally represented in the form of a graph. There are two kinds of data - quantitative and qualitative. Quantitative data is more structured, continuous, and discrete with statistical data whereas qualitative is unstructured where the data cannot be analyzed.

Principles of Graphical Representation of Data

The principles of graphical representation are algebraic. In a graph, there are two lines known as Axis or Coordinate axis. These are the X-axis and Y-axis. The horizontal axis is the X-axis and the vertical axis is the Y-axis. They are perpendicular to each other and intersect at O or point of Origin. On the right side of the Origin, the Xaxis has a positive value and on the left side, it has a negative value. In the same way, the upper side of the Origin Y-axis has a positive value where the down one is with a negative value. When -axis and y-axis intersect each other at the origin it divides the plane into four parts which are called Quadrant I, Quadrant II, Quadrant III, Quadrant IV. This form of representation is seen in a frequency distribution that is represented in four methods, namely Histogram, Smoothed frequency graph, Pie diagram or Pie chart, Cumulative or ogive frequency graph, and Frequency Polygon.

Advantages and Disadvantages of Graphical Representation of Data

Listed below are some advantages and disadvantages of using a graphical representation of data:

• It improves the way of analyzing and learning as the graphical representation makes the data easy to understand.
• It can be used in almost all fields from mathematics to physics to psychology and so on.
• It is easy to understand for its visual impacts.
• It shows the whole and huge data in an instance.
• It is mainly used in statistics to determine the mean, median, and mode for different data

The main disadvantage of graphical representation of data is that it takes a lot of effort as well as resources to find the most appropriate data and then represent it graphically.

Rules of Graphical Representation of Data

While presenting data graphically, there are certain rules that need to be followed. They are listed below:

• Suitable Title: The title of the graph should be appropriate that indicate the subject of the presentation.
• Measurement Unit: The measurement unit in the graph should be mentioned.
• Proper Scale: A proper scale needs to be chosen to represent the data accurately.
• Index: For better understanding, index the appropriate colors, shades, lines, designs in the graphs.
• Data Sources: Data should be included wherever it is necessary at the bottom of the graph.
• Simple: The construction of a graph should be easily understood.
• Neat: The graph should be visually neat in terms of size and font to read the data accurately.

Uses of Graphical Representation of Data

The main use of a graphical representation of data is understanding and identifying the trends and patterns of the data. It helps in analyzing large quantities, comparing two or more data, making predictions, and building a firm decision. The visual display of data also helps in avoiding confusion and overlapping of any information. Graphs like line graphs and bar graphs, display two or more data clearly for easy comparison. This is important in communicating our findings to others and our understanding and analysis of the data.

Types of Graphical Representation of Data

Data is represented in different types of graphs such as plots, pies, diagrams, etc. They are as follows,

Related Topics

Listed below are a few interesting topics that are related to the graphical representation of data, take a look.

• x and y graph
• Frequency Polygon
• Cumulative Frequency

Examples on Graphical Representation of Data

Example 1 : A pie chart is divided into 3 parts with the angles measuring as 2x, 8x, and 10x respectively. Find the value of x in degrees.

We know, the sum of all angles in a pie chart would give 360º as result. ⇒ 2x + 8x + 10x = 360º ⇒ 20 x = 360º ⇒ x = 360º/20 ⇒ x = 18º Therefore, the value of x is 18º.

Example 2: Ben is trying to read the plot given below. His teacher has given him stem and leaf plot worksheets. Can you help him answer the questions? i) What is the mode of the plot? ii) What is the mean of the plot? iii) Find the range.

Solution: i) Mode is the number that appears often in the data. Leaf 4 occurs twice on the plot against stem 5.

Hence, mode = 54

ii) The sum of all data values is 12 + 14 + 21 + 25 + 28 + 32 + 34 + 36 + 50 + 53 + 54 + 54 + 62 + 65 + 67 + 83 + 88 + 89 + 91 = 958

To find the mean, we have to divide the sum by the total number of values.

Mean = Sum of all data values ÷ 19 = 958 ÷ 19 = 50.42

iii) Range = the highest value - the lowest value = 91 - 12 = 79

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Practice Questions on Graphical Representation of Data

Faqs on graphical representation of data, what is graphical representation.

Graphical representation is a form of visually displaying data through various methods like graphs, diagrams, charts, and plots. It helps in sorting, visualizing, and presenting data in a clear manner through different types of graphs. Statistics mainly use graphical representation to show data.

What are the Different Types of Graphical Representation?

The different types of graphical representation of data are:

• Stem and leaf plot
• Scatter diagrams
• Frequency Distribution

Is the Graphical Representation of Numerical Data?

Yes, these graphical representations are numerical data that has been accumulated through various surveys and observations. The method of presenting these numerical data is called a chart. There are different kinds of charts such as a pie chart, bar graph, line graph, etc, that help in clearly showcasing the data.

What is the Use of Graphical Representation of Data?

Graphical representation of data is useful in clarifying, interpreting, and analyzing data plotting points and drawing line segments , surfaces, and other geometric forms or symbols.

What are the Ways to Represent Data?

Tables, charts, and graphs are all ways of representing data, and they can be used for two broad purposes. The first is to support the collection, organization, and analysis of data as part of the process of a scientific study.

What is the Objective of Graphical Representation of Data?

The main objective of representing data graphically is to display information visually that helps in understanding the information efficiently, clearly, and accurately. This is important to communicate the findings as well as analyze the data.

• Math Article

Graphical Representation

Graphical Representation is a way of analysing numerical data. It exhibits the relation between data, ideas, information and concepts in a diagram. It is easy to understand and it is one of the most important learning strategies. It always depends on the type of information in a particular domain. There are different types of graphical representation. Some of them are as follows:

• Line Graphs – Line graph or the linear graph is used to display the continuous data and it is useful for predicting future events over time.
• Bar Graphs – Bar Graph is used to display the category of data and it compares the data using solid bars to represent the quantities.
• Histograms – The graph that uses bars to represent the frequency of numerical data that are organised into intervals. Since all the intervals are equal and continuous, all the bars have the same width.
• Line Plot – It shows the frequency of data on a given number line. ‘ x ‘ is placed above a number line each time when that data occurs again.
• Frequency Table – The table shows the number of pieces of data that falls within the given interval.
• Circle Graph – Also known as the pie chart that shows the relationships of the parts of the whole. The circle is considered with 100% and the categories occupied is represented with that specific percentage like 15%, 56%, etc.
• Stem and Leaf Plot – In the stem and leaf plot, the data are organised from least value to the greatest value. The digits of the least place values from the leaves and the next place value digit forms the stems.
• Box and Whisker Plot – The plot diagram summarises the data by dividing into four parts. Box and whisker show the range (spread) and the middle ( median) of the data.

General Rules for Graphical Representation of Data

There are certain rules to effectively present the information in the graphical representation. They are:

• Suitable Title: Make sure that the appropriate title is given to the graph which indicates the subject of the presentation.
• Measurement Unit: Mention the measurement unit in the graph.
• Proper Scale: To represent the data in an accurate manner, choose a proper scale.
• Index: Index the appropriate colours, shades, lines, design in the graphs for better understanding.
• Data Sources: Include the source of information wherever it is necessary at the bottom of the graph.
• Keep it Simple: Construct a graph in an easy way that everyone can understand.
• Neat: Choose the correct size, fonts, colours etc in such a way that the graph should be a visual aid for the presentation of information.

Graphical Representation in Maths

In Mathematics, a graph is defined as a chart with statistical data, which are represented in the form of curves or lines drawn across the coordinate point plotted on its surface. It helps to study the relationship between two variables where it helps to measure the change in the variable amount with respect to another variable within a given interval of time. It helps to study the series distribution and frequency distribution for a given problem.  There are two types of graphs to visually depict the information. They are:

• Time Series Graphs – Example: Line Graph
• Frequency Distribution Graphs – Example: Frequency Polygon Graph

Principles of Graphical Representation

Algebraic principles are applied to all types of graphical representation of data. In graphs, it is represented using two lines called coordinate axes. The horizontal axis is denoted as the x-axis and the vertical axis is denoted as the y-axis. The point at which two lines intersect is called an origin ‘O’. Consider x-axis, the distance from the origin to the right side will take a positive value and the distance from the origin to the left side will take a negative value. Similarly, for the y-axis, the points above the origin will take a positive value, and the points below the origin will a negative value.

Generally, the frequency distribution is represented in four methods, namely

• Smoothed frequency graph
• Pie diagram
• Cumulative or ogive frequency graph
• Frequency Polygon

Merits of Using Graphs

Some of the merits of using graphs are as follows:

• The graph is easily understood by everyone without any prior knowledge.
• It saves time
• It allows us to relate and compare the data for different time periods
• It is used in statistics to determine the mean, median and mode for different data, as well as in the interpolation and the extrapolation of data.

Example for Frequency polygonGraph

Here are the steps to follow to find the frequency distribution of a frequency polygon and it is represented in a graphical way.

• Obtain the frequency distribution and find the midpoints of each class interval.
• Represent the midpoints along x-axis and frequencies along the y-axis.
• Plot the points corresponding to the frequency at each midpoint.
• Join these points, using lines in order.
• To complete the polygon, join the point at each end immediately to the lower or higher class marks on the x-axis.

Draw the frequency polygon for the following data

Mark the class interval along x-axis and frequencies along the y-axis.

Let assume that class interval 0-10 with frequency zero and 90-100 with frequency zero.

Now calculate the midpoint of the class interval.

Using the midpoint and the frequency value from the above table, plot the points A (5, 0), B (15, 4), C (25, 6), D (35, 8), E (45, 10), F (55, 12), G (65, 14), H (75, 7), I (85, 5) and J (95, 0).

To obtain the frequency polygon ABCDEFGHIJ, draw the line segments AB, BC, CD, DE, EF, FG, GH, HI, IJ, and connect all the points.

Frequently Asked Questions

What are the different types of graphical representation.

Some of the various types of graphical representation include:

• Line Graphs
• Frequency Table
• Circle Graph, etc.

Read More:  Types of Graphs

What are the Advantages of Graphical Method?

Some of the advantages of graphical representation are:

• It makes data more easily understandable.
• It saves time.
• It makes the comparison of data more efficient.

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Very useful for understand the basic concepts in simple and easy way. Its very useful to all students whether they are school students or college sudents

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Discover 20 Essential Types Of Graphs And Charts And When To Use Them

Table of Contents

1) What Are Graphs And Charts?

2) Charts And Graphs Categories

3) 20 Different Types Of Graphs And Charts

4) How To Choose The Right Chart Type

Data and statistics are all around us. It is very likely that you have found yourself looking at a chart or graph at work, in the news, sports, media, advertising, and many other places at some point in your life. That is because graphical representations of data make it easier to convey important information to different audiences. That said, there is still a lack of charting literacy due to the wide range of visuals available to us and the misuse of statistics . In many cases, even the chart designers are not picking the right visuals to convey the information in the correct way. 

So, how do you make sure you are using and understanding graphs and charts in the right way? In this post, we will provide you with the necessary knowledge to recognize the top 20 most commonly used types of charts and graphs and give insights into when and how to use them correctly. Each type of chart will have a visual example generated with datapine’s professional dashboard software . 

This knowledge will be valuable if you are a data visualization designer, a business user looking to incorporate visual analytics into his/her work, or just an average viewer looking to learn more about the topic. Let’s start this journey by looking at a definition. 

What Are Graphs And Charts?

A graph or chart is a graphical representation of qualitative or quantitative data. It uses different symbols such as bars, lines, columns, tables, box plots, maps, and more, to give meaning to the information, making it easier to understand than raw data. 

As you probably already know, multiple kinds of graphs and charts are widely used in various fields and industries such as business decision-making or research studies. These visual tools are used to find relationships between different data sets and extract valuable conclusions from them. In some cases, they are built by hand, and in others, most commonly, they are built using visualization tools. 

That said, the type of chart or graph you use will vary depending on the aim of the analysis. For instance, percentages are better viewed through a pie or bar chart while data that is changing over time is better viewed over a line chart. For that reason, it is important to have a clear understanding of the different chart types to make sure you are using the right one. 

Below we will discuss the graph and chart categories. These categories will build a solid foundation that will help you pick the right visual for your analytical aims. Let’s dive into them. 

Charts And Graphs Categories

As mentioned, asking the right questions will form the foundations for choosing the right types of visualization graphs for your project, strategy, or business goals. The fundamental categories that differentiate these questions are based on:

• Relationship : Understanding connections between different data points can significantly help discover new relevant insights. For instance, in the medical field, analyzing relationships between diseases and gene interactions can help discover a treatment for a particular disease. When it comes to visuals, a few graphics can help you easily identify and represent relationships. Scatter plots are valuable when you want to represent smaller data sets of two variables while bubble graphs are best for larger information with three or more variables. 
• Distribution: In statistics, distribution refers to the possibility of the occurrence of an outcome. To understand this, scientists and analysts use charts to represent the frequency distribution of the data and extract conclusions from it. For this purpose, they use line charts to analyze trends, scatter plots to highlight similarities across variables, and histograms to see the frequency distribution of a single variable across categories. 
• Composition : The purpose of business graphs and charts for composition is to compare parts to a whole in absolute numbers and normalized forms, usually a percentage. It is one of the most common and traditionally used visualization categories and it is usually limited by the simplicity of the chart types. Common composition graphs include pies, tree maps, and stacked bar charts. 
• Comparison: As its name suggests, this category refers to the comparison of multiple variables or multiple categories within a single variable. When comparing information it is fundamental to pick a chart that will make it easier to understand the differences. These differences can be within multiple elements, for example, top-selling products, or over time, such as the development of sales for different products over a year. For this purpose, tables, spiders, lines, columns, or area graphs are always a good choice.

To get a clearer impression, here is a visual overview of which chart to select based on what kind of data you need to show:

**click to enlarge**

Discover 20 Different Types Of Graphs And Charts

Now that you understand the key charting categories you are ready to dive into the main types of graphs and when to use them. Here, we will focus on the 20 most common types of visuals to represent your data in the most meaningful way possible. Each chart type has a visual example generated with datapine .

1) Number Chart

When to use 

A real-time number chart is essentially a ticker that will give you an immediate overview of a particular KPI . At a glance, you can see any total such as sales, percentage of evolution, number of visitors, etc. This is probably the easiest visualization type to build with the only consideration being the period you want to track. Do you want to show an entire history or simply the latest quarter? It is crucial to label the period clearly so your audience understands what story you are telling. Adding a trend indicator compares your number to the previous period (or to a fixed goal, depending on what you are tracking).

Other considerations

Number charts are often the first thing people see and are the quickest to read, so if there are too many, your narrative can get diluted. Using too many can also make your dashboard a little superficial. If you want more in-depth information, limit the number of number graphs and leave room for other types of data visualization that drill down a little deeper.

When you add a trend indicator, we suggest you compare numbers from the same period. For example, if you are tracking total sales for the current quarter, compare that data to the same quarter last year (or last period – depending on your story). If you select a target manually (perhaps you have no accurate past data), be sure to set realistic goals to be able to get on top of your KPI management practice. Again, remember to label the trend indicator clearly so your audience knows exactly what they are looking at.

2) Line Chart

The purpose of a line chart is to show trends, accelerations (or decelerations), and volatility. They display relationships in how data changes over a period of time. In our example above, we are showing Revenue by Payment Method for all of 2019 . Right away, you can see that the credit card payments were the highest and that everything took a dip in September. The takeaways are quick to register yet have depth.

Too many lines (variables) can make your chart complicated and hard to decipher. You may also find your audience constantly referencing the legend to remind them which one they are looking at. If you have too many variables, it’s time to consider a second (or even third) chart to tell this story.

When it comes to layout, keep your numbers relevant. When you set up your axis scale, keep it close to the highest data point. For example, if we had set the y-axis above to track all the way to 200K (when our highest data point is just over 90K), our chart would have been squished and hard to read. The top half would have been wasted space, and the data crammed. Let your data breathe a little!

One more thing!

A great feature of line graphs is that you can combine them with other types of visualizations, such as bar graphs. Using a double y-axis, one for the bar graph and one for the line, allows you to show two elements of your story in one graph. The primary y-axis below shows orders (bar graph), and the secondary y-axis is sales totals (line). The metrics are different and useful independently, but together, they tell a compelling story.

Maps are great at visualizing your geographic data by location. The information on a map is often displayed in a colored area map (like above) or a bubble map. Because maps are so effective at telling a story, they are used by governments, media, NGOs, nonprofits, public health departments – the list goes on. Maps aren’t just for displaying data; they also direct action. This was seen most recently through the Zika outbreak. Mapping the spread of the disease has helped health officials track it and effectively distribute resources where they are most needed.

Even if you aren’t saving the world from Zika, maps can help! For example, they are great at comparing your organization’s sales in different regions.

Everyone loves maps. However, that doesn’t mean you always need to display one. If the location isn’t a necessary part of your analytics story, you don’t need a map. They take up a lot of room, so only use them when necessary. Also, don’t just fill your maps with data points. Clickhole did a good job of satirizing this common data visualization type by placing 700 red dots on a map. Filling your map with data points doesn’t tell a compelling story; it just overwhelms the audience.

4) Waterfall Chart

This extremely useful chart depicts the power of visualizing data in a static, yet informative manner. It shows the composition of data over a set time period, illustrating the positive or negative values that help in understanding the overall cumulative effect. The decrements and increments can cause the cumulative to fall below or above the axis at various points, causing a clear overview of how the initial value is affected. It is often used in financial departments for analytical purposes, usually depicting the changes in revenue or profit. For example, your MRR ( monthly recurring revenue ), new revenue, upsell, lost, and current revenue. In our example above, we can conclude that our current revenue increased in our set time period.

Waterfall charts are static in their presentation so if you need to show dynamic data sets, then stacked graphs would be a better choice. Also, showing the relationship between selected multiple variables is not optimal for waterfall charts (also known as Cascade charts), as bubble plots or scatter plots would be a more effective solution.

5) Bar Graphs

There are three key types of bar graphs that we will cover in this section: Horizontal, Grouped and Stacked. Although all are in the same chart family, each serves a distinct purpose. Let’s discuss each of them in detail below. 

a) Horizontal Bar Graphs

Horizontal charts are perfect for comparative ranking, like a top-five list. They are also useful if your data labels are really long or complex. Keep them in an order that makes sense, though. Either list by value (like we did above) or, if that’s not the strength, choose a logic for the labels that makes sense, like listing them alphabetically.

Because time is best expressed left to right, it’s better to leave showing an evolution for the column chart. Also, like many charts, when you have too many values, a horizontal bar graph quickly becomes cluttered.

b) Grouped bar graph

When to Use 

Grouped bar charts follow the same logic as horizontal bars, except that they show values for two variables instead of one. The two variables are often displayed in disparate colors to help differentiate them from each other. It is recommended to use this chart type when you want to compare elements within a specific category or across other categories on the chart. For instance, in our example generated with a customer service analytics tool, we can see customer service tickets by channel divided between the total and solved ones. In this case, the grouped chart can help compare the values between the total and unsolved tickets as well as compare the number of solved tickets across channels and extract conclusions.  

Just like with the horizontal one, you need to be careful not to add too many categories into this graph type as they can make it look cluttered. The chart becomes difficult to read with the increase in categories, therefore, it is not the best when it comes to relationship or distribution analysis. 

c) Stacked bar chart

When to Use

A stacked bar chart is a variation of the traditional bar graph but, instead of dealing with one categorical variable, it deals with two or more. Each bar is divided into multiple subcategories that are compared with each other usually using a percentage. In the example above, the chart is comparing management and non-management positions (first categorical variable) that are being occupied by female and diverse employees vs male employees (second categorical variable). This allows the users of the bar to not only focus on the comparisons between the bars themselves but also extract conclusions based on the subcategories from each individual bar. 

When building a successful stacked bar graph it is important to carefully decide which of the two categorical variables will be the primary one. The primary one is the most important one and it will define the overall bar lengths. The secondary one will define the subcategories. Usually, if you are dealing with time ranges or numerical values, these make the best primary variables. However, it will vary from case to case. 

6) Column Graphs

When to use

Column charts are the standard for showing chronological data, such as growth over specific periods, and for comparing data across categories  (you can see this in the example where the accounts payable turnover is being compared based on date ranges). In general, for these kinds of charts, the categories are typically displayed on the horizontal axis while the numerical values are displayed vertically using rectangular columns. The size of the columns is proportional to the values displayed on the chart and their height allows people to easily extract conclusions at a glance. Unlike the bar chart which can display larger or more complex datasets, the column chart has a size limitation making it best to display smaller data. This makes it the go-to visualization for anyone looking for an easy and understandable way to display their information. 

Aside from the obvious design mistakes like using too many colors or too many categories, other things you want to make sure of are: if there is no natural order for the data (e.g. age categories or time ranges), it is recommended to order the values from higher to lowest or lowest to highest. Additionally, the y-axis should always start at 0, otherwise, the height of the columns can become misleading.  

c) Grouped column chart 

Just like the grouped bar chart, the grouped column chart compares two categorical variables instead of one using vertical columns instead of horizontal bars. The purpose of this graph is to see how the subcategories from the secondary variable change within each subcategory of the primary variable. Comparisons can be done within-group or between groups depending on the aim of the analysis.  In our sales data analysis example, Amount of Sales per Channel and Country (last year) , it is clear that we are comparing six regions and five channels. The color coding keeps the audience clued into which region we are referencing, and the proper spacing shows the channels (good design is at the heart of it all!). At a glance, you can see that SEM was the highest-earning channel, and with a little effort, the Netherlands stands out as the region that likely enjoyed the highest sales.

An important consideration when it comes to this graphic is to not use it to compare totals within the different levels of the categorical values. For this purpose, it is better to use a stacked column chart which we will discuss below.  

b) Stacked Column Chart

Stacked charts handle part-to-whole relationships. This is when you are comparing data to itself rather than seeing a total – often in the form of percentages. In the example above, the story isn’t about the total number of customers aged 15-25, but that 22% of the customers were 15-25 in the first quarter of 2014 (and 26% in Q4). The numbers we are working with are relative only to our total.

When showing single part-to-whole relationships, pie charts are the simplest way to go. Twenty-two percent of our customers are 15-25, leaving the other 78% to fit into the pie somehow. People get pie charts. They’re easy. But what if we want to show the same information over different periods? This would be a multiple part-to-whole relationship, and for this, we use a stacked bar graph. Again, we are telling the story of the percentage of customers in a certain age range, per quarter . The total number of each isn’t relevant here (although that information is used in the calculations). With proper spacing, we see each quarter clearly, and the color coding shows that overall, 46-55-year-olds are the most difficult customers to attract.

Aesthetically speaking, when you have too much data, columns become very thin and ugly. This also leaves little room to properly label your chart. Imagine we had 10 different age ranges per column. Some results, if not most, would be only slivers. To make your chart easy to understand, use good colors, proper spacing, and a balanced layout. This invites people to look at your chart and even enjoy it. A pretty chart is a much nicer way to consume data than squinting at a table.

7) Pie Charts

The much-maligned pie chart has had a bad couple of years. In fact, it has become pretty cliché to talk about how bad pie charts are. We understand the pie chart doesn’t do a lot, but it does do some things quite well. Pie charts are useful when demonstrating the proportional composition of a particular variable over a static timeframe. Let’s look at some particular cases:

• When the parts add up to 100%: The “part-to-whole relationship” is built right into it a pie chart in an obvious way. At a glance, any user knows a pie chart is splitting a population into parts and that the total of those parts equals 100%.
• When approximating is okay: The pie chart is particularly effective when eyeballing values are enough to get the conversation going. Also, it’s easier to estimate the percentage value of a pie chart compared to, let’s say, a bar chart. That’s because pies have an invisible scale with 25%, 50%, 75%, and 100% built-in at four points of the circle. Our eyes can easily decipher these proportions, driving the conversation about what variables do and don’t take up most of the pie. Your audience doesn’t have to guess the proportions – you can easily add data labels or build the sister of the pie chart, the donut chart, to display additional information.
• When there aren’t many proportions to the variable or they are combined: Pie charts are great when answering questions like, “What two largest suppliers control 65% of the market?”

Your audience isn’t always going to be comprised of data scientists. Accordingly, your presentation should be tailored to your particular audience. This brings us to another pie chart strength: people are familiar with pie charts. Any audience member will feel comfortable interpreting what the pie chart is presenting. As a bonus, circles generate more positive emotions: our brains like to look at circles over sharp corners. In the end, a pie chart simplifies the data story and encourages the audience.

Data visualization guru Edward Tufte famously declared that “pie charts are bad, and the only thing worse than one pie chart is lots of them.” We already talked about the pros of pie charts and why we don’t adhere to this strict no-pie-chart philosophy. We should also state that there are plenty of instances where you should not use a pie chart. First off, pie charts portray a stagnate time frame, so trending data is off the table with this visualization method. Make sure your audience understands the timeframe portrayed and try to document or label this applied filter somewhere.

Pie charts are also not the best types of data charts to make precise comparisons. This is especially true when there are multiple small pieces to the pie. If you need to see that one slice is 1% larger than another, it’s better to go with a bar chart. Another thing about multiple pieces to your pie – you don’t want too many. Pie charts are most effective when just displaying two portions. They lose presentation value after six segments. After six, it is hard for the eyes to decipher the slice's proportion. It also becomes difficult to label the pie chart, and valuable online dashboard /reporting real estate is often wasted in the process.

This brings us to the last issue: circles take up space. If you are using multiple pie charts in a dashboard, it is probably best to combine the data in one chart. We recommend checking out the stacked bar chart for these cases. You can also have a look at the different pie charts that are commonly used and explore the disadvantages of pie charts .

8) Gauge Charts

Gauge charts , also known as dial charts or speedometer charts, use needles and colors to show data similar to reading on a dial/speedometer, and they provide an easily digested visual. They are great for displaying a single value/measure within a quantitative context, such as to the previous period or to a target value. The gauge chart is often used in executive dashboards and reports to display progress against key business indicators. All you need to do is assign minimum and maximum values and define a color range, and the gauge chart will display an immediate trend indication.

Gauge charts are great for KPIs and single data points. After that, they can get a bit messy. With only one data point, you can’t easily compare different variables. You also can’t trend data using gauge charts. All of this makes taking actionable insight from a gauge chart difficult. Furthermore, they take up a lot of space – if your live dashboard has precious real estate, it may not be most efficient to fill it with multiple gauge charts. Using one chart to summarize multiple KPIs, you will likely get more bang for your buck.

9) Scatter Plot

Scatter plot is not only fun to say – it’s what you need when looking for the correlation in a large data set. The data sets need to be in pairs with a dependent variable and an independent variable. The dependent (the one the other relies on) becomes the y-axis, and the independent – the x-axis. When the data is distributed on the plot, the results show the correlation to be positive, negative (each to varying degrees), or nonexistent. Adding a trend line will help show the correlation and how statistically significant it is.

Scatter plots only work when you have a lot of data points and a correlation. If you are only talking about a few pieces of information, a scatter plot will be empty and pointless. The value comes through only when there are enough data points to see clear results. If you only have a little data or if your scatter plot shows no correlation at all, this chart has no place on your business dashboard. 

10) Spider Chart

Spider charts, or radar charts, are comparative charts used when multivariate data is displayed with three or more quantitative variables (aspects). This is useful when you want to evaluate two or more “things” using more than three aspects, all of which are similarly quantifiable. It’s certainly a mouthful, but it’s simple when you put it into use. Spider charts are great for rankings, appraisals, and reviews. For example, the three “things” we are comparing in our e-commerce example above are regions: Australia, Europe, and North America. The aspects we are comparing against are products sold are Cameras, TVs, Cell Phones, Games, and Computers. Each variable is being compared by how many units were sold – between 0 and 500. Europe is clearly outselling in all areas, and Australia is particularly weak in Cameras and Cell Phones. The concentration of strengths and weaknesses is evident at a glance.

This is not the easiest data analysis chart to pull off, but it really impresses when done correctly. Using this chart if you have more than five values in your dimension (five “things” to evaluate) makes it hard to read, which can make it pointless altogether. Whether you use solid lines or shaded areas, too many layers are difficult to interpret. Naturally, it is not a choice when you want to show time (the whole circular thing...).

We know – tables aren’t technically a graph. But sometimes, you really just need a table to portray your data in its raw format. With a table, you can display a large number of precise measures and dimensions. You can easily look up or compare individual values while also displaying grand totals. This is particularly beneficial when your audience needs to know the underlying data or get into the “weeds.” Tables are also effective if you have a diverse audience where each person wants to look at their own piece of the table. They are also great at portraying a lot of text or string values.

Remember – just because you are using a table doesn’t mean it can’t be visually pleasing. You can use various colors, border styles, font types, number formats, and icons to highlight and present your data effectively.

There are many reasons to use a table, but there are also many instances where different types of charts are a better choice. It all comes down to our eyes and brain. Tables interact primarily with the verbal system – we read tables. This reading includes processing the displayed information in a sequential fashion. Users read down columns or across rows of numbers, comparing one number to another. The keywords here are reading, processing, and time. Tables take longer to digest.

Graphs, on the other hand, are perceived by our visual system. They give numbers shape and form and tell a data story. They can present an immense amount of data quickly and in an easy-to-consume fashion. If data visualization is needed to identify patterns and relationships, a table is not the best choice. Also, while it is fun to get creative with colors, formatting, and icons, make sure your formatting and presentation choices are increasing your perception. The tables are hard enough to read as is!

12) Area Charts

The area chart is closely related to the line chart. Both chart types depict a time-series relationship, show continuity across a dataset, and are good for seeing trends rather than individual values. That said, there are some key differences between the two. Because of these differences, “when to use area charts” does not equal “when to use line charts.”

Line charts connect discrete but continuous data points through straight line segments. This makes them effective for facilitating trend analyses. Area charts technically do the same, except that the area below the plotted lines is filled with color. In this case, an un-stacked area chart is the same thing as a line chart – just with more coloring. The problem you run into here is occlusion: when you start comparing multiple variables/categories in an unstacked area chart, the upper layers obscure the lower layers. You can play around with transparency, but after three variables, un-stacked area charts are hard to read.

This brings us to the most commonly used area chart: the stacked area chart. Like stacked bar charts, stacked area charts portray a part-to-whole relationship. The total vertical of a stacked area chart shows the whole, while the height of each different dataset shows the parts. For example, a stacked area chart can show the sales trends for each region and the total sales trend. There are two different stacked area chart types you can use to portray the part-to-whole relationship.

Traditional Stacked Area Chart: The raw values are stacked, showing how the whole changes over time.

Stacked Percentage Area Chart: Percentages are stacked to show how the relationship between the different parts changes over time. This is best used to show the distribution of categories as parts of a whole where the cumulative total is less important.

As we hinted earlier, for the most part, you should stay away from un-stacked area charts. If you are just comparing 2-3 different variables that don’t obscure each other, then go ahead. But in general, they are often messy and don’t follow data visualization and dashboard design best practices . When it comes to stacked area charts, don’t use them when you don’t need to portray a part-to-whole relationship – use a line graph instead. Also, if you are trying to compare 7+ series, a stacked area graph becomes hard to read. In this case, you should once again turn to the line graph.

13) Bubble Plots

Bubble charts, or bubble graphs, are among the best types of data graphs for comparing several values or sets of data at a glance. If you’re looking to show the relationship between different product categories, revenue streams, investment risks, costs, or anything similar, bubble charts or plots are incredibly effective.

For instance, our example bubble plot showcases the relationship between a mix of retail product categories, primarily the number of orders and profit margin.

Here, you can tell that the TV & Home Theater product category has the highest number of orders (around 3,000 as you can see from the number scale on the left) as well as the highest profit margin, and therefore, it is the biggest bubble on the chart. Comparatively, the camera category shows the lowest number of orders in addition to the smallest profit margin and naturally is the smallest bubble on the chart.

The bubble plot is extremely powerful for visualizing two or more variables with multiple dimensions. And here, the bigger the bubble, the higher the profit margin. Not only are bubble plots visually stimulating, but they are also incredibly effective when building a comparative narrative for a specific audience.

It's difficult to go too far wrong with bubble charts, but the most common mistake with these types of business charts is focusing on varying the “radius” of the values rather than the “area” they take up on the chart. Doing so sometimes makes the bubbles on the plot disproportionate to the graph, making the information misleading at a glance. In short, your bubbles should be accurate in terms of size compared to the values. Get this right, and you’ll get the results you deserve. 

14) Boxplot 

Just like the histogram, the box plot is a graph that is used to represent the distribution of numeric data using boxes and lines. Each box is composed of five elements also known as the “Five-number summary” which are the minimum, first quartile, median (second quartile), third quartile, and maximum. Each of these elements represents a value and how it is distributed within the data set. Anything outside these values would be considered an outlier. An outlier is any value that is extremely high or extremely low compared to the nearest data point. Outliers (which are usually plotted as dots in the chart) need to be identified because they can affect the end result of the analysis and box plots are the best visuals to do so.  

Just like other types of charts on this list, box plots are not the best choice when it comes to big data sets. Their visual simplicity makes it hard to see details about the distribution results which makes it more difficult to interpret, especially when dealing with complex data. Plus, this chart works at its best when comparing different groups (as seen in our example above). So, if you are trying to look at the distribution of one single group a histogram is a better choice. 

15) Funnel chart 

As its name suggests, a funnel chart is a visualization type used to show how data moves or flows through a specific process. They are commonly utilized to display sales, recruitment, or order fulfillment funnels where the values are often decreasing as the funnel becomes smaller. This can be seen in the example above in which the number of potential clients decreases at each stage of the sales funnel. This is a natural progression that happens because not every person that shows interest in the opportunities stage will end up buying the product or subscribing to the service. 

In some cases, the sizes of the sections of the funnel chart are plotted proportionately with the value they are representing. This means the top section is 100% and the rest will represent their corresponding percentage with their size. This is not the case with our example in which the sections are sized to match the funnel shape, not the values contained in each section. 

Funnel graphics are very specific visuals that can only be used in particular cases. You should only use it if your data goes through a sequence of stages and the values are decreasing with each stage. Plus, they are only useful to represent a single variable which means they cannot be used to visualize relationships between variables. A good alternative for a funnel graph is a bar or column chart. 

16) Bullet chart 

A bullet chart is a variation of a bar or column chart but it provides some extra visual elements to give more context to the data. It is usually used for performance tracking to make comparisons against a goal or other relevant values and it is composed of three key elements. A single measure is represented by a darker shade bar with a length that represents the performance of that value, qualitative ranges are represented by lighter shades in the background, and a target or comparative measure which is represented with a small line that is perpendicular to the orientation of the graph. Bullet charts are great alternatives to gauge charts, especially when you are working with a KPI dashboard and don’t want to take up too much space from it. 

It is important to note that bullet charts are complex visuals that might be challenging to understand for non-technical audiences. In some cases, some people might choose to remove the shaded background to focus only on the actual value against the target or remove the target and focus on the qualitative ranges to make the chart friendlier to analyze.    This variation is also known as an overlapping bullet chart and it can be done using columns and bars, as we will see in our two examples below.

1. Overlapping bars bullet chart 

As we saw with different graph types previously, the bullet chart can be vertical (using columns) or horizontal (using bars). It is recommended to use bars when you want to display more categories or longer category names to avoid making the visual cluttered. In the example above, we can see the number of orders by product of the current year compared to a target. In this case, due to the number of products, a bar bullet graph is the best choice as it contains a lot of information without affecting the readability of the data.

2. Overlapping columns bullet chart 

On the other side, a bullet column chart is a better choice when you want to organize categories from left to right or when you have fewer categories to show. In this case, we can see the number of orders by product category. Given that product categories are fewer than the actual number of products, it is a good choice to pick columns to represent this data. In a traditional bullet chart, the number of orders by a quarter could be added for additional context as qualitative measures.

17) Treemap chart 

A treemap is a chart type used to display hierarchical data through rectangles that decrease their size as the value decreases, this process is also referred to as nesting. It is used to display large amounts of raw data in a visually appealing way that allows users to easily extract valuable conclusions at a glance. Its name comes from the shape of a tree, as the chart can be divided into multiple categories with different “branches” and “sub-branches”. Each of these categories should have a different color and the dimensions of the rectangles are based on the size of the data being displayed. 

Given that a treemap is used to visualize massive amounts of raw data, they can display an infinite amount of subcategories (or sub-branches) which can make them harder to understand. However, in most cases, users can drill down into the different categories to dig deeper into the data and answer different questions that might arise. 

If you are not trying to show hierarchical data then you should stay away from treemaps. Just like it happens with pie charts, this visualization is simply showing parts-to-whole relationships, therefore, it becomes useless for other purposes. You should also avoid treemaps if the data being displayed is too close in size. This defeats the purpose of the graph which is to easily identify the largest item from a specific category. A few alternatives for treemaps include column charts and scatter plots.

18) Stream charts 

A stream graph is considered a variation of the area chart with the difference that, while the area chart is plotted with a fixed x-axis, the stream graph has values displayed around a central axis. Hence, the flowing river-like shape. They are frequently used to identify trends and patterns in big datasets with multiple categories and evaluate how they change over time. Just like with other kinds of charts on this list, the width and length of the streams are proportional to the values being displayed. The colors can represent different categories or other specific criteria. In our example above, we can see the number of orders by product categories each month. The width of each stream can provide valuable insights into the performance of each category. For instance, from June to August orders for TVs and Home Theaters decreased a lot compared to other months so some conclusions need to be drawn.

In general, if your aim is to use the chart to deeply analyze the data and extract conclusions from it, then the stream is not your best option. They are often cluttered with a lot of information which can lead to legibility issues. This can happen especially when you have smaller categories that end up looking way too small compared to bigger ones. For that reason, it is best to use stream charts as interactive visuals instead of static or printed ones. 

19) Word Cloud 

A word cloud is a straightforward type of graph that displays a set of words concerning a specific topic. The words are arranged in different directions and the sizes of the words will vary depending on specific criteria. For example, if a word cloud is generated based on a text from product reviews, the size of the words can be influenced by the number of times each word is mentioned within the text. On the other hand, if you are generating a word cloud of all the countries in the world, the names of the countries can be bigger or smaller depending on their population. From an analytical perspective, word clouds don’t provide a lot of value apart from being an engaging and visually appealing way of presenting a topic or supporting discussions. 

There is no general rule when it comes to colors on a word cloud. Some might use different colors to provide meaning to certain words while others might use standard colors to match their branding. Whichever case you are using, the rule of not adding too many colors to avoid overcrowding the visual still applies when it comes to word clouds. 

20) Progress chart 

As its name suggests, this chart is used to track the progress of a specific activity or scenario usually in a percentage form. It can be represented using bars or columns and is often tracked against a set target, as seen in the graph examples below, in which you can see a colored area representing the completed percentage and a lighter shade representing the remaining percentage to complete 100%. Progress graphs are wildly popular when tracking the development of a project as they provide a clear overview of the status of different tasks. They are also valuable visuals when you are trying to show any kind of percentage value or progress against a target. 

Progress charts are very straightforward and don’t provide a lot more information than the development of a metric. If you want to gain more insights you can explore using a bullet chart as they provide more context to the data. 

• Progress bar graph 

The progress bar chart is used to track the progress of a specific activity or metric using horizontal bars. The example above is tracking the percentage of purchases in time and budget from a procurement department. Ideally, the end goal for each category would be 100% as this means all purchases are made on the expected time and budget. However, this is not always the case and the progress bar is a great way to see how far from the expected target the values actually are. In this case, the average is represented by a darker color of green, and the remaining percentage to reach 100% is represented by a lighter shade.  

• Progress column graph 

The progress column chart is a type of progress chart that uses columns to represent different data values. In this case, our example is showing the average time to fill a position by the department where each department has a predefined target they are expected to reach. In this case, the target value is represented by a dark purple dot, in other cases, it could be represented with a lighter shade of the same purple from the column. Using a progress chart to represent this metric is a great way to compare the different departments and see if any of the processes need to be optimized to better reach the expected target. 

How To Choose The Right Chart Type: 9 Essential Questions To Ask

To go further into detail, we have selected the top 9 questions you need to consider to ensure success from the very start of your journey.

1. What story do you want to tell?

At its core, data charts are about taking data and transforming it into actionable insight by using visuals to tell a story. Data-driven storytelling is a powerful force as it takes stats and metrics and puts them into context through a narrative that everyone inside or outside of the organization can understand.

By asking yourself what kind of story you want to tell with your data and what message you want to convey to your audience, you’ll be able to choose the right data visualization types for your project or initiative. And ultimately, you’re likely to enjoy the results you're aiming for.

For more on data storytelling, check out our full guide for dashboard presentation and storytelling.

2. Who do you want to tell it to?

Another key element of choosing the right data visualization types is gaining a clear understanding of who you want to tell your story to – or in other words, asking yourself the question, “ Who is my audience ?”

You may be aiming your data visualization efforts at a particular team within your organization, or you may be trying to communicate a set of trends or predictive insights to a selection of corporate investors. Take the time to research your audience, and you’ll be able to make a more informed decision on which data visualization chart types will make the most tangible connection with the people you’ll be presenting your findings to.

3. How big is your data? 

As you probably learned from our list of the essential types of charts and when to use them, the size of your data will significantly affect the type of visualization you decide to use. Some charts are not meant to be used with massive amounts of data due to design aspects while others are perfect for displaying larger information. 

For example, pie charts are not good if you are trying to show multiple categories. For that purpose, a scatter plot works best. Another example is with column and bar charts. Bar charts use horizontal bars that make it easier to represent larger data sets. On the other side, column charts are limited by size due to their vertical orientation, making them better for smaller data. 

4. What is the type of data you are using? 

Another important question to ask yourself is what type of data you are using. As we saw at the beginning of the post, there are 4 key categories when it comes to data visualization: composition, distribution, relationship, and comparison. There are also qualitative and quantitative data that can be better represented using a particular graphic. For this reason, it is important to carefully define the type of data you are using before thinking about visualizing it. In the following questions, we will see what you need to ask yourself based on the mentioned categories. 

5. Are you looking to analyze particular trends?

Every data visualization project or initiative is slightly different, which means that different data visualization chart types will suit varying goals, aims, or topics.

After gaining a greater level of insight into your audience as well as the type of story you want to tell, you should decide whether you're looking to communicate a particular trend relating to a particular data set, over a predetermined time period. What will work best?

• Line charts
• Column charts
• Area charts

6. Do you want to demonstrate the composition of your data?

If your primary aim is to showcase the composition of your data – in other words, show how individual segments of data make up the whole of something – choosing the right types of data visualizations is crucial in preventing your message from becoming lost or diluted.

In these cases, the most effective types of visual charts include:

• Waterfall charts
• Stacked charts
• Map-based graphs (if your information is geographical)

7. Do you want to compare two or more sets of values?

While most types of data visualizations will allow you to compare two or more trends or data sets, there are certain graphs or charts that will make your message all the more powerful.

If your main goal is to show a direct comparison between two or more sets of information, the best choice would be:

• Bubble charts
• Spider charts
• Columned visualizations
• Scatter plots

Data visualization is based on painting a picture with your data rather than leaving it sitting static in a spreadsheet or table. Technically, any way you choose to do this count, but as outlined here, there are some charts that are way better at telling a specific story.

8. Is timeline a factor?

By understanding whether the data you’re looking to extract value from is time-based or time-sensitive, you’ll be able to select a graph or chart that will provide you with an instant overview of figures or comparative trends over a specific period.

In these instances, incredibly effective due to their logical, data-centric designs, functionality and features are:

• Dynamic line charts

9. How do you want to show your KPIs?

It’s important to ask yourself how you want to showcase your key performance indicators as not only will this dictate the success of your analytical activities but it will also determine how clearly your visualizations or data-driven stories resonate with your audience.

Consider what information you’re looking to gain from specific KPIs within your campaigns or activities and how they will resonate with those that you’ll be sharing the information with - if necessary, experiment with different formats until you find the graphs or charts that fit your goals exactly.

Here are two simple bonus questions to help make your data visualization types even more successful:

• Are you comparing data or demonstrating a relationship?
• Would you like to demonstrate a trend?

At datapine, data visualization is our forte. We know what it takes to make a good dashboard – and this means crafting a visually compelling and coherent story.

"Visualization gives you answers to questions you didn’t know you had." – Ben Shneiderman

Design-thinking In Data Visualization

When it comes to different data visualization types, there is no substitute for a solid design. If you take the time to understand the reason for your data visualization efforts, the people you’re aiming them at, and the approaches you want to take to tell your story, you will yield great results.

Here at datapine, we’ve developed the very best design options for our dashboard reporting software , making them easy to navigate yet sophisticated enough to handle all your data in a way that matters.

With our advanced dashboard features , including a host of global styling options, we enable you to make your dashboard as appealing as possible to the people being presented with your data.

Your part in creating an effective design for the different types of data charts boils down to choosing the right visualization to tell a coherent, inspiring, and widely accessible story. Rarely will your audience understand how much strategic thought you have put into your selection of dashboards – as with many presentational elements, the design is often undervalued. However, we understand how important this is, and we’re here to lend a helping hand.

In this guide, we covered different types of charts to represent data, explored key questions you need to ask yourself to choose the right ones, and saw examples of graphs to put their value into perspective. By now, you should have a better understanding of how each type of visual works and how you can use them to convey your message correctly. 

To summarize, here are the top types of charts and their uses:

• Number Chart - gives an immediate overview of a specific value .
• Line Chart - shows trends and changes in data over a period of time .
• Maps - visualizes data by geographical location.
• Waterfall Chart - demonstrates the static composition of data.
• Bar Graphs - used to compare data of large or more complex items .
• Column Chart - used to compare data of smaller items. 
• Gauge Chart - used to display a single value within a quantitative context.
• Pie Chart - indicates the proportional composition of a variable.
• Scatter Plot - applied to express relations and distribution of large sets of data.
• Spider Chart - comparative charts great for rankings, reviews, and appraisals.
• Tables - show a large number of precise dimensions and measures .
• Area Chart - portrays a part-to-whole relationship over time .
• Bubble Plots - visualizes 2 or more variables with multiple dimensions.
• Boxplot -  shows data distribution within multiple groups.
• Funnel Chart - to display how data moves through a process.
• Bullet Chart - comparing the performance of one or more primary measures .
• Treemap - to plot large volumes of hierarchical data across various categories.
• Stream Graph - shows trends and patterns over time in large volumes of data.
• Word Cloud - to observe the frequency of words within a text.  
• Progress Chart - displays progress against a set target or goal.

But in our hyper-connected digital age, there are many more different kinds of graphs you can use to your advantage. Putting everything together in a professional business dashboard is even better. These visual tools provide centralized access to your most important data to get a 360-view of your performance so you can optimize it and ensure continuous growth.

Complete with stunning visuals, our advanced online data visualization software can make it easy for you to manipulate your data and visualize it using professional dashboards. The best part is, you can try it for a 14-day trial , completely free!

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2.1: Introduction

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Learning Objectives

By the end of this chapter, the student should be able to:

• Display data graphically and interpret graphs: stemplots, bar charts, frequency polygons, histograms, etc.

Once you have collected data, what will you do with it? Data can be described and presented in many different formats. For example, suppose you want to find a change in temperature in a particular city over time. Looking at all the raw data can be confusing and overwhelming. A better way to look at that data would be to create a graph that displays the data in a visual manner. Then patterns can more easily be discerned.

In this chapter, you will study graphical ways to describe and display your data. You will learn to create, and more importantly, interpret a variety of graph types, and you will learn when to use each type of graph.

A statistical graph is a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data clusters and where there are only a few data values. Newspapers and the Internet use graphs to show trends and to enable readers to compare facts and figures quickly. Statisticians often graph data first to get a picture of the data. Then, more formal tools may be applied.

Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar graph, the histogram, the stem-and-leaf plot, the frequency polygon (a type of broken line graph), the pie chart, and the box plot. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs, as well as frequency polygons, and time series graphs.

This book contains instructions for constructing some graph types using Excel.

Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] .

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Two common types of graphic displays are bar charts and histograms. Both bar charts and histograms use vertical or horizontal bars to represent the number of data points in each category or interval. The main difference graphically is that in a  bar chart  there are spaces between the bars and in a  histogram  there are not spaces between the bars. Why does this subtle difference exist and what does it imply about graphic displays in general?

Displaying Data

It is often easier for people to interpret relative sizes of data when that data is displayed graphically. Note that a  categorical variable  is a variable that can take on one of a limited number of values and a  quantitative variable  is a variable that takes on numerical values that represent a measurable quantity. Examples of categorical variables are tv stations, the state someone lives in, and eye color while examples of quantitative variables are the height of students or the population of a city. There are a few common ways of displaying data graphically that you should be familiar with. 

A  pie chart  shows the relative proportions of data in different categories.  Pie charts  are excellent ways of displaying categorical data with easily separable groups. The following pie chart shows six categories labeled A−F.  The size of each pie slice is determined by the central angle. Since there are 360 o  in a circle, the size of the central angle θ A  of category A can be found by:

CK-12 Foundation -  https://www.flickr.com/photos/slgc/16173880801  - CCSA

A  bar chart  displays frequencies of categories of data. The bar chart below has 5 categories, and shows the TV channel preferences for 53 adults. The horizontal axis could have also been labeled News, Sports, Local News, Comedy, Action Movies. The reason why the bars are separated by spaces is to emphasize the fact that they are categories and not continuous numbers. For example, just because you split your time between channel 8 and channel 44 does not mean on average you watch channel 26. Categories can be numbers so you need to be very careful.

CK-12 Foundation -  https://www.flickr.com/photos/slgc/16173880801  - CCSA

A  histogram  displays frequencies of quantitative data that has been sorted into intervals. The following is a histogram that shows the heights of a class of 53 students. Notice the largest category is 56-60 inches with 18 people.

A  boxplot  (also known as a  box and whiskers plot ) is another way to display quantitative data. It displays the five 5 number summary (minimum, Q1,  median , Q3, maximum). The box can either be vertically or horizontally displayed depending on the labeling of the axis. The box does not need to be perfectly symmetrical because it represents data that might not be perfectly symmetrical.

Earlier, you were asked about the difference between histograms and bar charts. The reason for the space in bar charts but no space in histograms is bar charts graph categorical variables while histograms graph quantitative variables. It would be extremely improper to forget the space with bar charts because you would run the risk of implying a spectrum from one side of the chart to the other. Note that in the bar chart where TV stations where shown, the station numbers were not listed horizontally in order by size. This was to emphasize the fact that the stations were categories.

Create a boxplot of the following numbers in your calculator.

8.5, 10.9, 9.1, 7.5, 7.2, 6, 2.3, 5.5

Enter the data into L1 by going into the Stat menu.

CK-12 Foundation - CCSA

Then turn the statplot on and choose boxplot.

Use Zoomstat to automatically center the window on the boxplot.

Create a pie chart to represent the preferences of 43 hungry students.

• Other – 5
• Burritos – 7
• Burgers – 9
• Pizza – 22

Create a bar chart representing the preference for sports of a group of 23 people.

• Football – 12
• Baseball – 10
• Basketball – 8
• Hockey – 3

Create a histogram for the income distribution of 200 million people.

• Below $50,000 is 100 million people
• Between $50,000 and $100,000 is 50 million people
• Between $100,000 and $150,000 is 40 million people
• Above $150,000 is 10 million people

1. What types of graphs show categorical data?

2. What types of graphs show quantitative data?

A math class of 30 students had the following grades:

3. Create a bar chart for this data.

4. Create a pie chart for this data.

5. Which graph do you think makes a better visual representation of the data?

A set of 20 exam scores is 67, 94, 88, 76, 85, 93, 55, 87, 80, 81, 80, 61, 90, 84, 75, 93, 75, 68, 100, 98

6. Create a histogram for this data. Use your best judgment to decide what the intervals should be.

7. Find the  five number summary  for this data.

8. Use the  five number summary  to create a boxplot for this data.

9. Describe the data shown in the boxplot below.

10. Describe the data shown in the histogram below.

A math class of 30 students has the following eye colors:

11. Create a bar chart for this data.

12. Create a pie chart for this data.

13. Which graph do you think makes a better visual representation of the data?

14. Suppose you have data that shows the breakdown of registered republicans by state. What types of graphs could you use to display this data?

15. From which types of graphs could you obtain information about the spread of the data? Note that spread is a measure of how spread out all of the data is.

Review (Answers)

To see the Review answers, open this  PDF file  and look for section 15.4. 

Additional Resources

PLIX: Play, Learn, Interact, eXplore - Baby Due Date Histogram

Practice: Types of Data Representation

Real World: Prepare for Impact

• School Guide
• Class 9 Syllabus
• Maths Notes Class 9
• Science Notes Class 9
• History Notes Class 9
• Geography Notes Class 9
• Political Science Notes Class 9
• NCERT Soln. Class 9 Maths
• RD Sharma Soln. Class 9
• Math Formulas Class 9
• CBSE Class 9 Maths Revision Notes

Chapter 1: Number System

• Number System in Maths
• Natural Numbers | Definition, Examples, Properties
• Whole Numbers - Definition, Properties and Examples
• Rational Number: Definition, Examples, Worksheet
• Irrational Numbers- Definition, Identification, Examples, Symbol, Properties
• Real Numbers
• Decimal Expansion of Real Numbers
• Decimal Expansions of Rational Numbers
• Representation of Rational Numbers on the Number Line | Class 8 Maths
• Represent √3 on the number line
• Operations on Real Numbers
• Rationalization of Denominators
• Laws of Exponents for Real Numbers

Chapter 2: Polynomials

• Polynomials in One Variable - Polynomials | Class 9 Maths
• Polynomial Formula
• Types of Polynomials
• Zeros of Polynomial
• Factorization of Polynomial
• Remainder Theorem
• Factor Theorem
• Algebraic Identities

Chapter 3: Coordinate Geometry

• Coordinate Geometry
• Cartesian Coordinate System
• Cartesian Plane

Chapter 4: Linear equations in two variables

• Linear Equations in One Variable
• Linear Equation in Two Variables
• Graph of Linear Equations in Two Variables
• Graphical Methods of Solving Pair of Linear Equations in Two Variables
• Equations of Lines Parallel to the x-axis and y-axis

Chapter 5: Introduction to Euclid's Geometry

• Euclidean Geometry
• Equivalent Version of Euclid’s Fifth Postulate

Chapter 6: Lines and Angles

• Lines and Angles
• Types of Angles
• Pairs of Angles - Lines & Angles
• Transversal Lines
• Angle Sum Property of a Triangle

Chapter 7: Triangles

• Triangles in Geometry
• Congruence of Triangles |SSS, SAS, ASA, and RHS Rules
• Theorem - Angle opposite to equal sides of an isosceles triangle are equal | Class 9 Maths
• Triangle Inequality Theorem, Proof & Applications

Chapter 8: Quadrilateral

• Angle Sum Property of a Quadrilateral
• Quadrilateral - Definition, Properties, Types, Formulas, Examples
• Introduction to Parallelogram: Properties, Types, and Theorem
• Rhombus: Definition, Properties, Formula, Examples
• Kite - Quadrilaterals
• Properties of Parallelograms
• Mid Point Theorem

Chapter 9: Areas of Parallelograms and Triangles

• Area of Triangle | Formula and Examples
• Area of Parallelogram
• Figures on the Same Base and between the Same Parallels

Chapter 10: Circles

• Circles in Maths
• Radius of Circle
• Tangent to a Circle
• What is the longest chord of a Circle?
• Circumference of Circle - Definition, Perimeter Formula, and Examples
• Angle subtended by an arc at the centre of a circle
• What is Cyclic Quadrilateral
• Theorem - The sum of opposite angles of a cyclic quadrilateral is 180° | Class 9 Maths

Chapter 11: Construction

• Basic Constructions - Angle Bisector, Perpendicular Bisector, Angle of 60°
• Construction of Triangles

Chapter 12: Heron's Formula

• Area of Equilateral Triangle
• Area of Isosceles Triangle
• Heron's Formula
• Applications of Heron's Formula
• Area of Quadrilateral
• Area of Polygons

Chapter 13: Surface Areas and Volumes

• Surface Area of Cuboid
• Volume of Cuboid | Formula and Examples
• Surface Area of Cube
• Volume of a Cube
• Surface Area of Cylinder (CSA and TSA) |Formula, Derivation, Examples
• Volume of Cylinder
• Surface Area of Cone
• Volume of Cone | Formula, Derivation and Examples
• Surface Area of Sphere | CSA, TSA, Formula and Derivation
• Volume of a Sphere
• Surface Area of a Hemisphere
• Volume of Hemisphere

Chapter 14: Statistics

• Collection and Presentation of Data

Graphical Representation of Data

• Bar graphs and Histograms
• Central Tendency
• Mean, Median and Mode

Chapter 15: Probability

• Experimental Probability
• Empirical Probability
• CBSE Class 9 Maths Formulas
• NCERT Solutions for Class 9 Maths: Chapter Wise PDF 2024
• RD Sharma Class 9 Solutions

In today’s world of the internet and connectivity, there is a lot of data available and some or the other method is needed for looking at large data, the patterns, and trends in it. There is an entire branch in mathematics dedicated to dealing with collecting, analyzing, interpreting, and presenting the numerical data in visual form in such a way that it becomes easy to understand and the data becomes easy to compare as well, the branch is known as Statistics . The branch is widely spread and has a plethora of real-life applications such as Business Analytics, demography, astrostatistics, and so on. There are two ways of representing data, 

• Pictorial Representation through graphs.

They say, “A picture is worth the thousand words”.  It’s always better to represent data in graphical format. Even in Practical Evidence and Surveys, scientists have found that the restoration and understanding of any information is better when it is available in form of visuals as Human beings process data better in visual form than any other form. Does it increase the ability 2 times or 3 times? The answer is it increases the Power of understanding 60,000 times for a normal Human being, the fact is amusing and true at the same time. Let’s look at some of them in detail. 

Types of Graphical Representations

Comparison between different items is best shown with graphs, it becomes easier to compare the crux out of the data pertaining to different items. Let’s look at all the different types of graphical representations briefly: 

Line Graphs

A line graph is used to show how the value of particular variable changes with time. We plot this graph by connecting the points at different values of the variable. It can be useful for analyzing the trends in the data predicting further trends. 

A bar graph is a type of graphical representation of the data in which bars of uniform width are drawn with equal spacing between them on one axis (x-axis usually), depicting the variable. The values of the variables are represented by the height of the bars. 

Histograms 

This is similar to bar graphs, but it is based frequency of numerical values rather than their actual values. The data is organized into intervals and the bars represent the frequency of the values in that range. That is, it counts how many values of the data lie in a particular range. 

Line Plot 

It is a plot that displays data as points and checkmarks above a number line, showing the frequency of the point. 

Stem and Leaf Plot 

This is a type of plot in which each value is split into a “leaf”(in most cases, it is the last digit) and “stem”(the other remaining digits). For example: the number 42 is split into leaf (2) and stem (4).  

Box and Whisker Plot 

These plots divide the data into four parts to show their summary. They are more concerned about the spread, average, and median of the data. 

It is a type of graph which represents the data in form of a circular graph. The circle is divided such that each portion represents a proportion of the whole. 

Graphical Representations used in Maths

Graphs in maths are used to study the relationships between two or more variables that are changing. Statistical data can be summarized in a better way using graphs. There are basically two lines of thoughts of making graphs in maths: 

• Value-Based or Time Series Graphs

Frequency Based

Value-based or time series graphs .

These graphs allow us to study the change of a variable with respect to another variable within a given interval of time. The variables can be anything. Time Series graphs study the change of variable with time. They study the trends, periodic behavior, and patterns in the series. We are more concerned with the values of the variables here rather than the frequency of those values. 

Example: Line Graph

These kinds of graphs are more concerned with the distribution of data. How many values lie between a particular range of the variables, and which range has the maximum frequency of the values. They are used to judge a spread and average and sometimes median of a variable under study. 

Example: Frequency Polygon, Histograms.

Principles of Graphical Representations

All types of graphical representations require some rule/principles which are to be followed. These are some algebraic principles. When we plot a graph, there is an origin, and we have our two axes. These two axes divide the plane into four parts called quadrants. The horizontal one is usually called the x-axis and the other one is called the y-axis. The origin is the point where these two axes intersect. The thing we need to keep in mind about the values of the variable on the x-axis is that positive values need to be on the right side of the origin and negative values should be on the left side of the origin. Similarly, for the variable on the y-axis, we need to make sure that the positive values of this variable should be above the x-axis and negative values of this variable must be below the y-axis. 

Advantages and Disadvantages of using Graphical System

Advantages: 

• It gives us a summary of the data which is easier to look at and analyze.
• It saves time.
• We can compare and study more than one variable at a time.

Disadvantage: 

It usually takes only one aspect of the data and ignores the other. For example, A bar graph does not represent the mean, median, and other statistics of the data. 

General Rules for Graphical Representation of Data

We should keep in mind some things while plotting and designing these graphs. The goal should be a better and clear picture of the data. Following things should be kept in mind while plotting the above graphs: 

• Whenever possible, the data source must be mentioned for the viewer.
• Always choose the proper colors and font sizes. They should be chosen to keep in mind that the graphs should look neat.
• The measurement Unit should be mentioned in the top right corner of the graph.
• The proper scale should be chosen while making the graph, it should be chosen such that the graph looks accurate.
• Last but not the least, a suitable title should be chosen.

Frequency Polygon

A frequency polygon is a graph that is constructed by joining the midpoint of the intervals. The height of the interval or the bin represents the frequency of the values that lie in that interval. 

Sample Problems

Question 1: What are different types of frequency-based plots? 

Answer: 

Types of frequency based plots:  Histogram Frequency Polygon Box Plots

Question 2: A company with an advertising budget of Rs 10,00,00,000 has planned the following expenditure in the different advertising channels such as TV Advertisement, Radio, Facebook, Instagram, and Printed media. The table represents the money spent on different channels. 

Draw a bar graph for the following data. 

Solution: 

Steps:  Put each of the channels on the x-axis The height of the bars is decided by the value of each channel.

Question 3: Draw a line plot for the following data 

Steps:  Put each of the x-axis row value on the x-axis joint the value corresponding to the each value of the x-axis.

Question 4: Make a frequency plot of the following data: 

Steps:  Draw the class intervals on the x-axis and frequencies on the y-axis. Calculate the mid point of each class interval. Class Interval Mid Point Frequency 0-3 1.5 3 3-6 4.5 4 6-9 7.5 2 9-12 10.5 6 Now join the mid points of the intervals and their corresponding frequencies on the graph.  This graph shows both the histogram and frequency polygon for the given distribution.

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17 Data Visualization Techniques All Professionals Should Know

• 17 Sep 2019

There’s a growing demand for business analytics and data expertise in the workforce. But you don’t need to be a professional analyst to benefit from data-related skills.

Becoming skilled at common data visualization techniques can help you reap the rewards of data-driven decision-making , including increased confidence and potential cost savings. Learning how to effectively visualize data could be the first step toward using data analytics and data science to your advantage to add value to your organization.

Several data visualization techniques can help you become more effective in your role. Here are 17 essential data visualization techniques all professionals should know, as well as tips to help you effectively present your data.

Access your free e-book today.

What Is Data Visualization?

Data visualization is the process of creating graphical representations of information. This process helps the presenter communicate data in a way that’s easy for the viewer to interpret and draw conclusions.

There are many different techniques and tools you can leverage to visualize data, so you want to know which ones to use and when. Here are some of the most important data visualization techniques all professionals should know.

Data Visualization Techniques

The type of data visualization technique you leverage will vary based on the type of data you’re working with, in addition to the story you’re telling with your data .

Here are some important data visualization techniques to know:

• Gantt Chart
• Box and Whisker Plot
• Waterfall Chart
• Scatter Plot
• Pictogram Chart
• Highlight Table
• Bullet Graph
• Choropleth Map
• Network Diagram
• Correlation Matrices

1. Pie Chart

Pie charts are one of the most common and basic data visualization techniques, used across a wide range of applications. Pie charts are ideal for illustrating proportions, or part-to-whole comparisons.

Because pie charts are relatively simple and easy to read, they’re best suited for audiences who might be unfamiliar with the information or are only interested in the key takeaways. For viewers who require a more thorough explanation of the data, pie charts fall short in their ability to display complex information.

2. Bar Chart

The classic bar chart , or bar graph, is another common and easy-to-use method of data visualization. In this type of visualization, one axis of the chart shows the categories being compared, and the other, a measured value. The length of the bar indicates how each group measures according to the value.

One drawback is that labeling and clarity can become problematic when there are too many categories included. Like pie charts, they can also be too simple for more complex data sets.

3. Histogram

Unlike bar charts, histograms illustrate the distribution of data over a continuous interval or defined period. These visualizations are helpful in identifying where values are concentrated, as well as where there are gaps or unusual values.

Histograms are especially useful for showing the frequency of a particular occurrence. For instance, if you’d like to show how many clicks your website received each day over the last week, you can use a histogram. From this visualization, you can quickly determine which days your website saw the greatest and fewest number of clicks.

4. Gantt Chart

Gantt charts are particularly common in project management, as they’re useful in illustrating a project timeline or progression of tasks. In this type of chart, tasks to be performed are listed on the vertical axis and time intervals on the horizontal axis. Horizontal bars in the body of the chart represent the duration of each activity.

Utilizing Gantt charts to display timelines can be incredibly helpful, and enable team members to keep track of every aspect of a project. Even if you’re not a project management professional, familiarizing yourself with Gantt charts can help you stay organized.

5. Heat Map

A heat map is a type of visualization used to show differences in data through variations in color. These charts use color to communicate values in a way that makes it easy for the viewer to quickly identify trends. Having a clear legend is necessary in order for a user to successfully read and interpret a heatmap.

There are many possible applications of heat maps. For example, if you want to analyze which time of day a retail store makes the most sales, you can use a heat map that shows the day of the week on the vertical axis and time of day on the horizontal axis. Then, by shading in the matrix with colors that correspond to the number of sales at each time of day, you can identify trends in the data that allow you to determine the exact times your store experiences the most sales.

6. A Box and Whisker Plot

A box and whisker plot , or box plot, provides a visual summary of data through its quartiles. First, a box is drawn from the first quartile to the third of the data set. A line within the box represents the median. “Whiskers,” or lines, are then drawn extending from the box to the minimum (lower extreme) and maximum (upper extreme). Outliers are represented by individual points that are in-line with the whiskers.

This type of chart is helpful in quickly identifying whether or not the data is symmetrical or skewed, as well as providing a visual summary of the data set that can be easily interpreted.

7. Waterfall Chart

A waterfall chart is a visual representation that illustrates how a value changes as it’s influenced by different factors, such as time. The main goal of this chart is to show the viewer how a value has grown or declined over a defined period. For example, waterfall charts are popular for showing spending or earnings over time.

8. Area Chart

An area chart , or area graph, is a variation on a basic line graph in which the area underneath the line is shaded to represent the total value of each data point. When several data series must be compared on the same graph, stacked area charts are used.

This method of data visualization is useful for showing changes in one or more quantities over time, as well as showing how each quantity combines to make up the whole. Stacked area charts are effective in showing part-to-whole comparisons.

9. Scatter Plot

Another technique commonly used to display data is a scatter plot . A scatter plot displays data for two variables as represented by points plotted against the horizontal and vertical axis. This type of data visualization is useful in illustrating the relationships that exist between variables and can be used to identify trends or correlations in data.

Scatter plots are most effective for fairly large data sets, since it’s often easier to identify trends when there are more data points present. Additionally, the closer the data points are grouped together, the stronger the correlation or trend tends to be.

10. Pictogram Chart

Pictogram charts , or pictograph charts, are particularly useful for presenting simple data in a more visual and engaging way. These charts use icons to visualize data, with each icon representing a different value or category. For example, data about time might be represented by icons of clocks or watches. Each icon can correspond to either a single unit or a set number of units (for example, each icon represents 100 units).

In addition to making the data more engaging, pictogram charts are helpful in situations where language or cultural differences might be a barrier to the audience’s understanding of the data.

11. Timeline

Timelines are the most effective way to visualize a sequence of events in chronological order. They’re typically linear, with key events outlined along the axis. Timelines are used to communicate time-related information and display historical data.

Timelines allow you to highlight the most important events that occurred, or need to occur in the future, and make it easy for the viewer to identify any patterns appearing within the selected time period. While timelines are often relatively simple linear visualizations, they can be made more visually appealing by adding images, colors, fonts, and decorative shapes.

12. Highlight Table

A highlight table is a more engaging alternative to traditional tables. By highlighting cells in the table with color, you can make it easier for viewers to quickly spot trends and patterns in the data. These visualizations are useful for comparing categorical data.

Depending on the data visualization tool you’re using, you may be able to add conditional formatting rules to the table that automatically color cells that meet specified conditions. For instance, when using a highlight table to visualize a company’s sales data, you may color cells red if the sales data is below the goal, or green if sales were above the goal. Unlike a heat map, the colors in a highlight table are discrete and represent a single meaning or value.

13. Bullet Graph

A bullet graph is a variation of a bar graph that can act as an alternative to dashboard gauges to represent performance data. The main use for a bullet graph is to inform the viewer of how a business is performing in comparison to benchmarks that are in place for key business metrics.

In a bullet graph, the darker horizontal bar in the middle of the chart represents the actual value, while the vertical line represents a comparative value, or target. If the horizontal bar passes the vertical line, the target for that metric has been surpassed. Additionally, the segmented colored sections behind the horizontal bar represent range scores, such as “poor,” “fair,” or “good.”

14. Choropleth Maps

A choropleth map uses color, shading, and other patterns to visualize numerical values across geographic regions. These visualizations use a progression of color (or shading) on a spectrum to distinguish high values from low.

Choropleth maps allow viewers to see how a variable changes from one region to the next. A potential downside to this type of visualization is that the exact numerical values aren’t easily accessible because the colors represent a range of values. Some data visualization tools, however, allow you to add interactivity to your map so the exact values are accessible.

15. Word Cloud

A word cloud , or tag cloud, is a visual representation of text data in which the size of the word is proportional to its frequency. The more often a specific word appears in a dataset, the larger it appears in the visualization. In addition to size, words often appear bolder or follow a specific color scheme depending on their frequency.

Word clouds are often used on websites and blogs to identify significant keywords and compare differences in textual data between two sources. They are also useful when analyzing qualitative datasets, such as the specific words consumers used to describe a product.

16. Network Diagram

Network diagrams are a type of data visualization that represent relationships between qualitative data points. These visualizations are composed of nodes and links, also called edges. Nodes are singular data points that are connected to other nodes through edges, which show the relationship between multiple nodes.

There are many use cases for network diagrams, including depicting social networks, highlighting the relationships between employees at an organization, or visualizing product sales across geographic regions.

17. Correlation Matrix

A correlation matrix is a table that shows correlation coefficients between variables. Each cell represents the relationship between two variables, and a color scale is used to communicate whether the variables are correlated and to what extent.

Correlation matrices are useful to summarize and find patterns in large data sets. In business, a correlation matrix might be used to analyze how different data points about a specific product might be related, such as price, advertising spend, launch date, etc.

Other Data Visualization Options

While the examples listed above are some of the most commonly used techniques, there are many other ways you can visualize data to become a more effective communicator. Some other data visualization options include:

• Bubble clouds
• Circle views
• Dendrograms
• Dot distribution maps
• Open-high-low-close charts
• Polar areas
• Radial trees
• Ring Charts
• Sankey diagram
• Span charts
• Streamgraphs
• Wedge stack graphs
• Violin plots

Tips For Creating Effective Visualizations

Creating effective data visualizations requires more than just knowing how to choose the best technique for your needs. There are several considerations you should take into account to maximize your effectiveness when it comes to presenting data.

Related : What to Keep in Mind When Creating Data Visualizations in Excel

One of the most important steps is to evaluate your audience. For example, if you’re presenting financial data to a team that works in an unrelated department, you’ll want to choose a fairly simple illustration. On the other hand, if you’re presenting financial data to a team of finance experts, it’s likely you can safely include more complex information.

Another helpful tip is to avoid unnecessary distractions. Although visual elements like animation can be a great way to add interest, they can also distract from the key points the illustration is trying to convey and hinder the viewer’s ability to quickly understand the information.

Finally, be mindful of the colors you utilize, as well as your overall design. While it’s important that your graphs or charts are visually appealing, there are more practical reasons you might choose one color palette over another. For instance, using low contrast colors can make it difficult for your audience to discern differences between data points. Using colors that are too bold, however, can make the illustration overwhelming or distracting for the viewer.

Related : Bad Data Visualization: 5 Examples of Misleading Data

Visuals to Interpret and Share Information

No matter your role or title within an organization, data visualization is a skill that’s important for all professionals. Being able to effectively present complex data through easy-to-understand visual representations is invaluable when it comes to communicating information with members both inside and outside your business.

There’s no shortage in how data visualization can be applied in the real world. Data is playing an increasingly important role in the marketplace today, and data literacy is the first step in understanding how analytics can be used in business.

Are you interested in improving your analytical skills? Learn more about Business Analytics , our eight-week online course that can help you use data to generate insights and tackle business decisions.

This post was updated on January 20, 2022. It was originally published on September 17, 2019.

About the Author

Graphs and graph representations

• vertices and edges
• directed vs undirected graphs
• labeled graphs
• adjacency and degree
• adjacency-matrix and adjacency-list representations
• paths and cycles
• topological sorting
• more graph problems: shortest paths, graph coloring

A graph is a highly useful mathematical abstraction. A graph consists of a set of vertices (also called nodes ) and a set of edges (also called arcs ) connecting those vertices. There are two main kinds of graphs: undirected graphs and directed graphs . In a directed graph (sometimes abbreviated as digraph ), the edges are directed: that is, they have a direction, proceeding from a source vertex to a sink (or destination ) vertex. The sink vertex is a successor of the source, and the the source is a predecessor of the sink. In undirected graphs, the edges are symmetrical.

Uses of graphs

Graphs are a highly useful abstraction in computer science because so many important problems can be expressed in terms of graphs. We have already seen a number of graph structures: for example, the objects in a running program form a directed graph in which the vertices are objects and references between objects are edges. To implement automatic garbage collection (which discards unused objects), the language implementation uses a algorithm for graph reachability .

• states of games and puzzles, which are vertices connected by edges that are the legal moves in the game,
• state machines, where the states are vertices and the transitions between states are edges,
• road maps, where the vertices are intersections or points along the road and edges are roads connecting those points,
• scheduling problems, where vertices represent events to be scheduled and edges might represent events that cannot be scheduled together, or, depending on the problem, edges that must be scheduled together,
• and in fact, any binary relation ρ can be viewed as a directed graph in which the relationship x ρ y corresponds to an edge from vertex x to vertex y.

What is the value of having a common mathematical abstraction like graphs? One payoff is that we can develop algorithms that work on graphs in general. Once we realize we can express a problem in terms of graphs, we can consult a very large toolbox of efficient graph algorithms, rather than trying to invent a new algorithm for the specific domain of interest.

There are many computational problems over graphs that are not known to be solvable in any reasonable amount of time. In particular, there is a large class of problems known as the NP complete problems that are not known to be efficiently solvable, but are known to be equivalent in complexity. If we could give an efficient algorithm for solving any one of them, then we would have efficient algorithms for all of them.

Vertices and edges

A graph consists of a set of vertices V and a set of edges E. If the graph is directed, the edges E are a set of ordered pairs (u,v) representing an edge with source vertex u and sink vertex v. When drawing graphs, this is usually represented as an arrow from u to v.

If the graph is undirected, then the edges are a set of sets of unordered pairs {u,v}. Alternatively, we can model an undirected graph as a directed graph with edges in both directions; i.e., if (u,v) ∈ E, then (v,u) ∈ E also. When drawing graphs, this is usually represented as a line between u and v without an arrowhead.

In some cases, edges from a vertex to itself or multiple edges between the same pair of vertices may be permitted. But usually the default is not to allow these things unless explicly stated otherwise.

Adjacency and degree

Two vertices v and w are adjacent if they are connected by an edge. The degree of a vertex is the total number of adjacent vertices. In a directed graph, we can distinguish between outgoing and incoming edges. The out-degree of a vertex is the number of outgoing edges and the in-degree is the number of incoming edgs.

The real value of graphs is obtained when we can use them to organize information. Both edges and vertices of graphs can have labels that carry meaning about an entity represented by a vertex or about the relationship between two entities represented by an edge. For example, we might encode information about the population of cities and distances between them as an undirected labeled graph:

Here, the vertices are labeled with a pair containing the name of the city and its population, and the edges are labeled with the distance between the cities.

A graph in which the edges are labeled with numbers is called a weighted graph . Of course, the labels do not have to represent weight; they might stand for distances, or the probability of transitioning from one state to another, or the similarity between two vertices, etc.

Graph representations

There are several ways to represent graphs in a computer program. Which representation is best depends on the application. For example, consider the following weighted directed graph with vertices {0,1,2,3} and directed edges with edge weights shown:

Adjacency matrix

An adjacency matrix represents a graph as a two-dimensional array. Each vertex is assigned a distinct index in {0,1,...,|V|-1}. If the graph is represented by the 2D array m , then the edge (or lack thereof) from vertex i to vertex j is recorded at m[i][j] .

The graph structure, ignoring the weights, can be represented by storing a boolean value at each array index. A directed edge from i to j is represented by a true (T) value in location m[i][j]. If there is no edge there, the value is false. For example, the edges in the example above are represented this matrix:

More compact bit-level representations for the booleans are also possible.

If there is some information associated with each edge, say a weight, we store that information in the corresponding array entry:

The space required by the adjacency matrix representation is O(V 2 ), so adjacency matrices can waste a lot of space if the number of edges |E| is much smaller than O(V 2 ). In particular, a graph with only O(V) edges is said to be sparse . For example, graphs in which either the in-degree or out-degree is bounded by a constant are sparse. Adjacency matrices are not as wasteful when the graphs they represent are dense (i.e., not sparse).

The adjacency matrix representation is time -efficient for some operations. Testing whether there is an edge between two vertices can clearly be done in constant time. However, finding all incoming edges to a given vertex, or finding all outgoing edges, takes time proportional to the number of vertices, even for sparse graphs.

Undirected graphs can also be represented with an adjacency matrix. The matrix will be symmetric around its main diagonal; that is, m[i][j]=m[j][i].

Adjacency list representation

Since sparse graphs are quite common, the adjacency list representation is often preferred. This representation keeps track of the outgoing edges from each vertex, typically as a linked list. For example, the graph above might be represented with the following data structure:

Adjacency lists are asymptotically space-efficient because they only use space proportional to the number of vertices and the number of edges. We say that they require O(V+E) space.

Finding the outgoing edges from a vertex is very efficient in the adjacency list representation too; it requires time proportional to the number of outgoing edges. However, finding the incoming edges to a vertex is not efficient: it requires scanning the entire data structure, requiring O(V+E) time.

When it is necessary to be able to walk forward on outgoing edges and backward on incoming edges, a good approach is to maintain two adjacency lists, one representing the graph as above and one corresponding to the dual (or transposed ) graph in which all edges are reversed. That is, there is a an edge (u,v) in the original graph if and only if there is an edge (v,u) in the transposed graph. Of course, this invariant must be maintained between the two adjacency list representations.

Testing whether there is an edge from vertex i to vertex j requires scanning all the outgoing edges, taking O(V) time in the worse case. If this operation needs to be fast, the linked list can be replaced with a hash table. For example, we might implement the graph using this Java representation, which preserves the asympotic space efficiency of adjacency lists while also supporting queries for particular edges:

Paths and cycles

Following a series of edges from a starting vertex creates a path through the graph, a sequence of alternating vertices and edges beginning and ending with a vertex (v 0 ,e 0 ,v 1 ,e 1 ,...,v n ) where e i = (v i ,v i+1 ) for all 0≤i≤n-1. The length of the path is the number of edges (that is, n). Note that n=0 is possible; this would be a path of length 0 consisting of just a single vertex and no edges. If no vertex appears twice in the path, except that possibly v 0 = v n , the path is called simple . If the first and last vertices are the same, the path is a cycle .

Some graphs have no cycles. For example, linked lists and trees are both examples of graphs in which there are no cycles. They are directed acyclic graphs , abbreviated as DAGs. In trees and linked lists, each vertex has at most one predecessor. In general, vertices of DAGs can have more than one predecessor.

Topological sorting

One use of directed graphs is to represent an ordering constraint on vertices. For example, vertices might represent events, and an edge (u,v) might represent the constraint that u must happen before v.

A topological sort of the vertices is a total ordering of the vertices that is consistent with all edges. That is, if (u,v) is an edge, then u must appear before v in the total ordering. A graph can be topologically sorted if and only if it has no cycles.

Topological sorts are useful for deciding in what order to do things. For example, consider the following DAG expressing what we might call the “men's informal dressing problem”:

A valid plan for getting dressed is a topological sort of this graph. In fact, any topological sort is in principle a workable way to get dressed. For example, the ordering (pants, shirt, belt, socks, tie, jacket, shoes) is consistent with the ordering on all the graph edges. Less conventional strategies are also workable, such as (socks, pants, shoes, shirt, belt, tie, jacket).

Does every DAG have a topological sort? Yes. To see this, observe that every finite DAG must have a vertex with in-degree zero. To find such a vertex, we start from an arbitrary vertex in the graph and walk backward along edges until we reach a vertex with zero in-degree. We know that the walk must generate a simple path because there are no cycles in the graph. Therefore, the walk must terminate because we run out of vertices that haven't already been seen along the walk.

This gives us an (inefficient) way to topologically sort a DAG:

Since finding the 0 in-degree node takes O(V) time, this algorithm takes O(V 2 ) time. We can do better, as we'll see shortly.

Other graph problems

Many problems of interest can be expressed in terms of graphs. Here are a few examples of important graph problems, some of which can be solved efficiently and some of which are intractable!

Reachability

One vertex is reachable from another if there is a path from one to the other. Determining which vertices are reachable from a given vertex is useful and can be done in linear time.

Shortest paths

For example, if a road map is represented as a graph with vertices representing intersections and edges representing road segments, the shortest-path problem can be used to find short routes. There are several variants of the problem, depending on whether one is interested in the distance from a given root vertex or in the distances between all pairs of vertices. If negative-weight edges exist, these problems become harder and different algorithms are needed.

Hamiltonian paths and the traveling salesman problem

The problem of finding the longest simple path between two nodes in a graph is, in general, intractable. It is related to some other important problems. A Hamiltonian path is one that visits every vertex in a graph exactly once. A Hamiltonian cycle is a cycle that visits every vertex exactly once. The ability to determine whether a graph contains a Hamiltonian path or a Hamiltonian cycle would be useful, but in general the best known algorithms for this problem require exponential time.

A weighted version of this problem is the traveling salesman problem (TSP), which tries to find the Hamiltonian cycle with the minimum total weight. The name comes from imagining a salesman who wants to visit every one of a set of cities while traveling the least possible total distance. This problem is at least as hard as finding Hamiltonian cycles and is not known to be solvable in any less than exponential time. However, finding a solution that is within a constant factor (e.g., 1.5) of optimal can be done in polynomial time under some reasonable assumptions. In practice, there exist good heuristics that allow close-to-optimal solutions to TSP even for large problem instances.

Graph coloring

Imagine that we want to schedule exams in k time slots such that no student has two exams at the same time. We can represent this problem using an undirected graph in which the exams are vertices, with the exams v 1 and v 2 connected by an edge if there exists at least one student who needs to take both exams. We can schedule the exams in the k slots if there is a k-coloring of the graph with k colors: a way to assign a color to each vertex with one of k colors such that no two adjacent vertices are assigned the same color. The chromatic number of a graph is the smallest number of colors needed to color it.

There is no known efficient algorithm for k coloring in general. There are some special classes of graphs for which coloring is efficient, and in practice, graph colorings close to optimal can be found, but in general the best known algorithms to solve the problem optimally take exponential time.