Venn diagrams visually represent relationships between concepts. They use circles to display similarities and differences between sets of ideas, traits, or items. Intersections indicate that the groups have common elements. Non-overlapping areas represent traits that are unique to one set. Venn diagrams are also known as logic diagrams and set diagrams.

The real strength of Venn diagrams comes from their ability to represent interesting relationships in reports and presentations. The alternative is to present lists of attributes and then explain the relationships, which can be tedious. Venn diagrams graphically organize the same information in an easy-to-digest format. Because these charts are so efficient, educators have used them extensively since the mid-20^{th} century as tools to present complex data.

An English logician, John Venn, popularized these diagrams in the 1880s by using them to teach set theory. However, a mathematician, Leonard Euler, created them back in the 1700s. Educators now use them in numerous contexts, such as statistics, probability, logic, business, education, and computer science, among many others.

In this post, learn how to interpret Venn diagrams using two- and three-circle versions. I’ll cover several specialized uses, proportional Venn diagrams and illustrating probabilities. I’ll close by showing an easy way to make your own Venn diagrams using widely available templates.

## How to Interpret Venn Diagrams

To interpret Venn diagrams, start by examining all the circles. The group of circles represents the “universe” of concepts that the chart explains. Each circle represents a set of items or an idea. Then notice the areas where the circles do not overlap. Non-overlapping sections indicate that the two groups do not share some characteristics. In other words, some properties are unique to a set. Finally, identify the areas where the circles overlap. Overlapping areas indicate that at least two concepts share some traits. Be sure to notice how many circles overlap.

Let’s get to some examples!

## Two-Circle Venn Diagram Example

In the Venn diagram below, the two circles tell us that we’re comparing and contrasting dolphins and fish.

Looking at the non-overlapping areas, we see that dolphins have the unique traits of breathing air, have live births, and are warm-blooded. On the other hand, fish breathe water, lay eggs, and are cold-blooded.

Then, looking at the overlapping area, we see that they share the traits of swimming, being aquatic creatures, and having fins.

That’s a simple example with two circles. But the similarities and differences are immediately apparent.

## Three-Circle Venn Diagram Example

Venn diagrams don’t need to apply to physical characteristics. The three-circle Venn diagram below shows how a company divides the roles between three areas of their company.

## Proportional Venn Diagram Example

Let’s move on to a more complex Venn diagram with three circles. Understanding the similarities and differences of the world’s top exporters might sound dull and complicated. However, the graph below shows the leading exports from the top exporting countries. Additionally, this is a proportional Venn diagram, also known as a scaled Venn diagram. The sizes of the circles are proportional to an attribute the circles represent, which is the annual dollar amount of exports in this example.

At a glance, the circles tell us that we’re comparing China, the United States, and Japan in terms of exports. China has the most exports—not quite twice as much as the United States and a lot more than Japan. The U.S. is in the middle. Of these three countries, the U.S. is unique in that petroleum is among its top exports. Japan is unique in having machines that produce other goods and seagoing ships among its leading exports. China has a diverse set of distinct exports. The three countries all have integrated circuits (IC) as one of their top exports. The U.S. and Japan both count vehicle parts and cars among their chief exports.

Even if understanding international exports is not your thing, you probably learned something new in a relatively painless way thanks to a Venn diagram!

## Using Venn Diagrams to Represent Probabilities

In math and statistics, you can use Venn diagrams to depict probabilities. Grasping how probabilities relate to events occurring separately, together, or not all can be complex. Venn diagrams make understanding these likelihoods much easier.

For this example, imagine we run a mountain bike rental shop in an area with rough terrain. Flat tires are a problem, and we want to understand the probabilities of the front, back, both, and no tires going flat during a rental. After collecting data, the owners display the probabilities in the chart below.

At a glance, the Venn diagram helps you understand the probabilities of various combinations of flat tires occurring!

### Exploring the Various Probabilities

The first thing to notice on the Venn diagram is that each circle represents the probability of a tire going flat. The value outside the circles indicates the likelihood of no flat tires. Collectively, all the probabilities sum to 1 (100%). This area is our entire sample space.

To find the probability for only the front tire going flat or only the rear tire on the Venn diagram, look at the values in the circles, 0.135 and 0.085, respectively. In probability theory terminology, these are the joint probabilities of one tire going flat and the other tire not going flat. The joint probability for both tires going flat (yikes!) is the intersection value of 0.015.

To calculate the probability that the front tire goes flat regardless of whether or not the rear goes flat, add the value in the Front circle to the intersection value, 0.135 + 0.015 = 0.15. You can do the same for the rear tire to obtain 0.10. Statisticians call these the marginal probabilities for the events.

On the chart, the total probability of any flat occurring is the sum of the front, rear, and intersection values, 0.135 + 0.15 + 0.085 = 0.235. In probability theory, this is the probability of the union of events. Conversely, the probability of no flat at all is 1 – 0.235 = 0.765, which is the value outside the circles.

The calculations for these probabilities go beyond this post, but you can see them in the Word file that contains my charts: Venn diagram examples. In that document, scroll to the bottom to see the calculations!

**Related post**: Probability Fundamentals

## Venn Diagram Maker

It’s easy to make your own Venn diagrams using Microsoft’s blank Venn diagram templates. I used Microsoft Word to create the ones in this post, but you can also use Excel, Outlook, and PowerPoint. The process for generating them is the same for all these applications.

You can download my Word file to see how I used the Venn diagram templates: Venn diagram examples.

- Go to the
**Insert**tab and click**Smart Shapes.** - In the box that appears, click
**Relationship**in the left pane. - Select one of the Venn diagram layouts and click
**OK**. - Now you need to add the text.
- To add the circle text, click the [Text] label in each circle.
- To add the intersection and other values, click
**Text Box**and choose**Draw Text Box**. - I had to format the text boxes to have no fill and no outline so only the text shows.

For more details, see Microsoft’s instructions for creating Venn diagrams.

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