In statistics, when we use the term **distribution**, we usually mean a **probability distribution**.

Good examples are the **normal distribution**, the **binomial distribution**, and the **uniform distribution**.

If this is your first time hearing the word **distribution**, don’t worry. Let’s just say that after reading this tutorial, you will have a higher chance of winning any game that involves rolling a die. To get a better understanding of what it is all about, we should start with a definition.

A

distributionis afunctionthat shows the possible values for a variable and how often they occur.

**Providing an Example of a Discrete Uniform Distribution**

Think about a fair die.

It has six sides, numbered from 1 to 6. Imagine that we roll the die. What is the probability of getting 1?

It is one out of six, so it must be one-sixth.

Try guessing what the probability of rolling a 2 is. Once again – one-sixth. The same holds true for 3, 4, 5 and 6.

We have an equal chance of getting each of the 6 outcomes.

Now, what do you think is the probability of getting a 7?

It is impossible to get a 7 when rolling a single die.

Therefore, the probability is 0.

**The Values that Make up a Distribution**

Let’s generalize. The **distribution** of an event consists not only of the input values that can be observed. It is actually made up of all possible values. So, the **distribution** of the event – rolling a die – will be given by the following table.

The probability of getting 1 is one-sixth, or 0.17. The probability of getting 2 is also 0.17, and so on.

**How to Tell if We Have Gone Through all Possible Values**

We are sure that we have exhausted all possible values when the sum of their probabilities is equal to 1 or 100%.

Similar to what we discussed about getting a 7, for all other values, the probability of occurrence is 0.

And that’s the probability **distribution** of rolling a die. By the way, it is called a **discrete uniform distribution**. All outcomes have an equal chance of occurring.

**The Visual Representation**

Each **probability distribution** has a visual representation. It is a graph** describing the likelihood of occurrence of every event. **You can see the graph of our example in the picture below.

**Important: **It is crucial to understand that the graph is __JUST__ a visual representation of a **distribution**.

Often, when we talk about **distributions**, we make use of the graph. That’s why many people believe that a **distribution** is the graph itself. However, this is __NOT__ true. A **distribution** is defined by the underlying probabilities and not the graph. The graph is just a visual representation.

After this short clarification, let’s explore a different example.

**Another Case in Point**

Think about rolling two dice.

What are the possible outcomes?

1 and 1, 2 and 1, 1 and 2, and so on.

In the picture below, you can see a table with all the possible combinations.

Say we are playing a game where we are trying to guess the sum of the two dice.

Can you guess the probability of getting a sum of 1? It’s 0, as this event is impossible.

**The Smallest Sum Possible**

The minimum sum we can get is 2. So, what’s the probability of getting a sum of 2? There is only one combination that would give us a sum of 2 – when both dice are equal to 1.

So, 1 out of 36 total outcomes, or 0.03.

Similarly, the probability of getting a sum of 3 is given by the number of combinations that give a sum of 3 divided by 36. If you think about it, 1 and 2, and, 2 and 1 are the only possibilities. Therefore, the probability is equal to 2 divided by 36. Or simply 0.06.

**The Graph Associated with the Distribution**

We can continue in this way until we have the full **probability distribution**.

The graph associated with it is shown below.

Looking at it we can easily understand that when rolling two dice, the probability of getting a 7 is the highest.

Moreover, we can also compare different outcomes such as the probability of getting a 10 and the probability of getting a 5.

As you can tell from the picture above, it’s less likely that we’ll get a 10.

**Other Distributions**

To sum up, there are various types of **distributions**. Usually, we will use a **graph** to visualize them. The examples that we discussed, were of discrete variables. However, they are not as common in inferences as the **continuous distributions**. The journey where we will be exploring them starts off with the **Normal distribution**.

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**Interested in learning more? You can take your skills from good to great with our statistics tutorials!**

**Next Tutorial: **Introducing the Normal Distribution