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A **probability distribution table** is a table that displays the probability that a random variable takes on certain values.

For example, the following probability distribution table tells us the probability that a certain soccer team scores a certain number of goals in a given game:

The left-hand column shows the number of goals and the right-hand column tells us the probability that the team will score this number of goals.

For example:

- The probability that the team scores exactly 0 goals is
**0.18**. - The probability that the team scores exactly 1 goal is
**0.34**. - The probability that the team scores exactly 2 goals is
**0.35**.

And so on.

**Properties of a Probability Distribution Table**

A probability distribution table has the following properties:

**1. All probabilities must add up to 1.Â **

For a probability distribution table to be valid, all of the individual probabilities must add up to 1. We can verify that the previous probability distribution table is valid:

Sum of probabilities = 0.18 + 0.34 + 0.35 + 0.11 + 0.02 = 1.

**2. The mean can be calculated.**

The formula to calculate the mean of a given probability distribution table is:

**Î¼ = Î£x * P(x)**

where:

**x:**Data value**P(x):**Probability of value

For example, consider our probability distribution table for the soccer team:

The mean number of goals for the soccer team would be calculated as:

Î¼ = 0*0.18Â +Â 1*0.34Â +Â 2*0.35Â +Â 3*0.11Â +Â 4*0.02Â =Â Â **1.45**Â goals.

**3. The standard deviation can be calculated.**

The formula to calculate the standard deviation of a given probability distribution table is:

**Ïƒ = âˆšÎ£(x _{i}-Î¼)^{2}Â * P(x_{i})**

where:

**x**Â The i_{i}:^{th}Â value**Î¼:**Â The mean of the distribution**P(x**Â The probability of the i_{i}):^{th}Â value

For example, hereâ€™s how to calculate the standard deviation of goals scored by the soccer team:

The standard deviation is the square root of the sum of the values in the third column:

Standard deviation = âˆš(.3785 + .0689 + .1059 + .2643 + .1301)Â =Â **0.9734**

**How to Visualize a Probability Distribution Table**

The easiest way to visualize the values in a probability distribution table is by using a histogram, which displays the values of the random variable along the x-axis and the probability of those values along the y-axis:

This lets us quickly visualize the probability values from the table.

In particular, we can see that there is a high probability that the team scores 2 goals or less while there is a tiny probability that the team scores as many as 4 goals.

**Additional Resources**

Probability Distribution Calculator

How to Find the Mean of a Probability Distribution

How to Find the Standard Deviation of a Probability Distribution