Wiki Page, Math 13 - Perez Sandbox : Module 11 - Probability Distributions for a Quantitative Variable (2 of 3)

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2026-01-17T21:15:11-08:00

In the previous discussion of Benford’s law and fraudulent tax returns, we examined the probability distributions for the quantitative variable “first digit in reported income.” Fraudulent tax returns had a higher average first digit and a much smaller standard deviation.

In that example, you may have wondered how we found the mean and standard deviation of the probability distribution. We will now work a few examples to demonstrate how to find the mean of a probability distribution. We will not require you to learn to calculate the standard deviation by hand. In general, we will use technology to do these calculations.

The Formula

The mean of the probability distribution is a weighted mean. Each value is weighted by its probability.

EXAMPLE Calculating the mean of a probability distribution for a quantitative variable

Students often change their major when they are in college. In this example, the variable is the number of times a student changes her major.

Using thousands of student records, we use our data to create the following empirical probability distribution from relative frequencies.

Probability distribution for number of times major is changed

X: # of changes in major | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
Probability | 0.135 | 0.271 | 0.271 | 0.180 | 0.090 | 0.036 | 0.012 | 0.003 | 0.002

What is the average number of times a student will change her major?

Before we dive into the calculation, let’s estimate the mean using the histogram of the probability distribution.

The median will be somewhere between 1 and 2. The right skew suggests that the mean will be a little larger, maybe closer to 2.

Now let’s calculate the mean. We first must calculate X*P(X) for each value of X. Then we add the values to get the mean.

The calculation X*P(X) is in the third row of the table.

Calculation of mean for number of times major is changed

X: # of changes in major | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
Probability | 0.135 | 0.271 | 0.271 | 0.180 | 0.090 | 0.036 | 0.012 | 0.003 | 0.002

X*P(X)

0(0.135)

0

1(0.271)

0.271

2(0.271)

0.542

3(0.180)

0.540

4(0.090)

0.360

5(0.036)

0.180

6(0.012)

0.072

7(0.003)

0.021

8(0.002)

0.016

Next step is to add the values of X*P(X) to get the mean

0 + 0.271 + 0.542 + 0.540 + 0.360 + 0.180 + 0.072 + 0.021 + 0.016 = 2.002

The mean number of times a student changes her major is 2.002.
The mean of a probability distribution is also called the expected value.

Why Is the Mean a Weighted Average?

The variable values are not equally likely to occur. Therefore, it makes sense to count more heavily those values of X that have a high probability. By multiplying each variable value times its probability, the formula weights the more likely outcomes more heavily and the less likely outcomes less heavily.

On the next page we will work another example.

Mark this page as done, then click the "Next" or > button to continue.

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