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

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

What is the connection between calculating the mean of a probability distribution (the expected value) and calculating the mean of a set of numbers (like we did in Unit 2)?

You may be wondering why the formula for the mean (aka expected value) does not require us to divide by the number of observations, like we would normally do when calculating a mean for a set of numbers.

In the next example we investigate the connection between calculating a mean of a set of numbers and calculating the mean of a probability distribution.

EXAMPLE The connection between expected value and the mean of a set of numbers

At Rushmore Community College, there have been complaints about how long it takes to get food from the college cafeteria.

The following table gives the total cafeteria wait times (rounded to the nearest 5 minutes) for 200 randomly selected students.

Here is a frequency table showing the data.

Cafeteria wait times

Time (minutes) | 5 | 10 | 15 | 20 | 25
Number of students | 30 | 52 | 62 | 40 | 16

Here is our formula for expected value:

To use our formula for calculating the expected value, we need to determine the empirical probabilities. To find the empirical probabilities, find the relative frequencies: divide each frequency (count) by the total number of observations.

For example, to calculate the probability that a student will have to wait 5 minutes to get their food we divide: (the number of students in the sample that waited 5 minutes) by (the total number of students in the sample) = 30 / 200 = 0.15.

Probability distribution for cafeteria wait times

X = Time (minutes) | 5 | 10 | 15 | 20 | 25
P(X) | 30 / 200 = 0.15 | 52 / 200 = 0.26 | 62 / 200 = 0.31 | 40 / 200 = 0.20 | 16 / 200 = 0.08

Now we use our formula to find the mean of the probability distribution (aka expected value) by finding the weighted mean.

We expect students to wait about 14 minutes on average in the cafeteria.

OK ... now the dive into understanding the connection between this calculation of the expected value and the usual way of calculating the mean of a set of numbers.

We will calculate the mean using the frequency table, instead of the probability distribution.

Cafeteria wait times

Time (minutes) | 5 | 10 | 15 | 20 | 25
Number of students | 30 | 52 | 62 | 40 | 16

We have 200 students. We have 30 students with a time of 5 minutes, 52 students with a time of 10 minutes, 62 students with a time of 15 minutes, and so on. To find the mean we have to add:

30 fives + 52 tens + 62 fifteens + 40 twenties + 16 twenty-fives

and then divide by 200.

Here is that calculation:

So the mean time for students to get their food in the cafeteria is 14 minutes (which we already know from our calculation of the expected value.)

Now let's take a closer look at the calculation we just did.

Notice that the large fraction can be broken up into a sum of five smaller fractions all with the denominator 200:

Okay, we are almost there. The last thing to do is rewrite each of these fractions like this:

Here is the same equation with the fractions expressed as decimals:

Do you recognize this equation? This is the same calculation that we did earlier for the weighted mean!

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