Supplement for the Lesson on Mean, Median, and Mode

In this lesson supplement we will explore the the three most common summaries of a collection of numbers or data.  These are called the mean, the median and the mode.  The mean is a fancy word for the average, the median is the middle number, and the mode is the number that occurs most frequently.

Finding the Mean

It is a common occurrence to have several numbers, called data, that we want to make sense of.  Most people are familiar with the word "average" and many know that to find the average, we add up all the numbers and divide by the total.  Another word for the average is the mean

To find the mean, just all up all the numbers and divide by the total number of numbers.  We can think of this as a three step process:

Step-by-Step Process to Find the Mean

Step 1:  Add up all the numbers.  The result is called the sum.

Step 2:  Count how many numbers there are.  This number is called the sample size.  We use the letter n for the sample size.

Step 3:  The mean is the sum divided by the sample size n.

Now lets go to an example.

Example 1

Over the past five weekends, Steve kept track of the total amount of money he spent.

\$20, \$50, \$70, \$80, \$30

Find the mean amount of money Steve spent over the past five weekends.

Solution

Step 1: Add the numbers to find their sum.

20 + 50 + 70 + 80 + 30 = 250

Step 2: How many numbers where there.

n = 5

Step 3: Divide the sum of the numbers by the sample size 'n'

250
=  50
5

Therefore the mean of these numbers is 50. We conclude that the mean amount of money that Steve spent over the past 5 weekends was \$50.

Now try one by yourself.  If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear.

Exercise 1

The chart below shows the times in minutes that three runners had in their three trials for the one mile track race.

 Trial 1 Trial 2 Trial 3 Trial 4 Enrique 8 7 7 9 Monique 9 6 6 7 Sam 10 12 9 11

What was Monique's mean time?

Finding the Median

Another way of describing the data is by looking at the middle number.  When there are an odd number of values, we can just find the value so that there are the same number of values above as there are below this middle value.  When there is an even number of values, there is an issue in there there is not one number that acts as a middle value.  Instead, the two middle numbers such that there are the same number of values above as below these two middle numbers.  As a compromise, we take the average of these two middle numbers.  We call this result the median of the data.  It might help to summarize this in a 3-step process.

Step by Step Process for Finding the Median

Step 1:  Put the numbers in numerical order from smallest to largest.

Step 2:  If there is an odd number of numbers, locate the middle number so that there is an equal number of values to the left and to the right.  If there is an even number of numbers locate the two middle numbers so that there is an equal number of values to the left and to the right of these two numbers.

Step 3:  If there is an odd number of numbers, this middle number is the median.  If there is an even number of numbers add the two middles and divide by 2.  The result will be the median.

Example 2

Rosa measured the weight in pounds of seven packages bags of oranges that were purchased at her fruit stand.  The weights are shown below.

 18, 10, 13, 10, 17, 11, 9

Find the median weight

Solution

Step 1: First, put the numbers in numerical order from smallest to largest.

9, 10, 10, 11, 13, 17, 18

Step 2: Notice that there are 7 numbers.  This is an odd number of values, so we locate the middle number.  The middle number is 11.  Notice that there are an equal number (3) of numbers to the left of 11 and to the right of 11.

9, 10, 10, 11, 13, 17, 18

Step 3: We are in the case that there are an odd number of values, so the median is this middle number.  That is, the median is 11.

Now try one by yourself.  If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear.

Exercise 2

Five of Chang-hee's homework scores for her math class have been graded.  Her scores are shown below.

 87, 92, 73, 88, 99

Find her median homework score.

Here is another example of finding the median.

Example 3

There are six brothers and sisters in Lupe's family.  Their ages are shown below

 22, 18, 11, 14, 20, 11

Find the median age of the brothers and sisters in Lupe's family.

Solution

Step 1: First, put the numbers in numerical order from smallest to largest.

11, 11, 14, 18, 20, 22

Step 2: Notice that there are 6 numbers.  This is an even number of values, so we locate the two middle numbers.  The two middle numbers are 14 and 18.  Notice that there are an equal number (2) of values to the left and to the right of these two middles.

11, 11, 14, 18, 20, 22

Step 3: We are in the case that there are an even number of values, so the median is this average of the two middle numbers.  Add these two middle numbers and divide by 2.

14 + 18
=  16
2

So the median age of the brothers and sisters in Lupe's family is 16.

Now try one by yourself.  If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear.

Exercise 3

Oscar priced eight desk lamps.  The prices are shown below.

 \$24.00, \$48.00, \$25.00, \$30.00, \$36.00, \$52.00, \$25.00, \$90.00

What is the median price?

Finding the Mode

A third number that we use to describe data is called the mode of the data.  The mode is the number or numbers that occur the most frequently.  Here is a 2-step process for finding the mode.

Two-Step Process for Finding the Median

Step 1:  Put the numbers in numerical order from smallest to largest.

Step 2:  Go down the list and determine if there is a number that appears in the list more than any other numbers.  If the is a tie for the most frequently occurring number, then just state that both numbers are the mode.

Example 4

The box below shows the number of customers that a restaurant has had during the past nine days

 90, 50, 70, 80, 30, 60, 50, 30, 50

What is the mode of these data?

Solution

Step 1: First, put the numbers in size place, smallest to largest.

30, 30, 50, 50, 50, 60, 70, 80, 90

Step 2: Now go down the list and determine if there is a number that appears in the list more than any other number.

30, 30, 50, 50, 50, 60, 70, 80, 90

Here that number is 50. Therefore, 50 is the mode.

Now try one by yourself.  If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear.

Exercise 4

The number of bookcases that each of 10 workers produced at a custom furniture factory is shown in the box below

 4, 6, 2, 4, 6, 5, 2, 7, 6, 3

What is the mode of these data?

Now for another example.

Example 5
The box below shows the number of trees growing in the backyard of each house on plumb street.

 7, 9, 9, 4, 11, 7, 9, 12, 3, 7

What is the mode of these data?

Solution

Step 1: First, put the numbers in size place, smallest to largest.

3, 4, 7, 7, 7, 9, 9, 9, 11, 12

Step 2: Now go down the list and determine if there is a number that appears in the list more than any other number.

3, 4, 7, 7, 7, 9, 9, 9, 11, 12

Here we see that both the numbers 7 and 9 appear three times. Therefore, both 7 and 9 are both modes

NOTE: There may be No mode, ONE mode, or multiple modes for a given sample of numbers.

Now try one by yourself.  If you want to see the answer, put the mouse on the yellow rectangle and the answer will appear.

Exercise 4

Sarah was studying how many pairs of Stellar Jays were living within each of 11 acres of land in the national forest.  Her data are shown in the box below

 5, 4, 5, 8, 2, 2, 1, 6, 5, 1, 2

What is the mode of these data?