Interpreting the Mean and Median of a Data Set: Corequisite Support Activity

What you’ll need to know

In this support activity you’ll become familiar with the following:

You will also have an opportunity to refresh the following skills:

In the next section of the course material and in the following activity, you will need to compare the mean and median of a quantitative variable and calculate mean and median by hand and using technology. This support activity will give you more practice calculating mean and median and will set the stage for interpreting the comparisons of mean and median in the upcoming section.

In Calculating Mean and Median of a Data Set: What to Know, you had a chance to calculate the mean and median of a small data set by hand. As you work through this activity, return to that section as needed to refresh the formula and the process.

Recall

Before you begin, recall the definitions of mean and median.

Core skill: 

Core skill:

Salaries

In this activity, we’ll be using the two data sets listed below. Suppose that the first data set lists the monthly salaries (in thousands of dollars) for all six employees at a company during the month of January. For example, Employee [latex]1[/latex] made [latex]\$4,000[/latex]in salary in January, Employee [latex]2[/latex] made [latex]\$6,000[/latex], and so on. We’ll consider this amount the regular salary per month for each of these employees.

Employee

Monthly Salary in January

(in thousands of dollars)
Employee 1 [latex]4[/latex]
Employee 2 [latex]6[/latex]
Employee 3 [latex]3[/latex]
Employee 4 [latex]5[/latex]
Employee 5 [latex]6[/latex]
Employee 6 [latex]3[/latex]

Then, suppose that one of the employees gets a large raise in February. The second data set lists the monthly salaries (in thousands of dollars) for the same six employees during the month of February. Can you locate which employee got the raise?

Employee

Monthly Salary in February

(in thousands of dollars)
Employee 1 [latex]4[/latex]
Employee 2 [latex]8[/latex]
Employee 3 [latex]3[/latex]
Employee 4 [latex]5[/latex]
Employee 5 [latex]6[/latex]
Employee 6 [latex]3[/latex]

question 1

Now consider just the data set of employee salaries from January. We’d like to know the median salary for the six employees for the month of January.

Calculating Median

Recall

To answer the question below, you’ll need to calculate the median of a data set containing an even number of values. You can refresh that information here if needed.

Core skill: 

question 2

Let’s use technology to verify the result you obtained for the median in Question 2.

Go to the Describing and Exploring Quantitative Variables tool at https://dcmathpathways.shinyapps.io/EDA_quantitative/ and confirm your answer using the online tool.

Step 1) Select the Single Group tab.

Step 2) Locate the drop-down menu under Enter Data and select Your Own.

Step 3) Under Do you have, select Individual Observations.

Step 4) Under Name of Variable, type “January Salaries (in thousands $)”.

Step 5) Under Enter observations, enter the data list, separated by spaces: “4 6 3 5 6 3.” The median will be among the Descriptive Statistics listed in the tool.

How did you do? Did your calculation match the one in the tool? Now consider what the median implies about the data. Remember that we think of the median as the 50th percentile.

question 3

 Calculating Mean

Recall

Do you recall how to calculate the mean of a data set?

 

question 4

Repeat the steps above to locate the mean in the tool under Descriptive Statistics then compare your answer to the technology calculation. Did they match?

question 5

Now, consider what the mean implies about the data. See the recall box at the top of the page for a hint.

question 6

question 7

We saw that the median and the mean employee salaries for January were the same. What understanding can we take from that information?

Interactive Example

What can we understand about the median and mean employee salaries for January being the same? Fill in the blanks to answer the following questions.

  1. The median of the data set implies that ____________ made more than [latex]\$4,500[/latex] in January and _________ made less.
  2. The mean of the data set implies that if the January salaries had been added up and evenly distributed across all six employees, each person would have received ________________.

Comparing Mean and Median

What happens to the mean and median if we change one of the values in the data set?

Recall that the data set of employee salaries from February includes a big raise from one employee. First calculate the median of this set to answer Question 8 below, then consider how we might expect the mean of the February salaries compares to the mean of the January salaries.

Before you get the median and the mean from the technology, or before you calculate the mean by hand, first think about what you think will be true about the February mean compared to the January mean and why.

question 8

question 9

question 10

Interactive example

Was the mean you calculated for February salaries higher, lower, or similar? What do you think caused that to be true? Click below for a discussion after you enter your answers to Questions 8 – 10.

Now let’s consider a slightly different question.

question 11

It may take some time before you really feel comfortable interpreting means and medians and understanding what they imply about a data set. A key idea to take from this activity is that, while the median stays relatively fixed in a data set if one value changes by a large amount, the mean does not. This tells us that the mean is sensitive to the presence of extreme values in the data set.

It’s okay if you need more practice to process the sensitivity of the mean. But if you feel comfortable calculating the means and medians in this activity by hand and using technology, please move on to the next section and activity.