CALCULATE: Finding the Mean and Median

Consider the following three sets of numbers.

Set 1: {1, 11, 21, 31, 41, 51, 61}

Set 2: {28, 29, 30, 31, 32, 33, 34}

Set 3: {0, 31, 31, 31, 31, 31, 62}

Hint:

What is the mean, median, and mode?

The mean is the number you get by dividing the sum of a set of values by the number of values in the set.

In contrast, the median is the middle number in a set of values when those values are arranged from smallest to largest.

The mode of a set of values is the most frequently repeated value in the set.

TEACHER TIP: This lesson assumes students have previously learned about median.  If you feel that you need a review of this concept, watch this video.

Describing a Data Set

Mean and median are measures of the center of a set of data, but they don’t tell us the whole story about the data.  The three sets in the intro are all very different but they had the same mean and median.

Box Plots and the 5 Number Summary

To get a clearer picture of a set of data, we can make a box plot.  A box plot is a visual representation of 5 numbers that represent the spread and skew of the data.  These numbers are:

• Minimum - the smallest number in the set

• Maximum - the largest number in the set

• Median - the middle number of the set

• 1st Quartile (Q1) - the number below which 25% of the data points are found.  You can find this by locating the middle number of the ordered data BELOW the median

• 3rd Quartile (Q3) -  the number below which 75% of the data points are found. You can find this by locating the middle number of the ordered data ABOVE the median

Example

As a part of a final project for school, Nick asked 11 of his classmates how much student loan debt they will have after graduating college and received the following responses:

Timor is looking into retirement investments and surveys 15 people about the age at which they plan to retire.  He got the following responses:

35        45        50        51        52        52        64        64        65        65        66        66        66        67        70

Analyzing a Box Plot

When looking at a set of data, we can ask some important questions:

1. How spread out is the data?  In the intro, the first set changed by 10s and spread from 1 to 61.  The second set only changed by 1s and was grouped around the center.  This is called dispersion.

2. Is there any data that looks like it doesn’t belong?  If you are a typical A student and most of your grades are in the 90s but you get one bad grade because you were sick that week, that one bad grade would stand out.  We call that type of data an outlier.

3. How is the data grouped?  If we looked at data on age of retirement, most of the data would be near the high end of ages but if we looked at the age that someone got their driver's license, most of the data would be near the low end.  This is called skewness. (You’ll learn more about this one in a couple of lessons)

Now that we have a visual representation of Nick’s data, let’s look at the conclusions that Nick can draw about his data set to include in his presentation.  Here’s the box plot again for reference.

A box plot breaks the data up into quartiles, which means 4 equal parts.  They might not look equal on the graph but each region represents 25% of the data:

his allows us to analyze where the data is located.   For example, Nick adds the following analysis to his presentation:

“75% of survey respondents said that they will have less than $42,000 of student loan debt when they graduate.”

Because $42,000 is Q3 of Nick’s data, there are three 25% sections below that data.

How can we use box plots to compare similar data sets? In this activity, you will work with your groupmates to create a box plot for your designated region then compare your results with another region to analyze the results.

Part 1: Create Your Box Plot

The ordered tables below represent average credit card debt of states in different regions of the US. Follow your teacher's directions to create a box plot for your assigned region.

Level 1

Comparing Student Loan Debt

The box plots below use the data on state average student loan debt for all 50 US states from 2016 and 2021. Use it to answer the questions that follow.