What you should learn

To find and interpret the mean, median, and mode of a set of data

NCTM Curriculm Standards2, 4 - 9

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Measures of Central Tendency

Mean

Median

Mode

Introduction: The table below shows the top ten charities in amount of contributions for a recent year.

When analyzing data, it is helpful to have one number that describes the entire set of data. Numbers known as measures of central tendency are often used to describe sets of data because they represent a centralized, or middle, value. Three of the most commonly used measures of central tendency are the mean, median, and mode.

Definiton of Mean: The mean of a set of data is the sum of the numbers in the set divided by the number of numbers in the set.

To find the mean of the charity data, find the sum of the dollar values and divide by 10, the number of numbers in the set.

The mean is $368.2 million. Notice that the amount collected by the Salvation Army, $726 million, is far greater than the amounts collected by the other charities. Because the mean is an average of several numbers, a single number that is so much greater than the others can affect the mean a great deal. In extreme cases, the mean becomes less representative of the values in a set of data.

The median is another measure of central tendency.

Definition of Median: The median of a set of data is the middle number when the numbers in the set are arranged in numerical order.

 

To find the median of the charity data, arrange the dollar values in order.

If there were an odd nubmer of dollar values, the middle one woudl be the median. However, since there is an even number of dollar values, the median is the average of the two middle values, 395 and 355.

The median is $375 milion. Notice that the number of values that are greater than the median is the same as the number of values that are less than the median.

A third measure of central tendency is the mode.

Definition of Mode: The mode of a set of data is the number that occurs most often in the set.

 

To find the mode of the charity, look for the number that occurs most often.

In this set, 407 appears twice. Thuse, $407 million is the mode of the data.

It is possible for a set of data to have more than one mode. For example, the set of data {2, 3, 3, 4, 6, 6} has two modes, 3 and 6.

Based on our results, the charity data has a mean $368.2 million, median $375 million, and a mode $407 million. As you can see, the mean, median, and mode are rarely the same value.

Exercise 1: When the Declaration of Independence was signed in 1776, George III was king of Great Britain. After George III, Great Britain had seven monarchs before Queen Elizabeth II was crowned queen in 1952. The table below shows the number of years that each monarch reigned. Find the mean, median, and mode of the data.

You can also determine the mean, median, and mode of a set of data by examining stem-and-leaf plots.

Exercise 2: The stem-and-leaf plot shows population data for the 20 largest cities in the United States to the nearest ten thousand. Find the mean, median, and mode of the data.

Activity: Exploration: Graphing Calculators

You can use a graphing calculator and the MEAN and MEDIAN functions to find the mean and median of a list of numbers. The format is as follows.

mean({a,b,c,d,...z})

median({a,b,c,d,...z})

To find the mean of the data in Exercise 2, access the LIST MATH funciton by pressing . Then choose 3 for mean or 4 for median. Enter the list of numbers, beginning with a left brace ({ ) and ending with a right brace (} ) and a right parenthesis. When you press , the mean or median appears on the screen.

Your Turn

Take time to work through the examples in this lesson using a grpahing calculator.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 17 - 31 odd, 32 - 37

Alternative Homework: Enriched: 16 - 26 even, 27 - 37

Extra Practice: Students book page 764 Lesson 3-7

Extra Practice Worksheet: Click Here.

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