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Mean, Median, Mode, and Range

Introduction

The word “average” is a description that can mean many different things.  It can describe many different measurable quantities in many different aspects of life.  Some examples are:

  • average speed
  • average temperature
  • scoring average
  • grade point average

The word “average” can also be used to describe quantities that cannot be measured mathematically.  The following examples use the word “average” to describe something as “ordinary” or “typical”:

  • average looking
  • average performance
  • feeling “about average”

Since this is a math lesson, we will focus on different ways to find the (measurable) average of a set of numbers.  Average can be represented by several different mathematical words.  The words mean, median, and mode are each mathematical words that can be used to describe the concept of “average.”  In this lesson, you will learn the definitions of each of these words (mean, median, and mode) as well as a variety of situations where average may be best represented by each word.

Lesson

Begin with the following problem:

Intro Problem: The girl’s on Alicia’s swim team swam a length of the pool and compared their times.  There were 10 girls and their times were 15, 15, 17, 18, 19, 19, 19, 21, 24, and 42 seconds.  What was their average time? 

The question “What was the average time?” is one that can be answered using several different mathematical interpretations of the word “average.”

  • The mean of a group of numbers is the sum of the values divided by the number of values
  • The median of a group of numbers is the “middle number” of the group when they are written from smallest to largest. 
  • The mode of a group of numbers is the value that occurs most often

 

In the intro problem, the mean, median, and mode can be found as follows:

Mean_median_mode_range_vis_1


 

Mean_median_mode_range_vis_2


 

Mean_median_mode_range_vis_3 

So the mean, median, and mode are 20.9, 19, and 19 respectively.  Note that the in finding the median there were two middle numbers so they were averaged to find the actual median here.  Most people think of the mean as the “average”, but in fact each of these three numbers can represent the average of the data.  We can take a closer look at the data to determine which of these averages is the best representation of the data.  A line plot can be used as a visual representation of the data.  Scroll over the line plot to see where the mean, median, and mode are on the number line.

Mean_median_mode_range_vis_4

The mean of a data set is generally thought of as the best way to analyze the average of a set.  However, in this case, it appears that the mean is higher than seven of the data points, about the same as one of them, and lower than the other two.  The median and mode each appear to represent a more centralized location among the data, which would make them better candidates to represent the average of the data set. 

It often helps to remember what the data represents in order to analyze it effectively.  In this case the data represents the amount of time (in seconds) that it takes for 10 girls to swim a length of the pool.  The average of this data could represent how long it takes the average girl to swim across the pool.  The fastest and slowest girls are certainly not average.  In this case, the slowest girl took about twice as long to swim the length as everyone else.  On the number line, all the numbers are clumped together except for the 42.  A number that is very different from all the others is called an outlier.  Outliers have no effect on the mode and a nominal effect on the median since they are not directly used to calculate either one.  They make a big difference in the mean since the size of the outlier is used directly in the calculation of the mean. 

There may be a reason one of the girls took so much longer than the others.  Maybe she had to stop during the length.  Perhaps it was her first day on the swim team.  Whatever the reason, her time is not a good representation of the average swimmer.  The mean of the other 9 girls can be found by adding up their sum and dividing by 9.  The sum of the times of the other 9 girls is 167, so the mean is (167 ÷ 9 =) 18.6. 

The mean of 18.6 is a better representation of the average time it took to swim the length.  The median and mode both remain at 19.  The most common representation for the “average” is the mean, so the mean should be used in most cases.  However, when the mean and median are far apart it is often the result of an outlier.  When they are far apart, it is usually the result of outlier(s) or a math mistake.  Take a closer look at the problem before moving on.

Mean, median, and mode are all measures of central tendency.  When comparing data, it can also be useful to find the distance between the highest and lowest numbers.  The range can be found by subtracting the lowest number from the highest number in a data set. 

 

Example 1:  William played in 8 basketball games and scored the following point totals:  13, 8, 5, 18, 11, 12, 8, 22.  Find the mean, median, mode, and range of his scores.

 

 Solution: 

   First put the numbers in order:  5, 8, 8, 11, 12, 13, 18, 22.

       Mean: 5 + 8 + 8 + 11 + 12 + 13 + 18 + 22 = 97.       97 ÷ 8 = 12.125.

       Median:  Middle numbers are 11 & 12, so the median is 11.5

       Mode:  The number 8 appears twice, so the mode is 8.

       Range:  22 – 5 = 17, so the range is 17. 

In basketball as in any other sport, the mean is generally used to represent the scoring average.  In this case, the mean and median are pretty close to each other, which is a good sign that we did the problem correctly.  There are no outliers here, so the best number to use for the average is the mean.  This data is most commonly represented by saying William’s scores an average of 12.125 points per game. 

Review

Find the mean, median, mode, and range for each problem.

1)  Heights of classmates (inches):  54, 56, 56, 57, 59, 64, 65

2)  Distances run in training (miles):  3, 3, 3, 5, 3, 3, 3, 7, 3, 3

3)  Golf scores: 90, 102, 88, 96, 91

 

Answer the question.

4)  What is an outlier?

5)  Which measure of central tendency is generally used to describe an average?

 

Scroll down for answers...

 

 

 

 

 

 

 

Answers:

 

1)  Mean is 58.7

     median is 57

     mode is 56

     range is 11

 

2)  Mean is 3.6

     Median is 3

     Mode is 3

     Range is 4

 

3)  Mean is 93.4

     Median is 91

     There is no mode

     Range is 14

 

4)  An outlier is a number in a data set that is far different from the other numbers in the set.


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