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Wize High School Grade 9 Math Textbook > Data Analysis

Measures of Central Tendency - Mean, Median, Mode

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Measures of Central Tendency

Wize Concept
Recall that a variable is an attribute that we measure, such as height, eye colour, age, etc.

When we are collecting numerical data for one variable, we will end up with a list of numbers called a data set. We often want to summarize or represent this data set using one single number. Sometimes this number is called the "average" or the "center".

There are three common ways of describing this "average" or "center", and these are called the measures of central tendency -- mean, median, and mode.

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Mean

The mean is what most people refer to as the "average".

Examples
  • How would you calculate your average mark across all 4 classes?
    you could add up all 4 marks in your 4 classes, then divide that total by 4 to find the "average"
  • How would you find the "average" height of the students in your math class?
    you could add up the heights of all students in your math class, then divide that total by the number of students in your math class to find the "average"


How to Calculate the Mean?

mean=sum of all numerical data pointstotal number of data points\boxed{\text{mean}=\dfrac{\text{sum of all numerical data points}}{\text{total number of data points}}}
We add up all of the numerical data, then divide by the number of data points there are.

Example #1 (No extreme values)
The times (in minutes) it takes a group of 5 students to complete a math problem are 2,  4,  6,  8,  102,~~4,~~6,~~8,~~10. Calculate the mean time it takes for these students to complete the math problem.

2+4+6+8+105=6\displaystyle \frac{2+4+6+8+10}{5}=6
So, the mean is 6 minutes.
Example #2 (With extreme values)
The heights (in inches) of 5 different plants are 2,  4,  6,  8,  1002,~~4,~~6,~~8,~~100. Calculate the mean height of these plants.

2+4+6+8+1005=24\displaystyle\frac{2+4+6+8+100}{5}=24
So, the mean is 24 inches.

Notice that 100 is much bigger than the other data points, so it caused the mean to be much larger (24 inches is taller than 4 out of 5 of the plants). 100 is called an extreme value or outlier.
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Median

The median is what we think of as the "middle" of our list of data.

Wize Tip
"Median" sounds like "middle. So, the median is the data that represents the middle


How to Calculate the Median?

  1. Sort the list of numerical data from lowest to highest
  2. If you have an odd number of data points, the median is the number that is right in the middle
  3. If you have an even number of data points, the median is the "average" or mean of the 2 numbers that are in the middle


Example #1 (Odd number of data)
Find the median of 2,  4,  10,  6,  82,~~4,~~10,~~6,~~8

2,  4,  6⏟middle,  8,  ,102,~~4,~~\underbrace{6}_\text{middle},~~8,~~,10
The median is the number right in the middle with an equal number of data on each side. So, the median is 6.


Example #2 (Even number of data)
Find the median of 6,  12,  8,  4,  10,  26,~~12,~~8,~~4,~~10,~~2

2,   4,  6,      8,⏟middle 2 numbers  10,   122,~~~4,~~\underbrace{6,~~~~~~8,}_\text{middle 2 numbers}~~10,~~~12
The median is in between the two numbers located in the middle. The mean of 6 and 8 is 6+82=7\dfrac{6+8}{2}=7, so, the median is 7.


Example #3 (Extreme value)
Find the median of 2,  100,  4,  8,  62,~~100,~~4,~~8,~~6

2,  4,  6⏟middle,  8,  1002,~~4,~~\underbrace{6}_\text{middle},~~8,~~100
The median is 6. Notice that it is not affected by an extreme value like 100.

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Mode

The mode is the numerical data that shows up most frequently. For some data sets, you can have multiple modes or no modes at all.

Wize Tip
"Mode" sounds like "most". So, the mode is the data that shows up the most.


*When we look at qualitative data, we can talk about the mode as well. The mode is the qualitative data that appears most frequently

Example #1 (Single mode)
The ages of 7 children on a playground are 4,  7,  8,  5,  4,  4,  74,~~7,~~8,~~5,~~4,~~4,~~7. Find the mode.

The number that shows up most frequently is 4. So, the mode is 4 years old.

Example #2 (Multiple modes)
The number of people in 8 households are 2,  4,  2,  3,  5,  4,  1,  62,~~4,~~2,~~3,~~5,~~4,~~1,~~6. Find the mode.

The numbers that show up most frequently are 2 and 4 (it's a tie!). So, there are 2 modes: 2 and 4 people.

Example #3 (No modes)
The amount of time (in hours/week) that 5 people exercise are 2,  1,  3,  5,  102,~~1,~~3,~~5,~~10. Find the mode.

Since all numbers show up with the same frequency, there is no mode.

Practice: Mode

For each of the following statements about mode, indicate whether it is True or False.

a) There can only be one mode is any given data set.

b) There is no mode for qualitative variables.

c) A data set will always have one or more modes.

d) If 75% of the class gave the professor a rating of "8", then "8" must be the mode.

Practice: Mean, Median, and Mode

Calculate the mean, median, and mode for the following two data sets.
12, 6, 3, 2, 5, 5, 2

Practice: Choosing Mean, Median, and Mode

For each of the following scenarios, decide if it's better to use the mean, median, or mode to describe the "center" or "average" of the data set. Justify your answer.

*You may select more than one answer if you think more than one measure of center is appropriate for that scenario.
A shoe store is preparing an inventory order for the next 3 months, and the manager of the shoe store needs to decide on how many pairs of each shoe size to order.

Practice: Mean

A survey was completed by 12 new college graduates and it was found that their average (mean) age is 24 years old. It turns out that some numbers were incorrectly entered. A number with value 27 was incorrectly entered as a 17 and a number with value 22 was incorrectly entered as an 20. Find the correct average (mean) age.