Minds On

Measures of central tendency

The hockey season just ended. The following numbers are the number of goals in each game for the Fantastic Fliers team. What number best represents how many goals they scored in each game?

7   7   9   10   5   7   8   3

Is there any score that doesn’t fit with the rest? Why or why not?

Record your thinking using a method of your choice.

Throughout this learning activity, you can record your thoughts digitally, orally, or in print.

Action

Understanding the measures of central tendency

To represent a diverse data set, we can use measures of central tendency. A measure of central tendency is any measure that estimates the center or average of a data set.

How do you calculate the range? Since the range is a measure of the spread between the highest and lowest values in the data set, you need to determine the difference between the highest number and lowest number.

7  7  9  10  5  7  8  3

Press ‘Hint’ to reveal an example.

The highest number is 10. The lowest is 3. 10 − 3 = 7. The range between the highest and lowest number is 7.

What are the measures of central tendency?

There are three different types that you can determine.

Mean is the average of a data set. It is found by adding all the numbers together in a data set and dividing by the amount of numbers in the set.

7  7  9  10  5  7  8  3

Press ‘Hint’ to reveal an example.

All the number added together equal 56. There are eight numbers total. 56 ÷ 8 = 7. The mean is 7.

Median is the value that falls in the middle of the data set when all the numbers are in consecutive order. If there is an even amount of data and two numbers fall in the middle, the average of those two numbers is calculated.

7  7  9  10  5  7  8  3

Reorder:3  5  7  7  7  8  9  10

Find middle:3  5  7  7  7  8  9  10

Press ‘Hint’ to reveal an example.

When finding the middle for this example, it would be 7 and 7. Finding the average: 7 + 7 ÷ 2 = 7. The median is 7.

Mode is the most common number in the data set.

7  7  9  10  5  7  8  3

Number Frequency
7 3
9 1
10 1
5 1
8 1
3 1

Press ‘Hint’ to reveal an example.

In this example there are three 7’s. As it is the most common number, the mode is 7.

In this example, the mean, median, and mode are all 7. That means that 7 is the best number to represent this data set. However, the answers will not always be the same for all three central tendencies.

Using the measures of central tendency

When should you use each measure? The mean is typically considered the best representation of a data set, but there are some circumstances where mode or median is preferred. Can you think of what these situations might be?

Press ‘Hint’ to reveal a suggested answer.

The median should be used if there are outliers in the data set. That means if there are a few numbers that are completely different from the rest of the data. Let’s add an outlier to our previous data set.

7   7   9   10   5   7   8   3   150

In this data set, the mean would be 22.8. Why might that not be the most appropriate representation of the data set? How does one number change the mean so much? The median of this data set would still be 7. Clearly, that is a better representation of these numbers than 22.8.

The mode should be used if the data is qualitative, rather than quantitative.

Quantitative data means that the data is recorded in numbers. When the data set is numerical, as it is in the example above, all measures of central tendency can be used.

Qualitative data means that the information recorded is a name or category. For example, if a survey was conducted to determine favourite seasons in one grade six class the results might be:

summer   summer   winter   autumn   summer   autumn

In this case, the median and mean cannot be calculated so determining the mode or most common/frequent answer (summer) would best represent the data set.

Analysing data sets

Analyze the following examples of data sets.

Choose two of the data sets and answer the questions in a method of your choice.

  1. A series of athletes were trying to complete an obstacle course. The following table records how everyone did and the percentages refer to the completion rate.
Rate Frequency
85% 12
87% 6
92% 4
53% 1
82% 3
  1. Which measure of central tendency would best represent this data set? Why?
  2. Calculate the mean, median, and mode and compare them.
  1. The following data set records the total tonnes of recycling in Country A from 2008 to 2018.

    723, 680, 691, 702, 710, 721, 721, 720, 706, 714, 729
    1. What is the range of the data set?
    2. Which measure of central tendency would best represent this data set? Why?

Press ‘Hint’ to reveal the a suggested answer.

  1. the range of the data set is 729-680 (the difference between the highest value and the lowest value) = 49 tonnes
  2. Mean - 7,817¸11 = 710.6 tonnes
    Median - 714 tonnes
    Mode - 721 tonnes
  1. The following data set records the amount of people that moved to four diverse cities.
Language Number of people
City A 1,185
City B 1,090
City C 0
City D 0
  1. Which measure of central tendency would best represent this data set? Why?
  2. Determine that measure of central tendency.

Press ‘Hint’ to reveal the a suggested answer.

  1. Mean - 2,275 ÷ 4 = 568.75 people
  2. Median - 545 people
  3. Mode - 0
  1. The following data set shows the number of goals made by the Toronto Maple Leafs in the last 10 games.
    3, 4, 5, 3, 4, 1, 3, 3, 3, 5
    1. What is the range of the data set?
    2. Which measure of central tendency would best represent this data set? Why?
    3. Calculate the mean, median, and mode and compare them.

Press ‘Hint’ to reveal the a suggested answer.

  1. The range of the data set is 5 - 1 = 4
  2. Mean - 34 ÷ 10 = 3.4
    Median - 3
    Mode - 3

Determining central tendencies from a graph


The following graph and table records the total precipitation (in mm) for each month of 2020 in Toronto.

Total Precipitation in Toronto in 2020
Months of the Year Precipitation (mm)
January 131.4
February 52
March 53. 8
April 41
May 43.2
June 49.8
July 67.6
August 91
September 40.8
October 60.4
November 65.8
December 64

Determine the mean, median, and mode of the data set. Which one best represents the amount of precipitation in Toronto in 2020?

Press ‘Hint’ to reveal the a suggested answer.

Mean - 760.8 ÷ 12 = 63.4

Median - 57.1

Mode - none

Consolidation

Digging deeper with the measures of central tendency

You have spent some time determining which measure of central tendency most represents different data sets. Using this knowledge, respond to the following questions in a method of your choice:

  • What is an example of a data set that is best represented by the mode? Why?
  • What is an example of a data set that is best represented by the median? Why?
  • What is an example of a data set that is best represented by the mean? Why?

Reflect

Reflect on and answer the following questions.

  • Why is it important to determine measures of central tendency?
  • Why does the context of the data set matter for determining which measure is the most representative?
  • Which measure is the most difficult to calculate? Easiest to calculate? Why?

Reflection

As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.

I feel…

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