Lesson 4

Lesson 4: Choosing a Measure of Central Tendency

 

Focus

Discover

 

In Focus you found your BMI. How could that number mislead someone to think that they are healthy or unhealthy? When considering the health of a person, many factors have to be considered—cholesterol levels, blood pressure, and hereditary patterns are just a few. Choosing just one number may give misconceptions of what is actually the case.

 

The following is a case in which it’s difficult to describe a situation with a single number.

 

Month	Average Depth of Snow on Ground (in cm)
January	19
February	21
March	16
April	3
May	0
June	0
July	0
August	0
September	0
October	1
November	5
December	12

Source: http://www.livingin-canada.com/climate-edmonton.html

 

Try This 1

 

Look at the set of data showing the monthly average depth of snow in Edmonton. Answer the following questions.

1. Which measure of central tendency (mean, median, or mode) would make it appear that Edmonton often has snow on the ground?
2. Which measure would make it seem like Edmonton rarely has snow on the ground?
3. Would any of the central tendencies you have learned provide you with a clear picture of how much snow fall there really is in Edmonton?

Save your responses in your course folder.

Explore

 

So far in this module you have looked at a variety of measures of central tendency, including mean, median, mode, and trimmed and weighted mean. When data is collected it is often summarized using one or more of these measures of central tendency. Although all of these measures represent an average, you have learned that each of them communicates slightly different information about a data set.

 

How do you know which measure of central tendency to use to best represent or describe the distribution of data?

 

The measure(s) of central tendency you would use requires careful consideration. Your choice is dependent on the data you have and what you want to communicate about it. Sometimes you may come across a set of data where all, or none, of the measures of central tendency are appropriate.

 

One of the first things you need to ask yourself when looking at a data set and deciding on which measure(s) of central tendency to use is whether there are any outliers. If so, then this may greatly influence the mean and may not be the best choice of measure. Also, if there are outliers, does it make sense to include them in the calculations? Remember that this will depend on what you are trying to communicate about the data.

 

Self-Check 1

 

Which measure(s) of central tendency would be the most appropriate for each of the following scenarios? Explain your selection.

1. You plan to buy a new bat for your softball team. What length of bat should you buy? Answer 
2. You would like to determine the typical income for people in your town. Answer 
3. You would like to know the typical height of students in a Grade 12 classroom. Answer 
4. A restaurant sells three sizes of ice-cream cones. How much does the store make per cone? Answer 
5. Using your previous knowledge, give an example of a scenario where the following measure of central tendency would be used. Explain the reason for each of your selections.
   1. arithmetic mean
   2. median
   3. mode
   
   Answer

6. Under what conditions would a trimmed or weighted mean be more appropriate than an arithmetic mean? Answer
   

In Self-Check 1 you looked at different situations and tried to decide which measure of central tendency is the most appropriate to use. In the next Self-Check you will think about how different measures of central tendency can be helpful to different people, depending on the focus the person has. If you choose a measure of central tendency that favours a particular viewpoint, then the results can be misleading to some people.

 

Self-Check 2

 

Revisit the 2011 Washington Capitals’ salaries table from Lesson 1. Use this table to answer questions 1 to 5.

 

2011 Salaries of Washington Capitals
Alexander Ovechkin	$9 000 000
Alexander Semin	$6 700 000
Brooks Laich	$6 500 000
Nicklas Backstrom	$6 000 000
Mike Green	$5 000 000
Dennis Wideman	$4 500 000
Roman Hamrlik	$3 500 000
Joel Ward	$3 000 000
Tom Poti	$3 000 000
Jeff Schultz	$2 500 000
Troy Brouwer	$2 350 000
Eric Fehr	$2 200 000
Mike Knuble	$2 000 000
Jason Chimera	$1 875 000
John Erskine	$1 500 000
Tomas Vokoun	$1 500 000
Michal Neuvirth	$1 150 000
Marcus Johansson	$900 000
Matt Hendricks	$850 000
Jeff Halpern	$825 000
John Carlson	$787 000
D. J. King	$650 000
Braden Holtby	$550 000
Jay Beagle	$525 000
Keith Aucoin	$525 000
Patrick McNeill	$525 000
Ryan Potulny	$525 000

 

1. Would other sports teams have a similar set of data? Explain why or why not. Answer
2. Calculate the mean, median, and mode for earnings. Answer
3. If you were the general manager of this team and had to report the average salary of your players to the following audiences, which measure would you use and why?
   1. ticket holders
   2. potential players
   3. your accountant
   
   Answer
4. Calculate two different trimmed means for this data. Answer
5. Does one of the trimmed means from question 5 better represent the central tendency of this data? If so, explain why. Answer
6. Give situations where one of the measures of central tendency is more useful than the others. Answer
7. What is another data set that would have very different values for the three measures of central tendency? Answer

Share 1

 

Share your results from Self-Checks 1 and 2 with a partner. Focus your discussion on the following questions.

1. Which measure of central tendency is an appropriate representation of the situation being explored?
2. Are there outliers present that would skew the mean?

Place a summary of your discussion in your course folder.

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Self-Check 3

Connect

 

Lesson 4 Assignment

Lesson 4 Summary

 

In this lesson you investigated the following question:

• Which measure of central tendency is the best for a given set of data?

Different sets of data can be interpreted using arithmetic mean, median, mode, weighted mean, and trimmed mean.

 

Which one do you use to best represent the data? Review “Average or Central Tendency: Arithmetic Mean, Median, and Mode.”

 

Source: Khan Academy (CC BY-NC-SA 3.0)

 

Based on your knowledge of these terms and carefully considering the data, your choices of how to interpret will result in a measure of central tendency that will best represent the data.