Susan interviewed the twenty five students in her class, asking each person how often they eat out. Most students replied between zero and three times. However, one student reported eating out for every single meal (21 meals a week). Data: 2, 0, 1, 0, 2, 2, 1, 0, 0, 2, 1, 3, 3, 3, 2, 1, 3, 21, 1, 1, 0, 3, 1, 2, 2

Which measure of central tendency will best convey how often the students typically eat out?

Possible Answers: Mean, Median, or Mode

The Scenario

Mean: The arithmetic average. Add up all of the values and divide by the number of scores.

Mean = 2, 0, 1, 0, 2, 2, 1, 0, 0, 2, 1, 3, 3, 3, 2, 1, 3, 21, 1, 1, 0, 3, 1, 2, 2 25 = 57 meals ‘eaten out’ 25 students

= 2.28 meals ‘eaten out’ per student

Mean

Consider what the mean would be without the outlier…

Mean = 2, 0, 1, 0, 2, 2, 1, 0, 0, 2, 1, 3, 3, 3, 2, 1, 3, 1, 1, 0, 3, 1, 2, 2 24 = 36 meals eat out 24 students

= 1.5 meals ‘eat out’ per student

Mean

College Student Income

US Dollars(in thousands)

Mean

US Dollars(in thousands)

Mean

College Student Income

US Dollars(in millions)

Mean

College Student Income

US Dollars(in millions)

Based on the Mean… College Students are Millionaires!

Mean

Mean

Mean – Uses all data, but is sensitive to outliers

Mean

Mode: The most frequently occurring value

Mode

Mode: The most frequently occurring value

Modes: 1, 2

Mode

Mode: The most frequently occurring value

Mode: 0

Mode

A small change in frequency can affect the mode(s)

Mode: The most frequently occurring value

Mode: 0

Mode

A small change in frequency can affect the mode(s)

Students Don’t Typically Eat Out

Mode – Perhaps the least robust. Easily affected by small changes in frequency

Mode

Median: The middle value in a ranked distribution. If there are an even number of values, then take the average of the middle two values.

Median

Median: The middle value in a ranked distribution. If there is an even number of values, then take the average of the middle two values.

Raw Data: 2, 0, 1, 0, 2, 2, 1, 0, 0, 2, 1, 3, 3, 3, 2, 1, 3, 21, 1, 1, 0, 3, 1, 2, 2

Median

Median: The middle value in a ranked distribution. If there is an even number of values, then take the average of the middle two values.

Raw Data: 2, 0, 1, 0, 2, 2, 1, 0, 0, 2, 1, 3, 3, 3, 2, 1, 3, 21, 1, 1, 0, 3, 1, 2, 2

Ranked: 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 21

Median

Median: The middle value in a ranked distribution. If there is an even number of values, then take the average of the middle two values.

Raw Data: 2, 0, 1, 0, 2, 2, 1, 0, 0, 2, 1, 3, 3, 3, 2, 1, 3, 21, 1, 1, 0, 3, 1, 2, 2

Ranked: 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 21

Median

Median: The middle value in a ranked distribution. If there is an even number of values, then take the average of the middle two values.

Raw Data: 2, 0, 1, 0, 2, 2, 1, 0, 0, 2, 1, 3, 3, 3, 2, 1, 3, 21, 1, 1, 0, 3, 1, 2, 2

Ranked: 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 21

Median: 2

Median

US Dollars(in millions)

Mean

Mean

Based on the Mean… College Students are Millionaires!

US Dollars(in millions)

Median

Median Based on the Median… College Students as a Group Aren’t Wealthy

US Dollars(in millions)

Median

Based on the Median… College Students as a Group Aren’t Wealthy

Median

Median – Does not use all data, but is robust; not affected by outliers

Median

Measures of Central Tendency – And Outliers

When there is an outlier, which measure of central tendency can we generally count on to give us the best measure of what is typical?

Which measure should Susan report?

Measures of Central Tendency – And Outliers

When there is an outlier, which measure of central tendency can we generally count on to give us the best measure of what is typical?

Which measure should Susan report?

Mean – Uses all data, but sensitive to outliers

Measures of Central Tendency – And Outliers

When there is an outlier, which measure of central tendency can we generally count on to give us the best measure of what is typical?

Which measure should Susan report?

Mean – Uses all data, but sensitive to outliersMode – Easily affected by small changes in frequency

Measures of Central Tendency – And Outliers

When there is an outlier, which measure of central tendency can we generally count on to give us the best measure of what is typical?

Which measure should Susan report?

Mean – Uses all data, but sensitive to outliersMode – Easily affected by small changes in frequencyMedian – Does not use all data, but is robust

Real World

Use

When there is an outlier, your reporting options are to report:

(1) Median, or

(2) Median and Mean

Measures of Central Tendency – And Outliers

Real World

Use

When there is an outlier, your reporting options are to report:

(1) Median, or

(2) Median and Mean

Measures of Central Tendency – And Outliers

If you think the outlier does not belong in thedata set (i.e., was an error)… then consider also reporting the mean without the outlier.

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References

References

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Appendix: Online Resources

Mean, Median, and Mode Song

From LearningUpGrade.com; posted on YouTube.Description of Video: A basic overview of how to determine the mean, median and mode. Includes music and animation. Length: 1m 33s.

View at tinyurl.com/yfsnmh9

Comparing the Properties of the Mean and the Medianat Principles & Standards for School Mathematics

Description of this Interactive Demonstration: Move the numbers around on the number line, and see the corresponding effect on the mean and median. How do outliers affect the mean and median?Length: Interactive Demonstration Participate at tinyurl.com/33tngr

Appendix: Online Resources

Statistics: The Average

Posted by khanacademy on YouTube. Description of Video: A more in depth, college level, introduction to the mean, median, and mode. Note – starts with a blank screen, which is then written upon…Length: 12m 35s

View at tinyurl.com/ykbbvmj

It’s Not Hard (Averages Song)

Posted on YouTube by jalapenojane.Description of Video: This is just for fun…. Covers mean, median, and mode in a way that may leave you laughing aloud.Length: 3m 49s

View at tinyurl.com/yhc885w

Appendix: Creative Commons License

Creative Commons Attribution-Share Alike 3.0 License

You are free to share (copy, distribution and transmit the work) and to remix (to adapt the work) this Powerpoint Presentation What to Report When There is an Outlier by Robert G. Kelley, Ph.D. on the condition that you provide attribution (you must attribute the work in a manner specified by the author or licensor – but not in any way that suggests that they endorse you use of the work) and share alike (if you alter, transform, or build upon this work, you may distribute the resulting work only under the same, similar or compatible license) this work.

Note that the online resources listed in the appendix are separate works from this Powerpoint presentation, and are not covered by this Creative Commons License.

Please attribute this work to: Robert G. Kelley, Ph.D. (www.miracosta.edu/home/rkelley)