Measures of Central Tendency – And Outliers

Report
The Scenario
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
Mean
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
College Student Income
US Dollars
(in thousands)
Mean
College Student Income
US Dollars
(in millions)
Mean
Based on the Mean… College
Students are Millionaires!
Mean
US Dollars
(in millions)
Mean
Mean – Uses all data, but is
sensitive to outliers
Mode
Mode: The most frequently occurring value
Mode
Mode: The most frequently occurring value
Modes: 1, 2
Mode
Mode: The most frequently occurring value
A small change in frequency
can affect the mode(s)
Mode: 0
Mode
Mode: The most frequently occurring value
Students Don’t Typically Eat Out
A small change in frequency
can affect the mode(s)
Mode: 0
Mode
Mode – Perhaps the least
robust. Easily affected by small
changes in frequency
Median
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
Mean
Based on the Mean… College
Students are Millionaires!
Mean
US Dollars
(in millions)
Median
Median
Based on the Median… College
Students as a Group Aren’t Wealthy
US Dollars
(in millions)
Median
Median
Based on the Median… College
Students as a Group Aren’t Wealthy
US Dollars
(in millions)
Median
Median – Does not use all data,
but is robust; not
affected by 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?
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?
Mean – Uses all data, but
sensitive to outliers
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?
Mean – Uses all data, but
sensitive to outliers
Mode – Easily affected by small
changes in frequency
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?
Mean – Uses all data, but
sensitive to outliers
Mode – Easily affected by small
changes in frequency
Median – Does not use all data,
but is robust
Which measure should Susan report?
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
Real
World
Use
When there is an outlier, your
reporting options are to report:
(1) Median, or
(2) Median and Mean
If you think the outlier does not belong in the
data set (i.e., was an error)… then consider
also reporting the mean without the outlier.
References
Posted on Flickr as fast food is the best!
by Ebruli. Available under Creative
Commons Attribution 2.0 Generic
License to share and remix.
Posted on Wikimedia Commons as
Earth Western Hemisphere white
background by Hansjorn. Available in
the Public Domain.
References
Posted on Flickr as Money! by Tracy O.
Available under Creative Commons
Attribution-Share Alike 2.0 Generic
License to share and remix.
Posted on Wikimedia Commons as Bill
Gates 2004 crop. Originally posted to
Flickr by deVos. Available under Creative
Commons Attribution-Share Alike 2.0
Generic License to share and remix.
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 Median
at 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)

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