This document discusses measuring central tendency through the mean, median, and boxplots. It provides definitions and formulas for calculating the mean and median of a data set. An example is given of commuting times for 20 New York workers, where the mean is calculated as 31.25 minutes and the median is 22.5 minutes. The key differences between the mean and median are explained. The document also introduces boxplots and five-number summaries as visual ways to understand the distribution of data.

1. AP Statistics Exploring Data
Describing Quantitative Data with Numbers
EdTech 541
Angie Kruzich
September 2014

2. Learning Objectives
MEASURE center using mean & median
CALCULATE mean
DETERMINE median
COMPARE mean & median
CONSTRUCT a boxplot

3. Measuring Center: The Mean
The most common measure of center
is the ordinary arithmetic average,
or mean, , (pronounced “x-bar”).

x

4. Calculate mean by adding all data
values and dividing by number of
observations.
If the n observations are x1, x2, x3,
…, xn, then:
x 
sum of observations
n

x1  x2  ... xn
n
Mean Definition

5. In mathematics, the capital Greek
letter Σ (sigma) is short for “add
them all up.” Therefore, the mean
formula can also be written:
x 
xi 
n
More Mean

6. Measuring Center: The Median
Another common measure of center is
the median. The median describes
the midpoint of a distribution.

7. Median Definition
It is the midpoint of a distribution such
that half of the observations are
smaller and the other half larger.

8. Finding Median
1. Arrange numbers from smallest to
largest.
2. The Median is the number in the
middle, unless…

9. Odd versus Even Numbers of Data

10. Interactive Quiz
Obtain an Nspire
classroom calculator
Log on
Your teacher will be
sending you a
document

11. Quiz Measuring Center
Calculate the mean
and median of the
commuting times
(in minutes) of 20
randomly selected
New York workers.
10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

12. Quiz Measuring Center
10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45
On page 1.1 finish
entering the
data in the
spreadsheet.
Press control right/left
arrow to change pages
on calculator.

13. Quiz Measuring Center
Read the
instructions
on page 1.2.

14. Quiz Measuring Center
On page 1.3 use
the calculator
page provided
to calculate
the mean.
Watch your
formatting!

15. Quiz Measuring Center
On page 1.4 and 1.5
enter your final
solutions.
Press control arrow
up when you are
done.

16. 10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45
0 5
1 005555
2 0005
3 00
4 005
5
6 005
7
8 5
Key: 4|5
represents a
New York
worker who
reported a 45-
minute travel
time to work.
M 
20  25
2
 22.5 minutes
Quiz Median Solution
To calculate the median:

17. Quiz Mean Solution
10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45
To calculate the mean:
x 
10 30 5  25 ... 40 45
20
 31.25 minutes

18. Comparing the Mean and the Median
If mean and median are close together,
then distribution is roughly symmetric.
If mean and median are exactly the same,
distribution is exactly symmetric.

19. Comparing the Mean and the Median
In a skewed distribution, the mean is
usually farther out in the long tail than is
the median.

20. Comparing the Mean and the Median
The mean and median measure center in different
ways.
Don’t confuse the “average” value of a variable
with its “typical” value.

21. The Five Number Summary
The mean and median tell us little about the tails
of a distribution.
The five-number summary of a distribution
consists of:

22. What are Quartiles?

23. Constructing Boxplots
Also known as box-and-whisker plots.
The five number summary gives us values to
construct a boxplot:
Minimum Q1 M Q3 Maximum

24. Constructing Boxplots

25. Constructing Boxplots
Consider our NY travel times data.
10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45
In your groups, discuss & construct a
boxplot for the data on your Nspires.

26. 10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45
5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85
Q M = 22.5 Q3= 42.5 1 Min=5 = 15
Max=85
Constructing Boxplots

27. Summary
• Mean is average
• Median is middle
• How to compare mean and median
• How to construct a boxplot

28. Resources
Images
• Slide 3 – Courtesy of Math is Fun
http://www.mathsisfun.com/definitions/mean.html
• Slide 6 – Courtesy of W3.org
http://www.w3.org/2013/11/w3c-highlights/
• Slide 7 – Courtesy of Knowledge Center
http://knowledgecenter.csg.org/kc/content/stats-101-mean-versus-median
• Slide 8 – Courtesy of Sparkle Box
http://www.sparklebox.co.uk/6771-6780/sb6779.html#.VCigcRaK18E
• Slide 10 – Courtesy of Underwood Distributing
http://www.underwooddistributing.com/shop/shop?page=shop.browse&c
ategory_id=109
• Slide 11 – Courtesy of Streetsblog USA
http://usa.streetsblog.org/2008/01/10/does-times-square-have-too-many-people-
or-just-too-many-cars/

29. Resources
Images
• Slide 18 – Courtesy of Profit of Education
http://profitofeducation.org/?p=2152
• Slide 19 and 20 – Courtesy of Data Analysis for Instructional Leaders
https://www.floridaschoolleaders.org/general/content/NEFEC/dafil/lesso
n2-5.htm
• Slide 21 – Courtesy of Penn State
https://onlinecourses.science.psu.edu/stat100/node/11
• Slide 23 and 25 – Courtesy of GCSE Math Notes
http://astarmathsandphysics.com/gcse-maths-notes/gcse-maths-notes-five-
figure-summaries-and-boxplots.html
Reference
Starnes, D., Yates, D., & Moore, D. (2011). The practice of statistics. New York,
New York: W.H. Freeman and Company.