Chapter 3
EDRS 5305
Fall 2005
Gravetter and Wallnau 5th edition
Central Tendency (defined)
► Definition
 A statistical measure to determine a single score
that defines the center of a distribution.
► Goal
 To find the single score that is most typical or
most representative of the entire group (i.e.
average).
Central Tendency (cont.)
► Data
is easier to understand;
► Problem
 No single standard procedure for determining
central tendency.
 No single measure will always produce a
central, representative value in every situation.
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure. 3.1
The difficulty in defining central tendency
Page 54
Mean, Median, Mode
► To
deal with the problems, statisticians have
developed three different methods for
measuring central tendency.
► How do you decide which one to use?
 Keep in mind – the general purpose of central
tendency is to find the single most
representative score.
Mean
► Arithmetic
average
► Add all the scores and divide by the number
of scores.
► For the average of a population use the
Greek letter mu, m (myoo)
► For the mean for a sample use X
(read as X-bar) or M
Mean
► The
mean for a distribution is the sum of
the scores divided by the number of scores.
► Formula
m = SC
N
Population Mean
M=SC
n
Sample Mean
Why Greek letters?
► Greek
► Our
letters to identify population values
own alphabet to identify sample values
M
Sample
n
n is used for the number of
scores in the sample
Example
For a population N=4 scores:
3, 7, 4, 6
m = S C = 20 = 5
N
4
Page 55
Alternative Definitions for Mean
► The
mean can be thought of as an amount each
individual would get if the total (S C) were equally
divided among all the individuals (N) in the
distribution.
► Example 3.2 pg. 55
n=6 boys
Buy 180 baseball cards
Each gets 30 cards
n = 4 boys
M = $5
$20 total
Do not know how much
each boy has
Alternative Definitions for Mean
► Define
the mean as a
balance point for the
distribution.
► Example 3.2 pg. 56
Weighted Mean
► Combining
two sets of scores and then finding the
overall mean for the combined group.
► Example pg. 57
► Because the samples are not the same size, one
will make a larger contribution to the total group
and therefore will carry more weight in
determining the overall mean.
► The overall mean is called the weighted mean.
Computing the mean from a
frequency distribution table
fX
Quiz score (X)
f
10
1
10
9
2
18
8
4
32
7
0
0
6
1
6
Characteristics of the Mean
► Every
score in the distribution contributes to
the value of the mean.
 Every score must be added into the total in
order to compute the mean.
► Changing
the value of the score will change
the mean
► Introducing a new score or removing a
score will change the value of the mean
Median
► The
score that divides a distribution exactly
in half.
► No symbols or notations
► Definition and computations are identical for
a sample and for a population
► Goal of a median is to determine the precise
midpoint of a distribution.
Example
► When
N is
3,
► When N is
3,
an odd number
5, 8, 10, 11
an even number
3, 4, 5, 7, 8
Median = 8
Median = 4.5
Median (cont.)
► Used
when a researcher wants to divide the
sample or population into two groups that
are exactly the same size.
► Median split
 Where one group is above the median line and
the other is below
 For example: one of high-scoring subjects and
one of low-scoring subjects
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 3.5
The median divides the area in the graph in half
Mode
► The
score or category that has the greatest
frequency
► No symbols or notation to identify the mode
► The definition is the same for either a
population or a sample distribution.
Mode (cont.)
► Can
be used to determine the typical or
average value for any scale of
measurement, including a nominal scale
(chapter 1)
► It is possible to have more than one mode
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Table 3.4
Favorite restaurants
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 3.6
A bimodal distribution
vs. multimodal
Selecting a Measure of
Central Tendency
► Could
be possible to compute two or three
measures of central tendency with a set of
data.
► Often get similar results.
Mean
► Mean
is the most preferred measure.
 Usually a good representative value
 Goal is to find the single value that best represents the
entire distribution.
► Mean
has the added advantage of being closely
related to variance and standard deviation (the
most common measures of variability)
► This relationship makes the mean a valuable
measure for purposes of inferential statistics
When to Use the Median
► Three
Page 68
situations in which the median serves
as a valuable alternative to the mean.
 Extreme scores or skewed distributions
 Undetermined values
 Open-ended distributions
When to Use the Mode
► Three
situations in which the mode is
commonly used as an alternative to the
mean, or is used in conjunction with the
mean to describe central tendency
 Nominal scales
 Discrete variables
 Describing shape
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 3.10
Central tendency and symmetrical distributions
Normal
Bimodal
Page 73
Rectangular
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 3.11
Central tendency and skewed distributions