Measures of Dispersion
Learning Objectives:
1. Explain what is meant by variability
2. Describe, know when to use, interpret and
calculate: range, variance, and standard
deviation
More Statistical Notation

X indicates the sum of squared Xs.
2
 Square
 Find

(X )
ea score (22+ 22)
sum of squared Xs =4+4=8
2 indicates
the squared sum of X.
 (2+2)2
2
Measures of Variability
A
… describe the extent to
B
C
which scores in a
0
8
6
distribution differ from
2
7
6
6
6
6
10
5
6
12
4
6
X=6
X=6
X=6
each other.
3
A Chart Showing the Distance
Between the Locations of Scores in
Three Distributions
4
Variability
Provides a quantitative measure of the
degree to which scores in a distribution
are spread out or clustered together
 Figure 4.1

Kurtosis

Kurtosis based on size of a distribution’s
tail.
 Leptokurtic:
thin or skinny dist
 Platykurtic: flat
 Mesokurtic: same kurtosis (normal
distribution)
Three Variations of the Normal
Curve
7
The Range,
Semi-Interquartile Range,
Variance, and Standard
Deviation
The Range

… indicates the distance between the two
most extreme scores in a distribution
 Crude
 Used
measurement
w/ nominal or ordinal data

Rangedifference btwn upper real limit of
max score and lower real limit of min
score

Range = highest score – lowest score
9
The Interquartile Range
Covered by the middle 50% of the
distribution
 Interquartile range= Q3-Q1
 Semi-Interquartile Range

 Half
of the interquartile range
10
Variance and Standard Deviation

Variance & standard deviation
communicate how different the scores in a
distribution are from each other

We use the mean as our reference point
since it is at the center of the distribution
and calculate how spread out the scores
are around the mean
11
The Population Variance and
the Population Standard
Deviation
Population Variance

The population variance is the true or
actual variance of the population of
scores.
(X )
X 
2
N
X 
N
2
2
13
Population Standard Deviation

The population standard deviation is the
true or actual standard deviation of the
population of scores.
(X )
X 
N
X 
N
2
2
14
Describing the Sample Variance
and the Sample Standard
Deviation
Sample Variance

The sample variance is the average of
the squared deviations of scores around
the sample mean
(X )
X 
2
N
SX 
N
2
2
16
Sample Variance
Variance is average of squared deviations
(usually large) & squared units
 Difficult to interpret
 Communicates relative variability

Standard Deviation
Measure of Var. that communicates the
average deviation
 Square root of variance

Sample Standard Deviation

The sample standard deviation is the
square root of the average squared
deviation of scores around the sample
mean.
(X )
X 
N
SX 
N
2
2
19
The Standard Deviation

… indicates

“average deviation” from mean,

consistency in scores,
&

how far scores are spread out around mean
larger the value of SD, the more the
scores are spread out around mean, and
the wider the distribution
20
Normal Distribution and the
Standard Deviation
21
Normal Distribution and the
Standard Deviation
Approximately 34% of the scores in a
perfect normal distribution are between the
mean and the score that is one standard
deviation from the mean.
22
The Estimated Population
Variance and the Estimated
Population Standard Deviation
Estimating the Population Variance
and Standard Deviation
2
X

The sample variance ( S ) is a biased
estimator of the population variance.

The sample standard deviation ( S ) is a
X
biased estimator of the population
standard deviation.
24
Estimated Population Variance

By dividing the numerator of the sample
variance by N - 1, we have an unbiased
estimator of the population variance.
( X )
X 
2
N
sX 
N 1
2
2
25
Estimated Population
Standard Deviation

By dividing the numerator of the sample
standard deviation by N - 1, we have an
unbiased estimator of the population
standard deviation.
(X )
X 
N
sX 
N 1
2
2
26
Unbiased Estimators
2
X

s is an unbiased estimator of

s X is an unbiased estimator of



2
The quantity N - 1 is called the degrees of
freedom
 Number
of scores in a sample that are free to
vary so that they reflect variability in pop
27
Uses of S X2 , S X , s X2 and s X


2
X
Use the sample variance S and the
sample standard deviation S to describe
X
the variability of a sample.
2
X
Use the estimated population variance s
and the estimated population s X
standard deviation for inferential purposes
when you need to estimate the variability
in the population.
28
Organizational Chart of
Descriptive and Inferential
Measures of Variability
29
Always..
Determine level of measurement
 Examine type of distribution
 Calculate mean
 Calculate variability

American Psychological
th
Association (5 ed)

Mean
M

Standard Deviation
 SD
Example

Using the following data set, find
The
 The
 The
 The

range,
semi-interquartile range,
sample variance and standard deviation,
estimated population variance standard deviation
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
32
Example Range

The range is the largest value minus the
smallest value.
17  10  7
33
Example
Sample Variance
2
(
X
)
X 2 
2
N
SX 
N
(246) 2
3406 
3406  3362
2
18
SX 

 2.44
18
18
34
Example
Sample Standard Deviation
(X )
X 
N
SX 
N
2
2
246 2
3406 
18  2.44  1.56
SX 
18
35
Example
Estimated Population Variance
( X )
X 
N
s X2 
N 1
2
2
(246) 2
3406 
3406  3362
2
18
sX 

 2.59
17
17
36
Example—Estimated Population
Standard Deviation
(X )
X 
N
sX 
N 1
2
2
246 2
3406 
18  2.59  1.61
sX 
17
37