Learning Objectives
In this chapter you will learn
about the importance of variation
how to measure variation
range
variance
standard deviation
Variation
Variation
–is the heart of statistics
–no variation, no need to do
statistical analysis
–the mean would describe the
distribution
Quest for Variation
Measures of dispersion
– consider the spread between scores
Calculations include
– range
– variance
– standard deviation
The Range
Range
– distance between the highest and
lowest score in a distribution.
Calculation
– Range = H minus L
• H = the highest score in the data set
• L = the lowest score in the data set
Variance and Standard
Deviation
Most commonly used measures of
dispersion
– based upon the distance of scores in a
distribution from the mean
– the mean is used as the central point
– first step is calculation of deviation scores
• how each score stands in relation to the mean
Calculation
Formula: x = (X minus the Mean)
– Table 4.2 reveals how this process
works
– we have the number of prior drug
arrests for five clients who appeared
in Tuesday’s drug court
Table 4.2: Number of Prior Arrests for Drug Offenses
Among Tuesday’s Drug Court Clients
X
f
fX
x ( x – X)
6
1
6
2
5
1
5
1
4
1
4
0
3
1
3
-1
2
1
2
-2
N=5
∑fX = 20
∑x = 0
Calculation
The mean number of prior drug
arrests was four
– next step is to calculate how far each score
(person in this case) stood in relation to this
mean of four.
The deviation score (x) is calculated
by subtracting the mean from each
score in the distribution
Calculation
Next step
– on the first line of Table 4.2 is the score 6.
If we subtract the mean, the deviation
score is 2
– this person had two more prior drug arrests
than the average person appearing before
the drug court on Tuesday
– repeat this process from each score in the
distribution
Check the Math
One characteristics of the mean
is that the sum of the deviations
from it equals zero (x = 0)
The sum of the deviations from
the mean equal zero
Characteristic
Positive deviation scores of
– the values above the mean
– are cancelled out
– by the negative deviation scores of the
values that fall below the mean
We are left with zero
– when we sum up the deviation scores in a
distribution
– except for rounding or calculation errors
Squaring the Distance
In order to remove the negative signs
– square every deviation score and thus
cancel out the negative numbers
– remember that a negative value times a
negative value equals a positive value
– squaring the deviation scores gives us a
total number we can work with – the
variance
xx
The Variance
A new frequency distribution
– based upon how each case stands in
relation to the mean
We can calculate another average
score – the variance
The variance is the mean of the
squared deviations from the mean
Variance
The previous definition is
– a verbal formula
– a description of how the variance is
calculated
The variance is a mean
– it represents the average squared
deviations
– that each score stands in relation to its
mean
Variance
The variance is
– a measure of the spread of scores in a
distribution around its mean
The larger the variance
– the greater the spread of scores around
the mean
The smaller the variance
– the more closely the scores are distributed
around the mean
Standard Deviation
We squared the deviation scores
around the mean in order to
clear the negative numbers
The standard deviation is the
square root of the variance
Standard Deviation
Standard deviation
– is a measure of dispersion of the scores
around the mean
The higher the standard deviation
– the greater the spread in the scores
The lower the standard deviation
– the closer the scores are on average from
the mean of the distribution
Formulae
Population variance
 
2
( x   )
N
Population standard deviation
Sample variance
2
s
2

Sample standard deviation
 
( x  x )
2
2
n 1
s 
s
2
Summary
Explaining variation is the basis for
statistical analysis
We begin with basic measures of
dispersion
– range
– variance
– standard deviation
SPSS
Open the SPSS
data file
Select
ANALYZE
Select
DESCRIPTIVE
STATISTICS
Select
FREQUENCIES
SPSS
Select the
variable from the
column on the
right
Highlight it
Mouse click the
ARROW button
to move it in the
VARIABLES
window
SPSS
Select the
appropriate
measures
of central
tendency
and, shape
and
dispersion
Mouse click
CONTINUE