Variability (chapter 4)
What is variability?
 Conceptually it is a quantitative measure of how much scores in
a distribution are spread out (high variability) or clustered
together (low variability)
Three important points:
 Variability provides an indication of how accurately the mean
describes the distribution
 Variability provides an indication of how well any individual
score represents the entire distribution
 When comparing different distributions, the variability of the
scores and the extent to which the distributions do or do not
overlap determines whether the distributions are reliably
different (inferential statistics – is condition A different from B)
Most common measures of variability:




Range
Interquartile and semi-interquartile ranges
Standard deviation and variance
Standard error of the mean (discussed later in the course)
1
Range:
 Distance between upper limit of a data set (x max) and the
lower limit (x min)
For discrete variables:
2,7,9,1,5
For continuous variables:
2,7,9,1,5
range = 9-1= 8
lower real limit = 0.5, upper real limit =9.5
range = 9.5 – 0.5 = 9
 Advantage: easy, simple way to describe the variability in a set
of data
 Disadvantage:
1) completely determined by 2 single values
in the data set (could be extreme scores)
2) neglects the rest of the data
Interquartile range:
 Distributions can be split up into 4 parts called quartiles
 Median is at the second quartile (Q2)
 Interquartile range is the difference between the 1st and 3rd
quartiles:
Interquartile range = Q3 – Q1
2
 Interquartile range is often expressed as semi-interquartile
range
(Q 3  Q1)
Semi-interquartile range =
2
For the following data set:
3, 4, 5, 7, 9, 10, 11, 13
3, 4 | 5, 7 | 9, 10 | 11, 13
Q1 = 4.5, Q2 = 8.0, Q3 = 10.5
 Advantage: less likely than range to be influenced by
extreme scores (based on middle 50% of
distribution)
 Disadvantage: does not take into account the actual distances
between all scores, therefore, it provides an
incomplete picture of the variability
3
Standard deviation (for a population)
 Most commonly used and most important measure of variability
 Uses the mean as a reference point and measures the variability
of every score from the mean
 Once the variability of every score from the mean is obtained,
the average of this value (divided by N) is what we term the
standard deviation (on average, how much do scores in the data
set differ from the mean)
Here’s where we start:
X
8
1
3
0
µ 
X- 
5
-2
0
-3
  12

3
N
4
(    )  0
 (    )  0 scores reflect distances of scores from the mean
but in this form cancel each other out
4
 To get around this problem we create an additional column
showing squared distances of each score from the mean (show
2
on board (    )
X
X- 
(   )2
8
1
3
0
5
-2
0
-3
25
4
0
9
Sums of squares
 sum of squared differences between individual scores and the
2
mean (    ) is called sums of squares or SS
 sums of squares can be expressed in 2 ways:
SS     
2
1) Definitional formula:
SS    2 
   2
2) Computational formula:
N
(memorize – particularly useful when data has fractions or
decimals)
Note: these formulas are equivalent


5
 The next step in calculating population standard deviation is to
work out the variance
 Population variance or  is the mean squared deviation of
the scores in the data set from the mean (on average what’s the
squared deviation of scores from the mean)
2
SS
 
N
2
 the value for variance will always be inflated (out of scale with
the original data set because of the squaring procedure)
 to return to the original scale of measurement we use a square
root procedure to give us the standard deviation 
standard deviation =
VARIANCE
 for a population:

SS
N
Note: you should always estimate sigma before you start to
calculate it (it will be somewhere between the closest and furthest
scores from the mean)
6
 see the text for how to calculate SS with a calculator (you can
also do this in separate steps by hand and with a calculator)
Standard deviation (for a sample):
We have similar definitional and computational formulas for sums
of squared with sample data as compared to population data:
 definitional formula:
SS   x  x 
2
 computational formula:
2



x
SS   x 2 
n
SS
S 
n 1
2
 sample variance:

n  1 also referred to as degrees of freedom is to correct for
bias in sample variation (explain)
 standard deviation for a sample:

df  n  1
S
S
SS
n 1
SS
df
7
Transformations:
 adding a constant to each score will not influence the standard
deviation
 multiplying each score by a constant will multiply the standard
deviation by the same constant
Example of reporting in the literature (p 101 Table 4.2)
Demonstration 4.1 (p104)
8