Dispersion
Outline
What is Dispersion?
I Ordinal Variables
1.Range
2.Interquartile Range
3.Semi-Interquartile Range
II Ratio/Interval Variables
1.Variance
2.Standard Deviation
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Significant Differences?
μ1= 40
μ2=60
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Significant Differences?
μ1= 40
μ2=60
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Dispersion is the Measure of Spread
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Measures of Dispersion
Ordinal
Interval/Ratio
Range
Variance
Interquartile Range
Standard Deviation
Semi-Interquartile Range (as well as range, I.R.
and S.I.R.)
Nominal Variables have no dispersion
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Range
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Range
• The range of a data set is the difference between
its maximum and minimum observations: Range =
Max – Min.
– Use Lower Real Limits: The Min is not merely the
lowest score its any score that could be rounded up to
the lowest score.
– Use Upper Real Limits: Likewise the Max is any score
that could be rounded down to the lowest score.
– For integer values this generally amounts to adding 0.5
to the highest to get the max, and subtracting 0.5 from
the lowest score to get the min.
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Quartiles
• Let n denote the number of observations.
Arrange the data in increasing order.
• The first quartile is at position (n + 1)/4.
• The second quartile is the median, which is at
position (n + 1)/2.
• The third quartile is at position 3(n + 1)/4.
• If a position is not a whole number, linear
interpolation is used to find the fraction
representing the quartile.
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Interquartile Range
• The interquartile range, denoted IQR, is
the difference between the first and third
quartiles; that is,
IQR = Q3 – Q1
• Roughly speaking, the IQR gives the range
of the middle 50% of the observations.
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The Interquartile Range
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Five Number Summary
• The five-number summary of a data set
consists of the minimum, maximum, and
quartiles written in increasing order: Min,
Q1, Q2, Q3, Max.
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Quartiles
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Box & Whiskers Plots
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Box & Whiskers Plots
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Box & Whiskers Plots
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Standard Deviation
68%
95%
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Standard Deviation
68%
95%
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Standard
Deviation
68%
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95%
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Standard Deviation of a Discrete
Random Variable
The population standard deviation of a discrete random variable X
is denoted by  and is defined by
  x 
N

2
Or the computational formula

x
N
2
 2
The variance, V, is the square of the standard deviation
V=2
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Variance is the Average Squared Deviation
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
-1
+1
+2
-6
+4
-15
-17 x 2
+6
+9 x 3
-20
+11
+14 x 2
+15
-22
-23
-27
+16
+18
+20
μ = 33
Average Deviation is Zero
Average Squared Deviation: V = Σ(x-μ)2/N
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Samples and Populations
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Population and Sample Variability
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Sample Standard Deviation
• For a variable x, the standard deviation of the
observations for a sample is called a sample
standard deviation. It is denoted by sx or, when no
confusion will arise, simply by s. We have
 x  M 
s
or
n 1
2
x



x

2
2
n 1
n
• where n is the sample size: n-1 is referred to as the
degrees of freedom
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Deviation from the Sample Mean
M
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Deviation From the Sample Mean
M
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Sample Variance and Standard
Deviation Using Conceptual Formula
M
x  M 
2
s
n 1
24

 6
4
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M
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Computational Columns Using
Conceptual Formula
M
M
 x  M 
s
n 1
2
353
s
 10.85
4 -1
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Computational Columns Using
Computational Formula
2
x 2  x 
s
s
n 1

3582
32, 394
4 1
n
4
32,394  32,041
s
4 1
353
s
 10.85
3
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APA Format For Mean and St.Dev
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Sample Standard Deviation
• Almost all of the observations in any data
set lie within three standard deviations to
either side of the mean
• 95% of the observations lie within two
standard deviations to either side of the
mean
• 68% of the observations lie within one
standard deviation to either side of the mean
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Sample Standard Deviation
68%
95%
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Summary of Descriptives
Central Tendency
1. Mode
Dispersion
1. --
2. Median
2. Interquartile range or
Semi-interquartile range
3. Variance or
Standard deviation*
3. Mean
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Again, The Basic Idea of
Experiments
1. Are there differences between means?
2. Is that difference large enough so that it is
not likely to be due to chance factors?
Answer:
It depends on how far apart the means are
and how much dispersion you have in
your variables
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Effect Size Compared to Random Variation
The
variability
within
samples is
small and it
is easy to
see the
5-point
mean
difference
between the
two
samples.
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Effect Size Compared to Random Variation
The 5point
mean
difference
between
samples is
obscured
by the
large
variability
within
samples.
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Significant Differences?
μ1= 40
μ2=60
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Significant Differences?
μ1= 40
μ2=60
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