#### Transcript 04 Dispersion

```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?
It depends on how far apart the means are
and how much dispersion you have in
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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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