anthony j greene1 dispersion outline what is dispersion? i ordinal variables 1.range 2.interquartile...
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![Page 1: Anthony J Greene1 Dispersion Outline What is Dispersion? I Ordinal Variables 1.Range 2.Interquartile Range 3.Semi-Interquartile Range II Ratio/Interval](https://reader035.vdocuments.us/reader035/viewer/2022070415/56649e6c5503460f94b6b098/html5/thumbnails/1.jpg)
Anthony J Greene 1
Dispersion
Outline
What is Dispersion?
I Ordinal Variables1.Range
2.Interquartile Range
3.Semi-Interquartile Range
II Ratio/Interval Variables1.Variance
2.Standard Deviation
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Anthony J Greene 2
Significant Differences?
μ1= 40 μ2=60
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Anthony J Greene 3
Significant Differences?
μ1= 40 μ2=60
![Page 4: Anthony J Greene1 Dispersion Outline What is Dispersion? I Ordinal Variables 1.Range 2.Interquartile Range 3.Semi-Interquartile Range II Ratio/Interval](https://reader035.vdocuments.us/reader035/viewer/2022070415/56649e6c5503460f94b6b098/html5/thumbnails/4.jpg)
Anthony J Greene 4
Dispersion is the Measure of Spread
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Anthony J Greene 5
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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Anthony J Greene 6
Range
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Anthony J Greene 7
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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Anthony J Greene 8
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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Anthony J Greene 9
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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Anthony J Greene 10
The Interquartile Range
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Anthony J Greene 11
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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Anthony J Greene 12
Quartiles
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Anthony J Greene 13
Box & Whiskers Plots
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Anthony J Greene 14
Box & Whiskers Plots
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Anthony J Greene 15
Box & Whiskers Plots
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Anthony J Greene 16
Standard Deviation
68%
95%
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Anthony J Greene 17
Standard Deviation
68%
95%
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Anthony J Greene 18
Standard Deviation
68%
95%
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Anthony J Greene 19
Standard Deviation of a Discrete Random Variable
The population standard deviation of a discrete random variable X is denoted by and is defined by
Or the computational formula
The variance, V, is the square of the standard deviation
V=2
N
x 2
22
Nx
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Anthony J Greene 20
Variance is the Average Squared Deviation
Average Deviation is Zero
Average Squared Deviation: V = Σ(x-μ)2/N
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
-6
-15
-17 x 2
-20
-22
-23
-27
+1+2
+4
+6
+9 x 3
+11+14 x 2
+15
+16
+18+20
μ = 33
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Anthony J Greene 21
Samples and Populations
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Anthony J Greene 22
Population and Sample Variability
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Anthony J Greene 23
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
• where n is the sample size: n-1 is referred to as the degrees of freedom
1or
1
222
n
nxx
n
Mxs
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Anthony J Greene 24
Deviation from the Sample Mean
M
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Anthony J Greene 25
Deviation From the Sample Mean
M
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Anthony J Greene 26
Sample Variance and Standard Deviation Using Conceptual Formula
M M
6
4
24
1
2
n
Mxs
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Anthony J Greene 27
Computational Columns Using Conceptual Formula
MM
85.101-4
353s
1
2
n
Mxs
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Anthony J Greene 28
Computational Columns Using Computational Formula
85.103
353
14
041,32394,32
1
14394,32
22
42358
s
s
s
nn
xxs
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Anthony J Greene 29
APA Format For Mean and St.Dev
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Anthony J Greene 30
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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Anthony J Greene 31
Sample Standard Deviation
68%
95%
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Anthony J Greene 32
Summary of Descriptives
Central Tendency
1. Mode
2. Median
3. Mean
Dispersion
1. --
2. Interquartile range or Semi-interquartile range
3. Variance orStandard deviation*
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Anthony J Greene 33
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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Anthony J Greene 34
Effect Size Compared to Random VariationThe variability within samples is small and it is easy to see the 5-point mean difference between the two samples.
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Anthony J Greene 35
Effect Size Compared to Random Variation
The 5-point mean difference between samples is obscured by the large variability within samples.
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Anthony J Greene 36
Significant Differences?
μ1= 40 μ2=60
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Anthony J Greene 37
Significant Differences?
μ1= 40 μ2=60