anthony j greene1 dispersion outline what is dispersion? i ordinal variables 1.range 2.interquartile...
TRANSCRIPT
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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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%
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
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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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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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
• 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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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 M
6
4
24
1
2
n
Mxs
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Computational Columns Using Conceptual Formula
MM
85.101-4
353s
1
2
n
Mxs
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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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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
2. Median
3. Mean
Dispersion
1. --
2. Interquartile range or Semi-interquartile range
3. Variance orStandard deviation*
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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 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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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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Significant Differences?
μ1= 40 μ2=60
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Significant Differences?
μ1= 40 μ2=60