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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