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- 1. Measures of Dispersion Greg C Elvers, Ph.D. 1
- 2. DefinitionMeasures of dispersion are descriptivestatistics that describe how similar a set ofscores are to each other The more similar the scores are to each other, the lower the measure of dispersion will be The less similar the scores are to each other, the higher the measure of dispersion will be In general, the more spread out a distribution is, the larger the measure of dispersion will be 2
- 3. Measures of DispersionWhich of thedistributions of scores 125has the larger 100 75dispersion? 50 25The upper distribution 0 1 2 3 4 5 6 7 8 9 10 125 100 That is, they are less 75 50 similar to each other 25 0 1 2 3 4 5 6 7 8 9 10 3
- 4. Measures of DispersionThere are three main measures ofdispersion: The range The semi-interquartile range (SIR) Variance / standard deviation 4
- 5. The RangeThe range is defined as the differencebetween the largest score in the set of dataand the smallest score in the set of data, X L -XSWhat is the range of the following data:4 8 1 6 6 2 9 3 6 9The largest score (XL) is 9; the smallestscore (XS) is 1; the range is XL - XS = 9 - 1 = 58
- 6. When To Use the RangeThe range is used when you have ordinal data or you are presenting your results to people with little or no knowledge of statisticsThe range is rarely used in scientific workas it is fairly insensitive It depends on only two scores in the set of data, XL and XS Two very different sets of data can have the same range: 1 1 1 1 9 vs 1 3 5 7 9 6
- 7. The Semi-Interquartile RangeThe semi-interquartile range (or SIR) isdefined as the difference of the first andthird quartiles divided by two The first quartile is the 25th percentile The third quartile is the 75th percentileSIR = (Q3 - Q1) / 2 7
- 8. SIR ExampleWhat is the SIR for the 2data to the right? 4 ← 5 = 25th %tile25 % of the scores are 6below 5 8 5 is the first quartile 1025 % of the scores are 12above 25 14 25 is the third quartile 20SIR = (Q3 - Q1) / 2 = (25 - ← 25 = 75th %tile 305) / 2 = 10 60 8
- 9. When To Use the SIRThe SIR is often used with skewed data as itis insensitive to the extreme scores 9
- 10. VarianceVariance is defined as the average of thesquare deviations: ∑ ( X − µ) 2 σ2 = N 10
- 11. What Does the Variance Formula Mean? First, it says to subtract the mean from each of the scores This difference is called a deviate or a deviation score The deviate tells us how far a given score is from the typical, or average, score Thus, the deviate is a measure of dispersion for a given score 11
- 12. What Does the Variance Formula Mean? Why can’t we simply take the average of the deviates? That is, why isn’t variance defined as: σ 2 ≠ ∑ ( X − µ) N This is not the formula for variance! 12
- 13. What Does the Variance Formula Mean? One of the definitions of the mean was that it always made the sum of the scores minus the mean equal to 0 Thus, the average of the deviates must be 0 since the sum of the deviates must equal 0 To avoid this problem, statisticians square the deviate score prior to averaging them Squaring the deviate score makes all the squared scores positive 13
- 14. What Does the Variance Formula Mean? Variance is the mean of the squared deviation scores The larger the variance is, the more the scores deviate, on average, away from the mean The smaller the variance is, the less the scores deviate, on average, from the mean 14
- 15. Standard DeviationWhen the deviate scores are squared in variance,their unit of measure is squared as well E.g. If people’s weights are measured in pounds, then the variance of the weights would be expressed in pounds2 (or squared pounds)Since squared units of measure are oftenawkward to deal with, the square root of varianceis often used instead The standard deviation is the square root of variance 15
- 16. Standard DeviationStandard deviation = √varianceVariance = standard deviation2 16
- 17. Computational FormulaWhen calculating variance, it is often easier to usea computational formula which is algebraicallyequivalent to the definitional formula: ( ∑ X) 2 ∑X ∑( X −µ) 2 − 2 Nσ2 = = N Nσ2 is the population variance, X is a score, µ is thepopulation mean, and N is the number of scores 17
- 18. Computational Formula Example X X2 X-µ (X-µ2 ) 9 81 2 4 8 64 1 1 6 36 -1 1 5 25 -2 4 8 64 1 1 6 36 -1 1 Σ 42 = Σ 306 = Σ 0 = Σ 12 = 18
- 19. Computational Formula Example ( ∑ X) 2 ∑X ∑( X −µ) 2 2 − N σ 2 σ = 2 = N N 2 12 306 − 42 = = 6 6 6 =2 306 − 294 = 6 12 = 6 =2 19
- 20. Variance of a SampleBecause the sample mean is not a perfect estimateof the population mean, the formula for thevariance of a sample is slightly different from the ( )formula for the variance of a population: 2 2 ∑ X −X s = N −1s2 is the sample variance, X is a score, X is thesample mean, and N is the number of scores 20
- 21. Measure of Skew Skew is a measure of symmetry in the distribution of scores Normal (skew = 0)Positive Negative SkewSkew 21
- 22. Measure of SkewThe following formula can be used todetermine skew: ( ∑ X− X ) 3 3 N s = ( ∑ X− X ) 2 N 22
- 23. Measure of SkewIf s3 < 0, then the distribution has a negativeskewIf s3 > 0 then the distribution has a positiveskewIf s3 = 0 then the distribution is symmetricalThe more different s3 is from 0, the greaterthe skew in the distribution 23
- 24. Kurtosis (Not Related to Halitosis) Kurtosis measures whether the scores are spread out more or less than they would be in a normal (Gaussian) distribution Mesokurtic (s4 = 3)Leptokurtic (s4 > Platykurtic (s43) < 3) 24
- 25. KurtosisWhen the distribution is normallydistributed, its kurtosis equals 3 and it issaid to be mesokurticWhen the distribution is less spread out thannormal, its kurtosis is greater than 3 and it issaid to be leptokurticWhen the distribution is more spread outthan normal, its kurtosis is less than 3 and itis said to be platykurtic 25
- 26. Measure of KurtosisThe measure of kurtosis is given by: 4 X−X ∑ ∑ (X − X) 2 s4 = N N 26
- 27. s2, s3, & s4Collectively, the variance (s2), skew (s3), andkurtosis (s4) describe the shape of thedistribution 27

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