# Phone bill cricket batting averages

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Cricket batting averages 0 10 20 30 40 50 60 30 40 50 60 70 80 90 100 Average Frequency

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8 Measures of variability l Keller Ex 3.2 compares rates of returns on 2 investments l Mean returns (%): A = 10.95, B = 12.76 l Should we chose to invest in B? l If not why not?
9 Measures of variability… l Range is a simple measure of variability l Range = maximum – minimum l For investment example ranges are: A: 63.00 – (– 21.95) = 84.95 B: 68.00 – (– 38.47) = 106.47 l Range is simple but potentially misleading 1 1 1 50 50  range = 49 1 10 20 40 50  range = 49 l Is variability the same here?

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10 Measures of variability… l Why not measure spread around location? l Variance is most common measure of variability l Measures average squared distance from the mean l Division by n 1 for sample variance relates to properties of estimators l Look ahead to Keller Ch 10 for justification ( 29 ( 29 1 variance Sample variance Population 1 2 2 1 2 2 n- x x s N μ x n i i N i i = = - = - = σ
11 Measures of variability… l Recall simple n = 5 sample with a = 2 (Slide 5)  0 0 1 3 6 sample mean = ? sample variance = ? l Variance is in squared units l Standard deviation is spread measured in original units l σ = l s = l Other measures of variability? 2 s 2 σ

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12 Measures of variability… l Again should we invest in A or B? l Mean returns: A: 10.95, B: 12.76 l Variance of returns A: 479.35, B = 786.62 l Standard deviation of returns A: s 2A = 479.35, s A = (479.35)1/2 = 21.89 B: s 2B = 786.62, s B = (786.62)1/2 = 28.05 l Calculations in investment data when n = 50? l Efficient methods exist for variance or use EXCEL
13 Measures of variability… l Using mean & standard deviation in combination we can standardize data l Create transformed variable with zero mean, unit variance & hence free of units of measurement l Calculate Z scores l (observation – mean) divided by standard deviation l For investment A maximum return is 63% which has a Z score of (63–10.95)/21.89 = 2.38 or 63% is 2.38 standard deviation units above mean return l Helpful because mean & standard deviation depend on units of measurement eg proportions or percentages l If returns in proportions (0.63 not 63%) how would Z score change?

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Coefficient of variation l Sometimes measure variation relative to location l Case 1: observations all in millions & standard deviation is 20  relatively little variability l Case 2: Observations all positive but less than 100  s =20 may be a lot of variability l (Sample) coefficient of variation, cv= l Provides measure of relative variability 14 x s /
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