Measures_of_Variability

Measures_of_Variability - Measures of Variability What is...

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Unformatted text preview: Measures of Variability What is Variability? Descriptive statistics that summarize the amount of the differences between scores Variability is a different dimension than central tendency! How much dispersion of the scores are there? Sample Distributions, N = 9 (A) 41 41 41 41 41 41 41 41 41 ∑ XA = 369 X = 41 (B) 38 39 40 41 41 41 42 43 44 ∑ XB = 369 X = 41 (C) 21 26 33 41 41 41 50 51 65 ∑ Xc = 369 X = 41 Comparing Distribution A, B, & C The means of all three distributions are the same… BUT, the scores in C have greater differences among them than B or A Plot of Distributions X X X X X X x X xxxxxxxx X X X X X X XX X Range Describes the distance in scale units between the largest and smallest scores in a distribution Range = XHighest – XLowest 100% of the scores fall within the number of scales units indicated by the range. In our sample distributions… Range of (A) = 41­41 = 0 Range of (B) = 44­38 = 6 – 100% of the scores fall at the same scale value – 100% of the scores fall within 6 scale units of each other – 100% of the scores fall within 44 scale units of one another Range of (C) = 65­21 = 44 Problems with the Range Insensitive! Based on only 2 scores Subject to bias of outlying or extreme scores 41 X X X 19 X X X X X 63 10 20 30 40 50 60 70 XA1 = 41 Range = 63­19 = 44 XA1 = 41 has same range as C, but it certainly has fewer differences among all of the scores! Interquartile Range (IQR) The range of values for the middle 50% of the scores in a distribution IQR = X75 – X25 More stable, less subject to bias than range, because does not use extreme, outlying scores Still insensitive – uses only 2 scores Interquartile Range 25 50 75 100 10 20 30 40 50 60 70 X25 = 30 X75 =51 IQR = 51­30 = 21 Equal to ½ of the IQR Semi­interquartile Range (SIQR) Indicates the number of scale units around the median than include 25% of the scores. SIQR = IQR/2 = (X75 – X25)/2 IQR 25 50 75 100 10 20 30 40 50 60 70 SIQR Variance Most widely used and useful measure of variability Four important properties – 1. sensitive – every score is used in its computation – 2. logical – value of variance increases as the difference between scores increases – 3. independent of the size of the scores (or the mean of the distribution). – 4. independent of the number of terms or scores in the distribution Variance Is an approximate average (or mean) of the squared deviation scores: S2 = ∑ (X­X)2 N Why don’t we use ∑ (X­X)/N? Because ∑ (X­X) = 0!! Sample Variance (S2) Formula Parts S2 = ∑ (X­X)2 = sum of squared deviation scores N number of scores Distribution B X 38 39 40 41 41 41 42 43 44 X 41 41 41 41 41 41 41 41 41 (X­X) (X­X)2 ­3 9 ­2 4 ­1 1 0 0 0 0 0 0 +1 1 +2 4 S2 =∑(X­X)2 =28 +3 9 S2 =28/9 = 3.11 Estimating the variance of a population from sample data S2 = ∑ (X­X)2 N­1 Denominator decreases by one, because the mean of the sample is known Called a degree of freedom or df (how many scores are “free to vary”) We lost one df because the mean is known Sample Variance S2 = ∑ (X­X)2 N Estimate of Population Variance S2 = ∑ (X­X)2 N­1 Sample Standard Deviation Esimate of Population – Variance S2 = ∑ (X­X)2 N ­ 1 – Standard Deviation S = ∑ (X­X)2 N­1 – Variance S2 = ∑ (X­X)2 N – Standard Deviation S = ∑ (X­X)2 N Values from Examples Variance Standard Deviation Dist A S2 = 0 S = 0 = 0 Dist B S2 = 3.11 S = 3.11 = 1.76 Dist C S2 = 160.66 S = 160.66 = 12.68 Variance Variance tells us the average squared deviations from the mean Standard deviation tells us the approximate average deviation from the mean ...
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