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 ...
View
Full Document
 '07
 Walsh
 Standard Deviation, Harshad number, scale units

Click to edit the document details