Notes

Outline

Statistics 4 Variance Sherril M. Stone, Ph.D. Department of Family Medicine OSU-College of Osteopathic Medicine

Variability of Data Range highest score minus the lowest score Outliers will skew your data Interquartile Range – 25th, 50th, 75th, 90th, etc 25th quartile – 25% of all scores fall below this score 75th quartile – 75% of all scores fall below this score Semi-interquartile – Q3–Q1   (middle 50%) – not affected by outliers                      2 These are CRUDE MEASURES OF VARIABILITY - does not tell the whole story about the raw data. A mean of 5.3 does not tell if most of the scores were similar. So, we have measures of variability, which tell us about the spread of the scores on the scale of measurement. Deviation the distance from the mean + is above mean, - is below the mean Deviation scores - MUST sum (S) to zero

Measures of Central Tendency Always report a measure of central tendency (Mean & SD) and a measure of variability to describe a set of scores Variation (SS) = Sum of Squares Variance (s2  or S2) = SS/N (deviation scores squared/N) Standard deviation (s or SD) = square root of variance Measures of variability provide a quantitative measure of the degree to which scores in a distribution are spread out or clustered together on the scale. If variance = 0 then all scores are the same, NO variability exists If variance is large - then scores were very far apart If variance is small – then scores were very close together

Variability Variation, variance, and standard deviation measures of variability report these measures with the mean   Population              Sample Variation: SS = S(X –m )2                SSx       SSx Variance: SS/N = mean of SS         s2        s2 Standard Deviation: square root of s2       s            s

Formulas Definitional 1. Find  N, SX, and mean 2. Subtract mean (X) from each X for deviation score 3. Square each deviation score (X– X)2 4. Sum the deviation scores S(X – X )2 = SS (aka Variation) 5. Variance (s2)  = SS/N 6. S (SD) = SS/N Computational 1. N, SX, and mean 2. Square each data score 3. Compute SS =    S X2  - (SX)2                                              N 4. Variance (s2) = SS/N 5. S (SD) =  SS/N

Definitional Formula X X - m (X - m)2 1 -1      1 N = 4 0 -2      4 m = 2 6 +4    16 1 -1      1 SX = 8      SX –m = 0       S(X –m)2 = 22

Computational Formula              SX2 – (SX)2 s2 =                    n                     N X X2 1 1    N = 4 0 0    X = 2 6 36 1 1   SX =  8       SX2 = 38 SS =  38 – 82/4    = 38 – 16 = 22

Degrees of Freedom Samples tend to underestimate the population data To correct sampling error we subtract 2 from n All scores are free to vary except 2 (e.g. when you know all but 1 data score then it is determined) s2 = Sample variance =  Sx2 =  SS =   35.00  = 11.67                         N-1    N-1      4 -1 s = Sample standard deviation =  s2  = 11.67 = 3.42

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