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Chapter 5 Measures of Variability Golf Analogy: Which golfer is more consistent? Each golfer hits 5 shots. The data are distance from the hole. Mean Variance Beginner Pro -20 1 20 2 1 -2 -15 1 16 3 0.4 1 320.3 3.5 Variability Variability is the spread or dispersion of a set of scores. Leptokurtic distribution has small dispersion. Platykurtic distribution has large dispersion. If the groups have a large amount of variance the difference between the means will have to be greater to find significance. Interquartile Range Difference between the 75th and 25th percentile. Variance is the average squared deviations from the mean. The sum of the deviations is ZERO Calculation of the Variance Standard Deviation The SD is the square root of the variance. The variance is in squared units The SD is in original data units Sigma is the population statistic Degrees of Freedom: The number of things that are free to vary Assume that the mean of four values is 5 Therefore the sum must equal 20 Let 2, 3, and 7 be the first three numbers What must the 4th value be so sum = 20? It must be 8 In this example the first 3 numbers are FREE TO VARY The df for a single data set is (N-1) [ see p 68 ] DF and Sampling Samples rarely contain the extreme values found in the population. A random sample of 100 university men from a population of 15,000 is unlikely to contain a subject with wt 350 or 100 although they exist in the population. The variability of the sample is almost never as large as the population. df is a correction factor so that a statistic is not a biased estimate of the parameter. When N is large and the distribution is close to normal there are 5-6 sds within the range 6 SDs