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Variability 1 Definition Variability provides a measure of the degree to which scores in a distribution are spread out or clustered together The more similar the scores are to each other, the lower the measure of dispersion will be The less similar the scores are to each other, the higher the measure of dispersion will be In general, the more spread out a distribution is, the larger the measure of dispersion will be 2 Measures of Dispersion Which of the distributions of scores has the larger dispersion? The upper distribution has more dispersion because the scores are more spread out That is, they are less similar to each other 3 1 Measures of Dispersion There are three main measures of dispersion: The range The interquartile range Variance / standard deviation 4 The Range The range is defined as the difference between the largest score in the set of data and the smallest score in the set of data, XMax- XMin What is the range of the following data: 4 8 1 6 6 2 9 3 6 9 The largest score (XMax) is 9; the smallest score (XMin) is 1; the range is XMax - XMin = 9 - 1 = 8 5 When To Use the Range The range is used when you have ordinal data or you are presenting your results to people with little or no knowledge of statistics The range is rarely used in scientific work as it is fairly insensitive It depends on only two scores in the set of data, XMax and XMin Two very different sets of data can have the same range: 1 1 1 1 9 vs 1 3 5 7 9 6 2 The Interquartile Range The interquartile range is defined as the difference of the first and third quartiles The first quartile is the 25th percentile The third quartile is the 75th percentile IR = Q3 - Q1 The semi-interquartile range is one half of the interquartile range 7 IR Example What is the IR for the data to the right? Lower half of the data: N=5; index of middle score = (N+1)/2=3 6 is the first quartile Upper half of the data: N=5; index of middle score = (N+1)/2=3 20 is the third quartile IR = (Q3 - Q1) = (20 - 6) = 14 8 IR Example n = 10 p = 25 i = (n X p) / 100 + 0.5 = 3 k = integer part of i = 3 f=i–k=0 Q1 = (1 – f) X Obsk + f X Obsk + 1 = (1 – 0) X 6 + 0 X 8 = 6 9 3 IR Example n = 10 p = 75 i = (n X p) / 100 + 0.5 = 8 k = integer part of i = 8 f=i–k=0 Q3 = (1 – f) X Obsk + f X Obsk + 1 = (1 – 0) X 20 + 0 X 30 = 20 IR = Q3 – Q1 = 20 – 6 = 14 10 When To Use the IR The IR is often used with skewed data as it is insensitive to the extreme scores 11 Variance of a Population Variance is defined as the average of the square deviations: 12 4 What Does the Variance Formula Mean? First, it says to subtract the mean from each of the scores This difference is called a deviate or a deviation score The deviate tells us how far a given score is from the typical, or average, score Thus, the deviate is a measure of dispersion for a given score 13 What Does the Variance Formula Mean? Why can’t we simply take the average of the deviates? That is, why isn’t variance defined as: This is not the formula for variance! 14 What Does the Variance Formula Mean? One of the definitions of the mean was that it always made the sum of the scores minus the mean equal to 0 Thus, the average of the deviates must be 0 since the sum of the deviates must equal 0 To avoid this problem, statisticians square the deviate score prior to averaging them Squaring the deviate score makes all the squared scores positive 15 5 What Does the Variance Formula Mean? Variance is the mean of the squared deviation scores The larger the variance is, the more the scores deviate, on average, away from the mean The smaller the variance is, the less the scores deviate, on average, from the mean 16 Standard Deviation When the deviate scores are squared in variance, their unit of measure is squared as well E.g. If people’s weights are measured in pounds, then the variance of the weights would be expressed in pounds2 (or squared pounds) Since squared units of measure are often awkward to deal with, the square root of variance is often used instead The standard deviation is the square root of variance 17 Standard Deviation Standard deviation = √variance Variance = standard deviation2 18 6 Computational Formula When calculating variance, it is often easier to use a computational formula which is algebraically equivalent to the definitional formula: σ2 is the population variance, X is a score, µ is the population mean, and N is the number of scores 19 Computational Formula Example 20 Computational Formula Example 21 7 Variance of a Sample Samples are usually less variable than the population Because the extreme values of the population are rare, they are less likely to appear in the sample If we used the formula for the variance of a population with a sample, the sample variance would systematically underestimate the population variance The estimate would be biased 22 Variance of a Sample Thus, to avoid being biased, we need to increase the size of the estimate by a little. That is we need to divide the sum of the squares by a slightly smaller number: s2 is the sample variance, X is a score, X is the sample mean, and N is the number of scores 23 Variance of a Sample Thus, to avoid being biased, we need to increase the size of the estimate by a little. That is we need to divide the sum of the squares by a slightly smaller number: s2 is the sample variance, X is a score, is the sample mean, and n is the number of scores 24 8 Sum of Squares (SS) & Degrees of Freedom (df) The numerator of the formula, Σ(X - )² , is called the sum of squares or SS The denominator of the formula, n – 1, is called the degrees of freedom or df The degrees of freedom are the number of scores that are free to take on any value after some restriction (such as specifying the value of the mean) has been placed on a set of data 25 Transformations of Scale Adding or subtracting a constant from all scores does not change the standard deviation Multiplying or dividing all scores by a constant causes the standard deviation to be multiplied or divided by the same constant 26 9