251descr2 2/10/06 (Open this document in 'Outline' view!)G. Measures of Dispersion and Asymmetry.1. Range2. The Variance and Standard Deviation of Ungrouped Data.a. The Population Variance - Definitional and Computational Formulas.b. The Sample Variance.251descr2 2/10/06c. The Coefficient of Variation.d. Chebyshef’s Inequality and the Empirical Rule3. The Variance and Standard Deviation of Grouped Data.4. Skewness and Kurtosis.251descr2 2/10/065. Reviewa. Grouped Data. See 251dscr_Db. Ungrouped Data. See 251dscr_DAppendix: Explanation of Sample Formulas (Not for student consumption until you know about expected value.) See 251dscr_B .Appendix: Explanation of Computational Formulas (The part about the variance is fairly easy, the rest is more difficult) See 251dscr_C .251descr2 2/10/06Appendix: Explanation of Chebyshef’s InequalityMeasures of InequalityCorrection?:251descr2 2/10/06 (Open this document in 'Outline' view!)G. Measures of Dispersion and Asymmetry.1. RangenumberlowestnumberhighestRange or midpointlowestmidpointhighest .Interquartile Range: 13 QQIQR . See 251descr2ex2 for example. 2. The Variance and Standard Deviation of Ungrouped Data.a. The Population Variance - Definitional and Computational Formulas. The definition of the population variance is ‘the average squared deviation of measurements from the mean.’ The definitional formula just realizes this definition. Definitional Nx22Computational 222Nx Standard Deviation =varianceb. The Sample Variance.Definitional 122nxxs Computational 1222nxnxs The computational formula is one of the most important formulas you will learn. Note that 2xis not the same as .2x For example, if x is 5,3,2, 3825945322222x, not 1001053222.Example: Use 5,3,2x Computational Method Definitional Methodx2xx xx 2xx 2 4 2 -1.33333 1.77778 3 9 3 -0.33333 0.11111 5 25 5 1.66667 2.7777810 38 10 0.00001 4.66667From this we find 33333.3310,38,102nxxxx and 66667.42xx Note that xx should be zero, but is not because of rounding. Now,if we use the computational method, we can use 3333.226667.41333333.333812222nxnxs (Some texts prefer 33333.2266666667.413103138112222nxnxs which gives us a littlemore accuracy for a little more work.) If we use the definitional method 33333.2266667.4122nxxs, but note that we had to do three subtractions instead of 1.251descr2 2/10/06 c. The Coefficient of Variation.meandeviationstdC.d. Chebyshef’s Inequality and the Empirical RuleChebyshef Inequality: 21kkxP or 211kkxkP . A z-scorexz is the same as .k(See explanation below)Empirical rule: (For Symmetrical Unimodal distributions only) 68% within one standard distribution of the mean, 95% within two and almost all (99.7%) within three.3. The Variance and Standard Deviation of Grouped Data.For grouped data generally substitute f for . 4. Skewness and Kurtosis.Define Population Skewness, the 3rd k-statistic, coefficients of Skewness; Population Kurtosis, the 4th k-statistic, the Coefficient of Excess; Leptokurtic, Platykurtic and Mesokurtic distributions. The usual measurement of skewness is often called the third moment about the mean .(The population variance is the second). The formula for population skewness is: Nx33.The corresponding sample statistic is the third k-statistic, 3321xxnnnk. The corresponding computational formulas are 3233231NxxN and 32332321xnxxxnnnk. To make grouped data formulas, put an f to the right of the sign. Positive values of these formulas imply skewness to the right, negative values to the left. Note that multiplying all the values of x by two would multiply the values of these coefficients by eight, but would not change the shape of the distribution. If we want to compare shapes, we need measurements that will not change if we multiply all values by a constant. Such a measure would be called the coefficient of relative skewness, with the formulas331331 and skg . Note that for the Normal distribution 01. Other measures of skewness are Pearson's measures of skewness, deviationstdemeanSK.mod1 and dev iationstdmedianmeanSK.32. These are roughly equivalent, since, for a moderately skewed distribution, medianmeanemean 3mod. It seems that 313 SK and that values between. 1 and -1 are considered to indicate moderate skewness.251descr2 2/10/06 Example:Profit Rate fx(midpoint) fx 2fx 3fx 9-10.99 3 10 30 300 3000 11-12.99 3 12 36 432 5184 13-14.99 5 14 70 980 13720 15-16.99 3 16 48 768 12288 17-18.99 1 18 18 324 5832 Total 15 202 2804 40024 So 15nf, 202fx, 28042fx, 400243fx, so that467.1315202nfxxand 981.514733.82115467.1315280412222nxnfxs, which means s 5 981 2 446. .. Csx 2 44613 4670182.... To measure skewness, use one of the following three results. 33233467.131522804467.133400241314152321xnfxxfxnnnk )13)(14(249.815 = 0.680, or Relative Skewness 046.446.2680.03331skg or Pearson's Measure of Skewness .2179.0446.214467.13.mod1 deviationstdemeanSK Note that, in this case, Pearson's Measure 1 and Relative Skewness contradict each other as to the direction of skewness.The measures of kurtosis are, for populations, 42234443641xxxNNx and, for samples, 243424131321nsnnxxnnnnnk. 4k can be considered an estimate of443. To get a measurement of shape use the Coefficient of Excess 442442or 3skg .
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