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Epidemiology 9509 sampling distributions (again) Epidemiology 9509 Principle of Biostatistics Chapter 7: Sampling Distributions (continued again) John Koval Department of Epidemiology and Biostatistics University of Western Ontario Epidemiology 9509 sampling distributions (again) Distribution of median for normal distribution 2 variance of X̄ is σn that of median, XM , is for n = 2m + 1 πσ2 4m for large samples, the ratio of the variance of the median to the variance of the sample mean is about π 2 ≈ 1.57 Epidemiology 9509 sampling distributions (again) Plot of distributions of Sample mean and Median Epidemiology 9509 sampling distributions (again) sampling from distribution of Sample Median 1. variance decreases as sample size increases 2. distribution is symmetric 3. distribution becomes ”Normal” as sample size increases 4. variance is larger than that of Sample Mean Question: Because it is easier to compute than the sample mean, should we use the sample median to estimate the centrality of the distribution? Epidemiology 9509 distribution of sample variance instead of calculating the sample mean, calculate the sample variance and look at its distribution sampling distributions (again) Epidemiology 9509 sampling distributions (again) distribution of sample variance (continued) following descriptive statistics Var Nsam Mean Std Dev Minimum Maximum ---------------------------------------------------xbar 10 2.0040000 0.3874715 0.7000000 3.3000000 xbar 30 1.9993000 0.2281213 1.3333333 2.8333333 xbar 100 2.0067300 0.1250946 1.6500000 2.3600000 xbar 1000 1.9997290 0.0399812 1.8890000 2.1500000 ---------------------------------------------------var 10 1.6202222 0.7428275 0.2222222 5.1111111 var 30 1.6055644 0.4191585 0.6160920 3.6965517 var 100 1.6098704 0.2260165 1.0079798 2.4544444 var 1000 1.6003043 0.0703617 1.4183373 1.8162162 ---------------------------------------------------still skewed for nsam = 100 requires several hundred for symmetry (normality) Epidemiology 9509 plots of distribution of sample variance sampling distributions (again) Epidemiology 9509 distribution of sample variance (nsam=30) sampling distributions (again) Epidemiology 9509 distribution of sample variance (nsam=100) sampling distributions (again) Epidemiology 9509 distribution of sample variance (nsam=1000) sampling distributions (again) Epidemiology 9509 sampling distributions (again) sampling from distribution of the sample variance 1. variance decreases as sample size increases 2. distribution is asymmetric 3. becomes more symmetric (Normal) as sample size increases 4. even as nsam = 100, still quit asymmetric Epidemiology 9509 sampling distributions (again) distribution of sample standard deviation as for variance but more symmetric at smaller sample sizes Epidemiology 9509 sampling distributions (again) Conclusion 1. distribution of most statistics become ”Normal” 2. some require very large sample size (100’s) 3. some (estimates of centrality) require smaller sample (30?) size to attain normality