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Probability and Statistics Grinshpan The distribution of the sample mean: an example Suppose that our data consists of numbers j = 1, . . . , 100000. sin(j), 2 10000 4000 0 Frequency The mean is μ ≈ 0.00 and the variance is σ ≈ 0.50. Here is a histogram: −1.0 −0.5 0.0 0.5 1.0 sin Let us generate 1000 samples of size n = 100 (without replacement) from this data, x1 , x2 , . . . , x100 , 200 100 0 Frequency and make a histogram for the observed values of the sample mean x1 + . . . + x100 . x= 100 −0.2 0.0 0.1 0.2 sample_mean The average value of x-values is approximately 0.0024, which is close to μ, as it should be. The variance of x-values is approximately 0.0052, which is close to σ 2 /100, as it should be. Does the histogram appear normal to you? It is surely more normal than the underlying distribution.