6.137 Median statistic.
When a distribution is symmetric, the mean and median will equal. So, when sampling from a symmetric population, it would seem that we would be indifferent in using either the sample mean or sample median for estimating the population mean. Let’s explore this question by simulation. With software, you need to generate 1000 SRS based on from the standard Normal distribution. The easiest way to proceed is to create five adjacent columns of 1000 rows of random numbers from the standard Normal distribution.
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For each row, find the mean and median of the five random observations. In JMP, define new columns using the formula editor, with the Mean function applied to the five columns and the Quantile function with the first argument as 0.5 and the other arguments being each of the five columns. In Minitab, this all can be done using the Row Statistics option found under Calc.
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Answers will vary. (a) They both should be close to 0. (b) The theoretical standard deviation is 0.4472. The estimated standard deviation should be close to this number. (c) This will be somewhat higher than 0.4472. (d) The standard deviation of the median statistic is larger than the standard deviation of the mean statistic. (e) D is associated with the mean, and B is associated with the median.