Question 6.137

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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  • Excel users: To generate a random number from the standard Normal distribution, enter “” in any cell. Use the convenience of the dragging the lower-right corner of a highlighted cell to copy and paste down the column and then across columns to get five columns of 1000 random numbers.
  • JMP users: With a new data table, right-click on header of Column 1 and choose Column Info. In the drag-down dialog box named Initialize Data, pick Random option. Choose the bullet option of Random Normal, which has the standard Normal as the default setting. Input the value of 1000 into the Number of rows box and then click OK. Repeat to get five columns of random numbers.
  • Minitab users: Do the following pull-down sequence: Calc → Random Data → Normal. The default settings is for the standard Normal distribution. Enter “1000” in the Number of rows of data to generate box and type “c1-c5” in the Store in column(s) box. Click OK to find 1000 random numbers in the five columns.

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.

  1. Find the average of the 1000 samples means and the average of the 1000 sample medians. Are these averages close to the population mean of 0?
  2. Find the standard deviation of the 1000 sample means. What is theoretical standard deviation? Is the estimated standard deviation close to the theoretical standard deviation?
  3. Find the standard deviation of the 1000 sample medians.
  4. Compare the estimated standard deviation of the mean statistic from part (b) with the standard deviation of the median statistic.
  5. Refer to the four bull’s-eyes of Figure 5.14 (page 280). In the estimation of the mean of a symmetric population, which bull’s-eye is associated with the sample mean statistic, and which bull’s-eye is associated with the sample median statistic?

6.137

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.