I randomly generated 10 datasets (samples, labeled X1 to X10) from the same population distribution of X, a random variable that is bimodal and non-symmetric. The distribution is similar to the one I showed at the end of class on August 30th. The question was whether the mean would be smaller, equal to, or larger than the median. I thought probably smaller but wasn’t completely sure. A student after class gave a good argument for why the median would be larger. So that you can see that the mean is smaller than the median for this kind of frequency distribution, I attach the results of the ten randomly generated datasets. This is not a rigorous proof but is an empirical demonstration of the event. Enjoy. X1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 13.23799.5% 13.23397.5% 11.93290.0% 11.04775.0% quartile 10.44950.0% median 9.53525.0% quartile 6.85910.0% 4.8262.5% 3.5240.5% 2.8410.0% minimum 2.839Moments Mean 8.713912Std Dev 2.3863579Std Err Mean 0.168741upper 95% Mean 9.0466619lower 95% Mean 8.3811621N 200X2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 12.54599.5% 12.54497.5% 12.03290.0% 11.17575.0% quartile 10.33650.0% median 9.47025.0% quartile 7.22510.0% 5.0142.5% 4.0120.5% 3.5940.0% minimum 3.593Moments Mean 8.7874722Std Dev 2.2784653Std Err Mean 0.1611118upper 95% Mean 9.1051777lower 95% Mean 8.4697667N 200X3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 12.11099.5% 12.11097.5% 11.89390.0% 11.07975.0% quartile 10.41550.0% median 9.49125.0% quartile 7.57510.0% 4.2232.5% 3.2730.5% 1.6010.0% minimum 1.601Moments Mean 8.6131393Std Dev 2.5671079Std Err Mean 0.1815219upper 95% Mean 8.9710927lower 95% Mean 8.2551859N 200X4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 12.52299.5% 12.52297.5% 12.02790.0% 11.21375.0% quartile 10.28950.0% median 9.37725.0% quartile 7.20810.0% 4.5102.5% 3.8350.5% 2.9750.0% minimum 2.975Moments Mean 8.6864321Std Dev 2.3780018Std Err Mean 0.1681501upper 95% Mean 9.0180168lower 95% Mean 8.3548473N 200X5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 12.73599.5% 12.73497.5% 12.14890.0% 11.01975.0% quartile 10.48350.0% median 9.64525.0% quartile 7.26110.0% 4.8942.5% 3.4220.5% 2.7930.0% minimum 2.792Moments Mean 8.7905661Std Dev 2.3719027Std Err Mean 0.1677188upper 95% Mean 9.1213004lower 95% Mean 8.4598318N 200X6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 12.25699.5% 12.25697.5% 11.92390.0% 11.06875.0% quartile 10.44550.0% median 9.54225.0% quartile 7.49410.0% 4.9042.5% 3.7800.5% 2.7820.0% minimum 2.781Moments Mean 8.812851Std Dev 2.3112338Std Err Mean 0.1634289upper 95% Mean 9.1351257lower 95% Mean 8.4905763N 200X7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 12.66799.5% 12.66797.5% 12.04490.0% 11.35675.0% quartile 10.64250.0% median 9.44625.0% quartile 7.42510.0% 4.7112.5% 3.8510.5% 2.8740.0% minimum 2.874Moments Mean 8.829134Std Dev 2.4210206Std Err Mean 0.171192upper 95% Mean 9.1667172lower 95% Mean 8.4915508N 200X8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 12.16199.5% 12.16197.5% 11.90190.0% 11.12475.0% quartile 10.63250.0% median 9.67625.0% quartile 7.29610.0% 4.9782.5% 3.8790.5% 3.2270.0% minimum 3.226Moments Mean 8.864749Std Dev 2.3302453Std Err Mean 0.1647732upper 95% Mean 9.1896746lower 95% Mean 8.5398233N 200X9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 12.77299.5% 12.76997.5% 12.00390.0% 11.38575.0% quartile 10.49350.0% median 9.52525.0% quartile 7.22510.0% 4.6582.5% 2.8260.5% 2.0250.0% minimum 2.023Moments Mean 8.7636786Std Dev 2.4962669Std Err Mean 0.1765127upper 95% Mean 9.111754lower 95% Mean 8.4156032N 200X10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantiles 100.0% maximum 13.30299.5% 13.29897.5% 12.40490.0% 11.29475.0% quartile 10.40150.0% median 9.50725.0% quartile 7.12310.0% 4.9162.5% 3.8580.5% 3.2580.0% minimum 3.258Moments Mean 8.8010752Std Dev 2.3764157Std Err Mean 0.168038upper 95% Mean 9.1324387lower 95% Mean 8.4697116N
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