### Five marbles

This term will mark the first term that statistics drops section 9.12 involving confidence intervals that are double the standard error. This was intended as a bridge back to section 2.4 and the concept of an unusual z-score being out beyond plus or minus two standard deviations. Then I would follow the next day with replacing the "two" with the t-critical value from the t-distribution for a 95% confidence interval. The complication has been that no matter how clearly I explain that the "two SE" confidence interval is only a temporary construct, students continue to construct 95% confidence intervals at plus and minus 2 SE for the rest of the term. And this works well enough when the sample size n exceeds thirty, but in the small data sets we often work with, n is often less than 30.

Five marbles is an exercise to show that the distribution of samples means is narrower than the distribution of the data. The exercise is a hand waving physical example. Each student is given five marbles. I then make certain that the students are aware of the average number of marbles per student and the total number of marbles present in the class, which is just five times the number of students. The students then get to keep or trade their marbles. Up to them. Keep all, keep some, or give away all. Their choice. I encourage the students to move around the room.

 Before Data After Data Sample means 5 5 5.8 5 8 5.2 5 6 4.8 5 6 4.2 5 4 5 7 5 5 5 5 5 5 5 4 5 5 5 5 5 5 5 5 5 4 5 3 5 4 5 5 5 4 5 5

In the nine o'clock section twenty students participated, data is tracked in a Google Sheets spreadsheet.  After trading marbles I have the students form groups of five students each and each group works out the average number of marbles per student in their group.

 Name Number of marbles before trading Number after trading Kwodjo 5 Abenaa 0 Kweku 7 Yaa 4 Kofi 3 Group average number of marbles after trading 3.8

Note that the groups will not necessarily obtain an average of five. At this point I ask the class what the average number of marbles per student in the class is. This stumps most classes. Very few students still have five marbles, and they are looking at group average that is not five. Then I ask them to shift their perspective: did any marbles enter or leave the room? Do we have the same number of students? 100 marbles. 20 students. The average, the population mean, must still be five!

In the earlier table from the nine o'clock section no group has obtained the population mean of five. And this is the beginning of confidence interval statistics: each sample mean is a point estimate of the population mean, but none of them are correct (in this instance, although the same was true at 8:00). This leads rather naturally to considering a range, an interval, in which the population mean might reasonably be expected to be found.

I also note that the sample means, of which we had four at 9:00, have a narrower maximum - minimum range than the original data: the sample means distribute more narrowly than the data did. This leads to the following chart.

A relative frequency histogram done as a smoothed line chart of the nine o'clock class data and means demonstrates nicely the narrower and taller distribution of the sample means. With the standard deviation of the means being the distance from the centerline to the inflection point, and that being arguably smaller than the standard deviation of the data, one can at least suggest how the standard error of the mean winds up less than the standard deviation for the data. At this level a rigorous derivation of the standard would not enhance student comprehension.

 Before Data After Data Sample means Sum: 100 100 20 Sample size n: 20 20 4 Average: 5 5 5 Standard deviation: 0 1.12390 0.6733 Standard error of the mean: 0.25131

One of the difficulties with this example is that the standard deviation of the sample means is not equal to the standard error calculated from the data. Bear in mind too that the sample means were from samples without replacement. For every mean that is high, there must be countervailing means that are low. The population mean remains five.

Board notes from class at the end of the 9:00 section. Of some interest is that a multiterm examination of the data currently does not provide as much additional insight.

 Data Sample means Sum: 700 n 140 31 mean 5 5.00 sx 4.76 1.80 SE 0.40

For a 700 distributed marbles to 140 students and 31 sample means (groups are not always of five students), the standard error of the mean calculated from the post-trade data and the standard deviation of the mean are even more separated. I cannot prove that these ought to converge. And the results do show that the standard deviation of the mean is smaller than the standard deviation of the data.