Crunch time for number

This article and the accompanying figure have great potential for classroom use at several levels. First, there are several statistical intuitions which can potentially be put to the test in discussion.

From reading the article students might believe 8 is the best number to choose for several reasons: "the Chinese say so", "it hasn't come up in the last 11 weeks so it should come up soon", or "overall it is the most likely from the data".

Second, with a probability simulator (and there is some nice software available to do this) it is possible to select samples without replacement from the 45 numbers many times, tally the results and compare them with the numbers in the figure.

Third, using the data in the table, a line plot (a horizontal axis with a range of 83 to 124 and X's placed for each of the 45 values) can be drawn to picture the distribution of the data set. Does it look unusual? Does it look normal? Calculating the mean and median will show they are very close to the same value, indicating symmetry in the distribution. What about the extreme values? How extreme are they? Calculating the standard deviation will show there is only one value at each end outside of two standard deviations. There are 67% of values within one standard deviation of mean, almost what one would expect of a normal distribution of values (68%).

Fourth, for those with a bit more sophistication it is possible to do a chi-square goodness-of-fit test to see how well the data set fits a uniform distribution (all values the same). This is rather tedious by hand or with a basic calculator but a nice exercise with a spreadsheet. Most text books will give the formula. The value of chi-square for this distribution is 34.237. With 44 degrees of freedom this value is not significant. In fact, 75% of the time we would expect a larger chi-square value from a data set with this uniform distribution. Hence although it may look varied to the casual eye, the data set is typical of a random even distribution of numbers 1 to 45. So there is no help in choosing your numbers.

Where to next?

Student Questions for this article
Newspaper article
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