Experiments with correlations and standard deviations The marks for a group of four students in two examinations were:

Mathematics: Ann: 60, Bill: 75, Sue: 85, Dan: 90

English: Ann: 50, Bill: 48, Sue: 49, Dan: 46

The standard deviations (stdevp) of the marks are 11.5 for maths and 1.5 for English. Pearsons correlation coefficient between the marks in the two subjects is -0.8. Work out Kendalls correlation coefficient. Is it similar to Pearsons coefficient? Assuming that these marks are reasonably typical of all students doing the two examinations, what comments would you make about the difference between the two standard deviations, and about the correlation? Would it be fair to add the marks in the two examination to give students an overall mark? You should be able to answer the following questions without a computer or calculator:

What would the standard deviation of the maths marks be if the marks were halved (that is, they became 30, 37.5, 42.5, 45)?

What if 30 was deducted from each mark?

What effect would both of these changes have on the correlation? Would it still be -0.8 in both cases?

Find the standard deviations of these marks: 60, 60, 75, 75, 85, 85, 90, 90 (compare with the maths marks above)

And the standard deviation of these: 70, 70, 70, 70.

If you have a computer available you should be able to check these. The final part of this exercise exploits one of the strengths of spreadsheets; the fact that they allow you to experiment: to change the data and see what effect this has on the answer. See if you can find six numbers whose mean is 30 and whose standard deviation is 10. Make the best guess you can, then work out the standard deviation (using the function stdevp), then try and adjust the guess and so on until the mean is 10 and the sd is 30 (both to the nearest whole number). How many steps did you need?