Deltoid

Crime rates go up and crime rates go down. Before seizing on some possibly coincidental factor such as gun training or gun control as the cause of the change, we need to establish if the change was unusual, i.e. statistically significant. The only attempt I have seen to establish this is in Kleck and Bordua's paper which claims that the change was significant since it exceeded two standard deviations. This is wrong. A rate two standard deviations from the mean would be significant, but changes exceeding two standard deviations occur 15% of the time for normal variates, nowhere near the 5% generally accepted as statistically significant.

Richard Earl Benedict said:

I don't understand you here. Are you saying that Kleck's claim of two standard deviations is incorrect, or that two standard deviations is not significant? In normally distributed data, according to the empirical rule, 95% of the values fall within two standard deviations of the mean. Kleck actually only needed a z > 1.96 to achieve this, so if his claim of a z > 2 is correct, he has his 95% confidence level. A change of two standard deviations can't be both significant and insignificant at the same time, as you claim above.

Yes, my explanation was pretty opaque. I'll try again:

A crime rate two standard deviations from the mean would be statistically significant. However, The 67 rape rate was only 0.9 standard deviations less than the 58-66 mean, so this rate is not statistically significant. However the 66 rate was 1.7 standard deviations above the mean, so the change from 66 to 67 was 2.6 standard deviations (of the 58-66 rate). This is NOT significant because the standard deviation of the changes in the rates does not equal the standard deviation of the rates. For this data, the standard deviation of the rates from 58-66 is 12, and the standard deviation of the changes from one year to the next in the years 58-66 is 16. The change from 66 to 67 does NOT exceed 2 standard deviations (of the 58-66 changes in the rate).

If this is still unclear: consider a sequence of independent normal variates with mean 0 and standard deviation 1. 5% of the time the values will be greater than 2 or less than -2. Now construct a new sequence by computing the changes from one value to the next. This new sequence is also normally distributed with mean 0, but has standard deviation sqrt(2). 15% of the time the values will be greater than 2 or less than -2. 5% of the time the values will be values will be greater than 2.8 or less than -2.8.