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.