Why does

accomplish this? Well, because that is the rate at which the standard deviation of a sum increases. For example, if

are independent from a normal distribution with mean

and variance

then

so that the standard deviation is

. So, we have the standard deviation of a sum increasing at a rate of

- or equivalently the standard deviation of

is decreasing at a rate of

; scaling by

makes it so that our standard deviation (of the test statistic) is going neither to 0 nor to

, but is staying constant.