Standard Deviation
Chance variation is conveniently measured by standard deviation, denoted by SD. Roughly, SD measures how far a typical occurrence of a random process will be from the average. For example, in many situations, about 68 percent of observations will be within one SD of the average. Often, the SD is difficult to compute, but there is simple formula for the SD of the number of occurrences of an dvent in repeated, independent trials, whether it be heads in coin tosses or "7" in dice rolls. If p is the chance of occurrence in one trial, then 1 - p is the chance of nonoccurrence, and the formula for the SD of, say, the number of occurrences in N trials is as follows:
Since the chance that a coin lands heads is 1/2, to find the SD for the number of heads in, say, 100 coin tosses, we get:
SD =/100 x 1/2 x 1/2=5
Since the average number of heads in 100 tosses equals 100 x 1/2 = 50, the observed number of heads in 100 tosses will typically be within one SD of the average, between 45 and 55.
When rolling a pair of dice, we have seen that the chance of rolling "7" is 1/6 and the chance of not rolling "7" is 5/6. Thus, for my computer simulation of 72,000 rolls,
SD = /72,000 x 1/6 x 5/6 = 100
Since the average number of occurrences of "7" in 72,000 dice rolls is 72,000 x 1/6 = 12,000, a typical number of "7"s in 72,000 rolls would be within one SD of the average, or between 71,900 and 72,100 occurrences. In my 10 computer simulations of 72,000 dice rolls, 8 were in the one SD range