Average and Standard Deviation of the Binomial Distribution
We can now use the results found in the previous example to ask the more general question: what is the averageand standard deviation of the number of heads found in N flips of a coin, if the probability for a head in one flip is p.
Although we have phrased this is the language of coin flips, we are really talking here about the result of any experiment in which the outcomes can be divided into two mutually exculsive groups, one which we can label as "win", the other which we label as "lose". We are then interested in the average and standard deviation of the number of wins in N plays.
In the previous example we found for two flips,
Now consider three flips. We can regard this as the following two experiments: experiment #1 is the number of heads in two flips of a coin, experiment #2 is the number of heads in one flip of a coin. By our general results for adding the reults of two random experiments we get,
Using our result thatand we conclude that
Now consider four flips. We regard this as the following two experiments: experiment #1 is the number of heads in three flips of a coin, experiment #2 is the number of heads in one flip of a coin. We then get,
In a similar way we can keep going to conclude that the average and standard deviation of the number of heads found in N flips of a coin is,
Since the standard deviation is a measure of the width of the spread in the probability distribution about the mean, we have now derived what we saw experimentally earlier: the mean grows proportional to N, while the width grows proportional to .
Note that for p=1/2,, and so . We can compare this to the "error" which we defined earlier as the distance from the point of maximum probability to the point with one half the maximum probability -- this we found to behave as . Since the coefficients of the are slightly different in these two expressions, the standard deviation and the "error" are not exactly the same quantity. However since they both increase as , they are measuring qualitatively the same thing. The precise meaning of the standard deviation in terms of the relative probability for various outcomes will now be discussed in the next section.