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Checking Out the Mean and Standard
Deviation of the Binomial
Because the binomial distribution is so commonly used, statisticians went
ahead and did all the grunt work to figure out nice, easy formulas for finding
its mean, variance, and standard deviation. (That is, they’ve already applied
the methods from the section “Defining a Random Variable” to the binomial
distribution formulas, crunched everything out, and presented the results to
us on a silver platter — don’t you love it when that happens?) The following
results are what came out of it.
If X has a binomial distribution with n trials and probability of success p on
each trial, then:
1. The mean of X is
.
Part III: Distributions and the Central Limit Theorem
.
2. The variance of X is
3. The standard deviation of X is .
For example, suppose you flip a fair coin 100 times and let X be the number of
heads; then X has a binomial distribution with n = 100 and p = 0.50. Its mean
is heads (which makes sense, because heads and tails
are 50-50). The variance of X is , which
is in square units (so you can’t interpret it); and the standard deviation is the
square root of the variance, which is 5. That means when you flip a coin 100
times, and do that over and over, the average number of heads you’ll get is
50, and you can expect that to vary by about 5 heads on average.
The formula for the mean of a binomial distribution has intuitive meaning. The
p in the formula represents the probability of a success, yes, but it also rep-
resents the proportion of successes you can expect in n trials. Therefore, the
total number of successes you can expect — that is, the mean of X — is .
The formula for variance has intuitive meaning as well. The only variability in
the outcomes of each trial is between success (with probability p) and failure
(with probability 1 – p). Over n trials, the variance of the number of successes/
failures is measured by . The standard deviation is just the
square root.
If the value of n is too large to use the binomial formula or the binomial table
to calculate probabilities (see the earlier sections in this chapter), there’s an
alternative. It turns out that if n is large enough, you can use the normal distri-
bution to get an approximate answer for a binomial probability. The mean and
standard deviation of the binomial are involved in this process. All the details
are in Chapter 9.

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