# Statistics: What is the major difference amongst these three binomial variance formulas and how to derive them

I am self-studying statistics as a high school senior, but I'm having trouble putting some formulas into better context - as in where do they come from?

I want to know why there are three binomial variance formulas and how to get them using the definition of variance:

$pq/n$, $pq$ and $npq$.

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The following may involve fancier machinery than you are comfortable with. Imagine repeating an experiment independently $n$ times, where each time the probability of success is $p$. Let random variable $X$ denote the total number of successes in the $n$ trials.

Then $X$ has binomial distribution.

A closely related random variable is $\overline{X}$, the average number of successes per trial. We have $\overline{X}=\frac{X}{n}$.

For $i=1$ to $n$, let $Y_i=1$ if we have a success on the $i$-th trial, and let $Y_i=0$ otherwise. The random variables $Y_i$ are called Bernoulli random variables.

A straightforward calculation shows that for each $i$, $E(Y_i)=1$ and $\operatorname{Var}(Y_i)=pq$. That's one of the items you asked about.

There is a theorem that says that the variance of a sum of independent random variables is the sum of the variance. Note that $$X=Y_1+Y_2+\cdots+Y_n.$$ Since each $Y_i$ has variance $1$, we conclude that $$\operatorname{Var}(X)=npq.$$ So we have shown that the variance of our binomial random variable $X$ is $npq$. That's another item you asked about.

Finally, there is a general theorem that says that if $k$ is a constant, and $W$ is a random variable, then the variance of $kW$ is $k$ times the variance of $W$. It follows that $$\operatorname{Var}(\overline{X})=\operatorname{Var}\left(\frac{X}{n}\right)=\frac{1}{n^2}\operatorname{Var}(X)=\frac{1}{n^2}(npq)=\frac{pq}{n}.$$

So the three variances you mentioned are variances of quite different (but related) random variables. But only one of the formulas, namely $npq$ gives the variance of the binomial.

The formula for the variance of $\overline{X}$ is very useful. It basically tells us how reliable $\overline{X}$ is as an estimator of $p$. So it is deeply connected with estimates of the reliability of a poll.

Remark: The variance of the binomial can be found in a more computational way, by using the definition of variance, and manipulation of binomial coefficients. We introduced the more abstract viewpoint because it makes things much simpler (after one gets used to it).

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thanks! I learned a lot from your explanations –  Person Aug 17 '13 at 4:02
You are welcome. I hope it is a useful beginning. –  André Nicolas Aug 17 '13 at 4:05
What is $q$ here? –  soandos Mar 11 '14 at 14:56