# Finding pooled variance

Find the variance of $S^2_p$ under the conditions; $\bar{x_1}, \bar{x_2}, s_1, s_2$ are the means and standard deviations of independent random samples of sizes $n_1$ and $n_2$ from normal populations with equal variances.

The answer to this is $\frac {\sigma^4}{n_1 +n_2- \lambda}$. I am confused as to how to acquire this answer.

## 1 Answer

The formula for the pooled variance given two samples $x_1^1,\dots, x_1^{n_1}$ and $x_2^1,\dots, x_2^{n_2}$ of sizes $n_1$ and $n_2$ respectively is given by $$S_p = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2}$$ where $s_1^2$ and $s_2^2$ are the sample variances of the respective samples.

You want $$Var(S_p) = \left(\frac{n_1-1}{n_1+n_2}\right)^2 Var(s_1^2) + \left(\frac{n_2-1}{n_1+n_2}\right)^2 Var(s_2^2)$$ where $$s_i^2 = \frac{1}{n_i-1} \sum_{j=1}^{n_i} (x_i^j - \overline{x}_i)^2 \mbox{ for } i=1,2.$$

You have that your samples are drawn from a normal distribution with equal variances, that is $$x_i^1,\dots, x_i^{n_i} \sim N(\mu_i,\sigma^2 ) \mbox{ for } i=1,2$$ and hence $$\overline{x}_i \sim N(\mu_i, \frac{\sigma^2}{n_i}) \mbox{ for } i=1,2.$$

Using the above information compute now $Var(s_i^2)$ for $i=1$ and $i=2$ under the assumptions above and plug it in the formula for $Var(S_p)$.

Good luck!

P.S.: By the way, it's common within statistics that capital letters denote random variables, for instance $X\sim N(\mu,\sigma^2)$ and small letters denote realizations of these random variables, that is $x$ is an observation from $X$ means that $x$ is a given value drawn from $X\sim N(\mu,\sigma^2)$. I tell you because here we have not been consistent with this notation.