# Relationship between variance of mean and mean variance

In ranked set sampling, we select $n$ random sets, each of size $n$. Then we choose the largest unit from the 1st set, 2nd largest from the 2nd set, and thus $n$th largest from the $n$th set. This sampling procedure was first introduced by McIntyre (1952). The reference is A method for unbiased selective sampling, using ranked sets. Australian Journal of Agricultural Research, 3(4), 385-390. In the Method section (page 2) of this paper, it is written that

The variance of the mean of five quadrats one from each subdistribution is one- fifth of the mean variance of these distributions. This may be contrasted with the variance of the mean of five random samples, that is, one-fifth of the variance of the parent population.

Can anyone please illustrate how does the variance of the mean of five quadrats one from each subdistribution equal one-fifth of the mean variance of these distributions?

And also, what does this sentence "This may be contrasted with the variance of the mean of five random samples, that is, one-fifth of the variance of the parent population." mean?

• Don't you think that the first part of the question ("one-fifth of the variance of the parents population") may refer to the variance of a sample mean instead of the mean variance of these distributions? – KenHBS Sep 6 '17 at 8:03

$$\newcommand{\Var}{\operatorname{Var}}$$

The article compares two different estimates for the true population mean, $\mu$.

The mean computed from five random samples $x_i$

$$\hat{\mu}_x = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5}$$

and the mean computed from five random quintiles $y_i$ (5 quintiles selected by five times randomly picking samples and ordering them and picking the 1st quintile from the 1st picking, the 2nd...)

$$\hat{\mu}_y = \frac{y_1 + y_2 + y_3 + y_4 + y_5}{5}$$

With those expressions we should be able to see what they mean. The variance of the stochastic variables $\hat{\mu}_x$ and $\hat{\mu}_y$ (they are not fixed values like $\mu$) can be described by the sum of the variances of the variables $x_i$ and $y_i$. Using the general summation law of variance of uncorrelated variables $\Var(a z_1 + b z_2) = a^2 \Var(z_1) + b^2 \Var(z_2)$ you get:

$$\Var(\hat{\mu}_x) = \frac{1}{25}\Var(x_1) + \frac{1}{25}\Var(x_2) + \frac{1}{25}\Var(x_3) + \frac{1}{25}\Var(x_4) + \frac{1}{25}\Var(x_5)$$

and

$$\Var(\hat{\mu}_y) = \frac{1}{25}\Var(y_1) + \frac{1}{25}\Var(y_2) + \frac{1}{25}\Var(y_3) + \frac{1}{25}\Var(y_4) + \frac{1}{25}\Var(y_5)$$

now notice that all $\Var(x_i)$ are the same thus:

$$\Var(\hat{\mu}_x) = \frac{1}{5}\Var(x_i)$$

and also notice that we have $\Var(y_i)<\Var(x_i)$ for each $i$ (the variance of a random sampled quintile is smaller than a random sample, which the Figure 1 in the reference shows very well). Thus:

$$\Var(\hat{\mu}_y) < \frac{1}{5}\Var(x_i)$$

which is the increased efficiency what they where looking for (yes you need five times more samples to get those quintiles, but that is what the explain at the end, where they state that the method is especially suited for situations where ordering is much easier than exact quantitive measurement).

And finally to get even closer to their statement we can use something like

\begin{align} \Var(\hat{\mu}_y) &= \frac{1}{5} \frac{\Var(y_1)+\Var(y_2)+\Var(y_3)+\Var(y_4)+\Var(y_5)}{5} \\&= \frac{1}{5}\overline{\Var(y_i)} < \frac{1}{5}\Var(x_i) \end{align}

(1) The equality in the second line relates to:

"The variance of the mean of five quadrats one from each subdistribution is one- fifth of the mean variance of these distributions."

(2) the inequality in the second line relates to:

"This may be contrasted with the variance of the mean of five random samples, that is, one-fifth of the variance of the parent population."