Question

What happens to the sample standard deviation when the sample size is increased?

Answer 1

It depends on the actual data added to the sample, but generally, the sample S.D. will approach the actual population S.D.

Explanation:

The formula for sample standard deviation is

`s=sqrt((sum_(i=1)^n (x_i-bar x)^2)/(n-1))`

while the formula for the population standard deviation is

`sigma=sqrt((sum_(i=1)^N(x_i-mu)^2)/(N-1))`

where

• `n` is the sample size,
• `N` is the population size,
• `bar x` is the sample mean, and
• `mu` is the population mean.

As `n` increases towards `N`, the sample mean `bar x` will approach the population mean `mu`, and so the formula for `s` gets closer to the formula for `sigma`.

Thus, as `n->N, s -> sigma`.

Note:

When `n` is small compared to `N`, the sample mean `bar x` may behave very erratically, darting around `mu` like an archer's aim at a target very far away. Adding a single new data point is like a single step forward for the archer—his aim should technically be better, but he could still be off by a wide margin. Thus, incrementing `n` by 1 may shift `bar x` enough that `s` may actually get further away from `sigma`.

It is only over time, as the archer keeps stepping forward—and as we continue adding data points to our sample—that our aim gets better, and the accuracy of `barx` increases, to the point where `s` should stabilize very close to `sigma`.