Sample size and the square root rule

What you will be able to do

• find the standard error of a sample share;
• use the square root rule to plan the size of a sample;
• explain why the size of the population hardly matters.

The standard error of a share

In the school, the true share of cyclists is $p = 0.35$. The sample share $\hat p$ has its own sampling distribution. It is centred at $p$, and its standard error is

For samples of $40$ students, $SE = \sqrt{\dfrac{0.35 \times 0.65}{40}} \approx 0.0754$. That is why the five friends missed by up to about $0.075$.

Four times as many, half the error

The sample size sits under a square root. So

Each time the sample is four times as large, the standard error halves. To make it ten times smaller, you need a hundred times as many people. This is the square root rule. Precision is expensive.

You should also see that $p(1 - p)$ is largest at $p = 0.5$. A share near one half is the hardest to pin down.

The picture shows $1000$ simulated sample shares for each of three sample sizes. The middle histogram is about half as wide as the top one, and the bottom one about half as wide again.

Try it in Python

The program draws $500$ samples for each of four sample sizes. It compares the spread of the sample shares with the rule.

The population size hardly matters

A random sample of $1000$ adults gives almost the same standard error for a city of a million as for a country of a hundred million. The formula has no population size in it. What matters is how many people you ask, not what fraction of the population they are.

There is a small exception. When the sample is a large part of the population, say more than a tenth, the standard error is a little smaller than the formula says. Asking $900$ of $1000$ people leaves little room for chance.

Planning a sample

Suppose you expect a share near $0.3$ and want a standard error of at most $0.02$. You need

Practise with new numbers

Common mistakes

• Doubling the sample to halve the error. Doubling divides the standard error by only $\sqrt 2 \approx 1.41$. You need four times the sample.
• Thinking a bigger population needs a bigger sample. For a sample that is a small part of the population, the population size hardly matters.
• Forgetting bias. A huge sample from a biased method has a tiny standard error and a large error. The rule measures chance only.

Remember these

• The standard error of a sample share is $\sqrt{\dfrac{p(1 - p)}{n}}$, and of a sample mean $\dfrac{\sigma}{\sqrt{n}}$.
• Both shrink like $\dfrac{1}{\sqrt{n}}$: four times the sample, half the error.
• The population size hardly matters unless the sample is a large part of it.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The true share is $0.5$. Find the standard error of $\hat p$ for samples of $25$.

2. A survey of $150$ people has a standard error of $0.04$. What standard error would a survey of $600$ people have?

3. You want a standard error $5$ times smaller than your pilot survey gave. By what factor must you multiply the sample size?

4. A random sample of $800$ people is taken in a city of $2$ million and another in a country of $50$ million. Which standard error is smaller?

5. A website asks its visitors to vote, and $100\,000$ people answer. Is its result more reliable than a random sample of $1000$ people?

References

• OpenStax, Introductory Statistics, Chapter 7, section on the central limit theorem for proportions, and Chapter 8 on choosing a sample size.