Sampling Variability and Sample Size: Quick Revision

A brief note on Sampling Variability and Sample Size. Read it before a test. Try each question at the end before you open its answer.

Words

• Parameter: a number for the whole population, such as $\mu$ or $p$. Fixed, usually unknown.
• Statistic: a number from one sample, such as $\bar x$ or $\hat p$. Known, but it changes from sample to sample.
• Sampling variability: the change of a statistic from sample to sample, by chance.
• Sampling distribution: the pattern of a statistic over all possible samples of one size.

Rules

• Centre: the sampling distribution of $\bar x$ is centred at $\mu$, and that of $\hat p$ at $p$.
• Standard error of a mean: $\dfrac{\sigma}{\sqrt{n}}$.
• Standard error of a share: $\sqrt{\dfrac{p(1 - p)}{n}}$.
• Square root rule: four times the sample halves the standard error.
• Shape: for samples of about $30$ or more, $\bar x$ is close to normal (the central limit theorem).
• Population size: hardly matters, unless the sample is more than about a tenth of the population.

A worked example

Seeds from a made-up packet sprout with true share $0.8$. Samples of $64$ seeds are planted.

1. Centre. The sample shares are centred at $0.8$.
2. Standard error. $\sqrt{\dfrac{0.8 \times 0.2}{64}} = \sqrt{0.0025} = 0.05$.
3. A larger sample. With $256$ seeds, four times as many, it halves to $0.025$.

Practice with new numbers

A program to try

The program works out the three answers of the worked example.

Mistakes to avoid

• Mixing up $\sigma$ and the standard error. One is the spread of single values, the other the spread of sample means.
• Dividing by $n$ in place of $\sqrt n$.
• Doubling the sample to halve the error. It takes four times the sample.
• Thinking size fixes bias. A large biased sample is still biased.

Check yourself

1. A sample of $50$ students has a mean height of $152$ cm. Is $152$ cm a parameter or a statistic?

2. Single values have standard deviation $20$ cm. Find the standard error of the mean of $25$ values.

3. The true share is $0.1$. Find the standard error of $\hat p$ for samples of $100$.

4. By what factor must you multiply a sample size to divide the standard error by $4$?

5. A population is skewed. Is the sampling distribution of the mean for samples of $3$ close to normal?

Sources

• OpenStax, Introductory Statistics, Chapter 7, the central limit theorem for sample means and for proportions.