Home Exam Structure Topics Past Papers Resources Homework About Contact
Home Exam Structure Topics Past Papers Resources Homework About Contact

Home / Probabilities & Estimation / Sampling Distributions

3.6a · Probabilities & Estimation · Sub-skill

Sampling Distributions

How the mean of repeated samples behaves, and why sample size affects how reliable an estimate is.

Build it up, step by step

Understanding sampling distributions

Click each step below to reveal it — work through them in order the first time round.

Step 1 · Sampling to estimate a population

1 of 4▾

Since we usually can't measure an entire population, we take a sample and use it to estimate population values (like the population mean). Different samples from the same population will give slightly different sample means — this natural variation is called sampling variability.

Step 2 · The sampling distribution of the mean

2 of 4▾

If you took many different samples (all the same size) from a population and calculated the mean of each, those sample means would themselves form a distribution — the sampling distribution of the mean. This distribution clusters around the true population mean.

Step 3 · Sample size and reliability

3 of 4▾

Larger samples give sample means that are, on average, closer to the true population mean — the sampling distribution becomes narrower as sample size increases. This is why bigger samples generally give more reliable estimates, though they cost more time/resources to collect.

Step 4 · Standard error

4 of 4▾

The spread of the sampling distribution of the mean is measured by the standard error, $SE = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. As $n$ increases, the standard error decreases.

Worked example

A population has standard deviation 12. Calculate the standard error of the mean for a sample of size 36, and for a sample of size 144. Comment on the difference.

n=36: SE = 12÷√36 = 12÷6 = 2
n=144: SE = 12÷√144 = 12÷12 = 1
Quadrupling the sample size (36→144) halves the standard error (2→1) — increasing sample size improves precision, but with diminishing returns, since SE depends on √n.

Test yourself

Past-paper style question

A researcher wants to estimate the mean weight of apples from a large orchard, known to have a population standard deviation of 15g.

(a) Calculate the standard error of the mean for a sample of 25 apples.
(b) The researcher increases their sample to 100 apples. Calculate the new standard error, and explain the effect of this change on the reliability of the estimate. [4 marks]

Show the answer

▾

(a) SE = 15÷√25 = 15÷5 = 3g.

(b) SE = 15÷√100 = 15÷10 = 1.5g. Increasing the sample size from 25 to 100 (4 times as many apples) halves the standard error (from 3g to 1.5g), meaning the sample mean is likely to be a more precise/reliable estimate of the true population mean weight.

Practice

Sampling Distributions worksheet

Five short questions on sampling distributions. Work through them, then reveal the mark scheme to check.

  1. Explain, in your own words, what is meant by the ‘sampling distribution of the mean.’
  2. A population has standard deviation 20. Calculate the standard error of the mean for a sample of size 16.
  3. Using the same population as Q2, calculate the standard error for a sample of size 100, and compare it to your answer to Q2.
  4. Explain why increasing sample size improves the reliability of an estimate, using the standard error formula.
  5. A student says ‘doubling the sample size halves the standard error.’ Explain why this statement is incorrect, using the formula for standard error.

Mark scheme

▾
  1. The sampling distribution of the mean is the distribution formed by taking many different samples (of the same size) from a population, calculating the mean of each sample, and looking at how those sample means themselves are distributed; it clusters around the true population mean.
  2. SE = 20÷√16 = 20÷4 = 5.
  3. SE = 20÷√100 = 20÷10 = 2. This is smaller than the answer to Q2 (5), showing the larger sample (100 vs 16) gives a more precise/reliable estimate of the population mean.
  4. Because SE = σ÷√n, increasing n increases the denominator, which decreases the standard error — a smaller standard error means the sample mean is likely to be closer to the true population mean, i.e. a more reliable estimate.
  5. Because SE is proportional to $\frac{1}{\sqrt{n}}$, not $\frac{1}{n}$; doubling n only decreases SE by a factor of about 0.71 (about a 29% reduction), not by half — you would need to quadruple the sample size to actually halve the standard error.
← Back to Probabilities & Estimation