# Sampling and estimation

> Mathematics · CIE A-Level 9709
> Source: https://www.owlsprep.com/study/cie-9709-u4-sampling-and-estimation/

This module covers core sampling terminology, standard sampling methods, unbiased population parameter estimates, and introductory sampling distributions, all mapped to CIE 9709 exam mark schemes.

**Prerequisites:** [Basic mean and variance calculation for ungrouped data](https://www.owlsprep.com/study/cie-9709-u4-descriptive-statistics/); [Fundamental probability rules](https://www.owlsprep.com/study/cie-9709-u4-probability-fundamentals/)

## Learning objectives

- Distinguish between population, sample, parameter and statistic for exam scenarios
- Select and justify appropriate sampling methods for given contexts
- Calculate unbiased estimates of population mean and variance correctly
- Apply basic sampling distribution rules to solve standard exam problems

## How often is this tested?

Based on OwlsPrep's analysis of official CIE 9709 S2 past papers (2016–2025): **Sampling and estimation** appears **122 times** in the last 10 years — **28.5%** of all S2 questions (122 of 428).

Most-tested forms: Type I / Type II errors; significance level; p-value (63), Hypothesis test for population mean using normal distribution (57), 总体均值与方差的无偏估计 (34), Hypothesis test for Poisson mean (one-tailed) (26).

## Core Sampling Terminology

All CIE exam questions on sampling start with identifying the difference between a population parameter and a sample statistic. A parameter describes a fixed, usually unknown, characteristic of the full population, while a statistic is a calculated value from sample data used to estimate that parameter.

**Unbiased Estimator** — An estimator $\hat{\theta}$ for population parameter $\theta$ is unbiased if the expected value of the estimator equals the true value of the parameter.

*Notation:* $E[\hat{\theta}] = \theta$

*Example:* Sample mean $\bar{x}$ is always an unbiased estimator of population mean $\mu$.

**Exam command terms**

CIE uses specific command terms for this topic:

- **State** — No justification required, 1 mark for correct term

- **Explain** — 2 marks, must link property to population representativeness

**Worked example:** A researcher calculates the average height of 50 randomly selected 12th graders to estimate the average height of all 12th graders in the country. Identify the parameter and the statistic in this scenario.

1. Step 1: Locate the full group of interest
2. The population is all 12th graders in the country, so the parameter is the true average height of this full population.
3. Step 2: Locate the calculated value from the subset
4. The sample is the 50 selected students, so the statistic is the average height calculated from these 50 students.

## Common Sampling Methods

CIE exams regularly ask you to compare 4 core sampling methods: simple random, stratified, systematic, and quota sampling. You must be able to describe each method, state one advantage and one disadvantage, and select the most appropriate method for a given scenario.

**Comparing methods**

Side-by-side comparison of required sampling methods:

- **Simple Random Sampling** — Every possible sample of size n has equal probability of being selected
  - Pros: Unbiased, easy to justify statistically
  - Cons: Requires full sampling frame, can over-represent small subgroups

- **Stratified Sampling** — Population split into distinct subgroups (strata), random samples taken from each stratum proportional to stratum size
  - Pros: Reduces sampling error, representative of all subgroups
  - Cons: Requires prior knowledge of stratum membership for all population members

- **Systematic Sampling** — Select every k-th member from the sampling frame, starting from a random integer between 1 and k
  - Pros: Fast and simple to implement in field work
  - Cons: Biased if population has periodic patterns matching interval k

- **Quota Sampling** — Interviewer selects non-random participants to meet pre-set subgroup quotas
  - Pros: Low cost, no sampling frame required
  - Cons: High risk of selection bias, not statistically random

> **tip**
>
> CIE mark schemes award full marks for stratified sampling justifications that explicitly mention 'reduced sampling variation compared to simple random sampling'.

**Worked example:** A school with 1200 students (450 in Year 12, 750 in Year 13) wants a stratified sample of 40 students for a survey. Calculate the number of students to select from each year group.

1. Step 1: Calculate the sampling proportion for the whole population
2. $$\text{Proportion} = \frac{\text{Total sample size}}{\text{Total population}} = \frac{40}{1200} = \frac{1}{30}$$
3. Step 2: Apply the proportion to each stratum
4. $$\text{Year 12 sample size} = 450 \times \frac{1}{30} = 15$$
5. $$\text{Year 13 sample size} = 750 \times \frac{1}{30} = 25$$
6. Final answer: 15 Year 12 students, 25 Year 13 students

## Unbiased Estimates of Mean and Variance

This is the highest weight sub-concept in this topic, appearing in almost every Statistics 6 and 7 paper. The sample mean is always an unbiased estimate of the population mean, but the raw sample variance (divided by n) is biased, so you must use the n-1 correction factor.

**Derivation:** Derive the unbiased sample variance formula

*Starting from:* Biased sample variance $\hat{\sigma}^2 = \frac{1}{n}\sum(x_i - \bar{x})^2$

1. Expand the squared difference term: $\sum(x_i - \bar{x})^2 = \sum x_i^2 - n\bar{x}^2$
2. Take expected value: $E[\sum(x_i - \bar{x})^2] = (n-1)\sigma^2$
3. To make the estimator unbiased, divide by n-1 instead of n

*Conclusion:* Unbiased sample variance: $s^2 = \frac{1}{n-1}\left(\sum x^2 - \frac{(\sum x)^2}{n}\right)$

**Worked example:** Calculate the unbiased estimates of population mean and variance for the sample data: 12, 15, 17, 21, 25

1. Step 1: Calculate sum of x and sum of x squared
2. $$\sum x = 12 + 15 + 17 + 21 + 25 = 90$$
3. $$\sum x^2 = 144 + 225 + 289 + 441 + 625 = 1724$$
4. Step 2: Unbiased mean estimate
5. $$\bar{x} = \frac{90}{5} = 18$$
6. Step 3: Unbiased variance estimate, use n-1 = 4
7. $$s^2 = \frac{1}{4}\left(1724 - \frac{90^2}{5}\right) = \frac{1}{4}(1724 - 1620) = 26$$

**Check your understanding**

Test your understanding before proceeding:

1. Which denominator gives an unbiased estimate of population variance?

   - n
   - n-1
   - n+1
   - n/2

   *Why:* The n-1 Bessel correction removes the bias from sample variance.

## Introduction to Sampling Distributions

The sampling distribution of a statistic is the probability distribution of all possible values of that statistic calculated from samples of the same size taken from the same population. For large n, the Central Limit Theorem tells us the sampling distribution of the sample mean will be approximately normal.

$$\text{If } X \sim N(\mu, \sigma^2) \text{ then } \bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$$

> **warning**
>
> CIE examiners regularly penalize students who forget to divide the population variance by the sample size when calculating probabilities for the sample mean.

**Worked example:** A population has mean 42 and variance 18. Find the variance of the sampling distribution of the sample mean for samples of size 9.

1. Step 1: Use the sampling distribution variance rule
2. $$\text{Var}(\bar{X}) = \frac{\sigma^2}{n} = \frac{18}{9} = 2$$
3. Final answer: Variance of the sample mean is 2

## Common pitfalls

- **Wrong:** Using n as the denominator for sample variance instead of n-1
  - Why it fails: CIE mark schemes deduct 2-3 marks for this error, even if all other steps are correct
  - Correct: Explicitly write the n-1 term in your variance calculation to confirm you are using the unbiased estimator
- **Wrong:** Confusing population parameters and sample statistics
  - Why it fails: 1 mark questions on identification give zero partial credit for mixing up the two terms
  - Correct: Label all population values with Greek letters and sample values with Roman letters to keep them distinct
- **Wrong:** Calling quota sampling 'random sampling' in answers
  - Why it fails: Quota sampling uses non-random selection, and CIE mark schemes explicitly reject this description
  - Correct: Name the exact sampling method requested, and only label methods as random if every member of the population has equal selection probability
- **Wrong:** Forgetting to divide population variance by n when calculating sample mean probabilities
  - Why it fails: This produces a completely wrong normal distribution, leading to zero marks for the entire probability calculation
  - Correct: Write out the full sampling distribution notation at the start of every question to avoid missing the variance scaling step
- **Wrong:** Rounding stratified sample sizes before summing
  - Why it fails: This can lead to a total sample size that does not match the required value
  - Correct: Calculate all stratum sample sizes first, then adjust the largest fractional value to make the total sample size exactly match the required number

## Cheatsheet

| Quantity | Formula | CIE Exam Requirement |
| --- | --- | --- |
| Unbiased population mean estimate | $\bar{x} = \frac{\sum x}{n}$ | Identical to raw sample mean |
| Unbiased population variance estimate | $s^2 = \frac{1}{n-1}\left(\sum x^2 - \frac{(\sum x)^2}{n}\right)$ | Denominator n-1 is mandatory |
| Systematic sampling interval | $k = \lfloor \frac{N}{n} \rfloor$ | k must be a positive integer |
| Stratified stratum sample size | $n_h = n \times \frac{N_h}{N}$ | Round to nearest whole number |
| Sampling distribution of sample mean | $\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$ | Variance divided by sample size n |

## What's next

Mastering sampling and estimation is the critical foundation for all subsequent inferential statistics topics in CIE 9709, which together make up ~30% of the total Statistics paper weight. You will reuse the unbiased variance formulas you learned here directly in every confidence interval and hypothesis test question for the rest of your course. To reinforce your mastery, first practice targeted questions on identifying biased vs unbiased estimators, then move to full past paper questions that combine sampling method selection with variance calculation. The linked modules below will build directly on this knowledge to extend your statistical inference skills.

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