What is the difference between a sample and a population in the context of Edexcel International A Levels Statistics 2?

In Statistics 2, a population refers to the entire set of individuals or items that we are interested in studying, while a sample is a subset of the population that is selected for analysis. Sampling is used because it is often impractical or impossible to study the entire population. By analyzing a sample, we can make inferences about the population as a whole.

Why is sampling important in statistical analysis for Edexcel International A Levels Mathematics?

Sampling is crucial because it allows researchers to gather data and make inferences about a population without needing to collect data from every individual. This is particularly important in Edexcel International A Levels Mathematics, where time and resources are limited. Sampling helps in estimating population parameters, testing hypotheses, and making predictions efficiently.

What are the different types of sampling methods covered in Edexcel International A Levels Statistics 2?

In Edexcel International A Levels Statistics 2, several sampling methods are discussed, including simple random sampling, systematic sampling, stratified sampling, and cluster sampling. Each method has its own advantages and is chosen based on the research objectives and the nature of the population being studied.

How does a sampling distribution differ from a sample distribution in Statistics 2?

A sample distribution refers to the distribution of data points within a single sample. In contrast, a sampling distribution is the probability distribution of a statistic (like the mean or variance) calculated from multiple samples of the same size drawn from the same population. Understanding sampling distributions is key in Edexcel International A Levels Statistics 2 as it helps in estimating the variability of sample statistics.

What role does the Central Limit Theorem play in sampling distributions for Edexcel International A Levels?

The Central Limit Theorem is a fundamental concept in Statistics 2, stating that the sampling distribution of the sample mean will approximate a normal distribution, regardless of the population's distribution, provided the sample size is sufficiently large. This theorem is crucial for making inferences about population parameters and is a cornerstone of statistical analysis in Edexcel International A Levels Mathematics.

Why is "Using $\sigma$ instead of $\sigma/\sqrt n$ when standardising $\bar X$" a common mistake in Sampling and Sampling Distributions?

Why it happens: Students forget that X has a different SD from individual X. How to avoid it: Always write ' X has variance ^2/n' as the first line. Use SE = /√ n in the z-formula. Source: WST02/01 examiner reports — top annual error

Why is "Stating 'by CLT' when the parent is normal" a common mistake in Sampling and Sampling Distributions?

Why it happens: Students apply CLT habitually. How to avoid it: If the parent is normal, X is EXACTLY normal — no approximation needed, NO 'by CLT'. Just write 'X is normal, so X is normal'.

Why is "Invoking CLT for very small $n$ (e.g. $n = 4$) from a skewed parent" a common mistake in Sampling and Sampling Distributions?

Why it happens: Misremembering the rule. How to avoid it: CLT is reliable for n 30 (rough rule). For smaller n, only the EXACT distribution (or normal parent) works. Edexcel mark schemes use n 30 as the threshold.

Why is "Reporting variance when SE was asked, or vice versa" a common mistake in Sampling and Sampling Distributions?

Why it happens: Confusion between ^2/n and /√ n. How to avoid it: VARIANCE of X: ^2/n (dimensions of X^2). STANDARD ERROR: /√ n (dimensions of X). Both are useful; know which the question asks for.

Why is "Vague description of simple random sampling" a common mistake in Sampling and Sampling Distributions?

Why it happens: Students give the spirit, not the method. How to avoid it: Include: numbering the population, using a random number generator, handling duplicates / out-of-range, continuing until n chosen. Four explicit steps. Source: WST02/01 examiner reports

Statistics 2Edexcel International A Levels Mathematics (XMA01-YMA01)

Sampling and Sampling Distributions

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Sampling and Sampling Distributions Study Notes — Edexcel IAL Mathematics S2 (WST02/01, 2018 spec — 2026 onwards)

A SAMPLE is selected from a POPULATION; a STATISTIC (e.g. $\bar X$ ) is computed from the sample; the SAMPLING DISTRIBUTION is the distribution of that statistic across all possible samples. The fundamental result: $\bar X$ has mean $\mu$ and variance $\sigma^{2}/n$ . The Central Limit Theorem extends this to a normal approximation for large $n$ regardless of parent.

What you’ll learn

Mapped to the Cambridge IGCSE XMA01-YMA01 syllabus (2026 onwards).

S2 6.1 — Distinguish population, sample, statistic, parameter.
S2 6.2 — Describe simple random sampling procedures.
S2 6.3 — Identify the sampling distribution of $\bar X$ .
S2 6.4 — Apply the CLT to compute probabilities involving $\bar X$ .
S2 6.5 — Understand the effect of sample size on the standard error.

Vocabulary — population, sample, statistic

Population = full set; sample = chosen subset; statistic = function of the sample.

Population. The complete collection of items / individuals about which we want information. Can be finite (the 1800 students at a college) or conceptually infinite (all possible flips of a coin).

Sample. A subset of the population, used to draw conclusions about the whole.

Statistic. A quantity computed from the sample. Examples:

Sample mean $\bar X = (1/n)\sum X_i$ .
Sample variance $s^{2} = (1/(n-1))\sum (X_i - \bar X)^{2}$ .
Sample median, IQR, range, etc.

Parameter. A quantity describing the population. Examples: $\mu$ (population mean), $\sigma^{2}$ (population variance), $p$ (population proportion). Usually UNKNOWN — we estimate them with statistics.

Why sample?

The population is too large to measure completely.
Measurement is destructive (testing battery lifetime).
Cost / time constraints.
Sometimes measuring everyone is impossible (e.g. all future production).

The fundamental tension. A sample is uncertain — different samples give different statistics. The sampling distribution captures this uncertainty.

Population: complete set.
Sample: subset.
Statistic: function of sample (random).
Parameter: function of population (fixed but unknown).

Simple random sampling — the procedure

Number, draw random integers, discard duplicates / out-of-range, repeat until $n$ .

Simple random sample (SRS). Every subset of size $n$ from the population has equal probability of being selected. Equivalently, every individual is equally likely to be included.

Procedure.

Number the population $1, 2, \ldots, N$ .
Generate random integers in $1$ - $N$ using a random number generator (calculator's $\text{Ran}\#$ , computer, random number tables).
Discard any number $> N$ and any duplicates.
Continue until $n$ distinct numbers obtained.
The individuals with the chosen numbers form the sample.

Why SRS is desirable.

Unbiased — every individual has equal chance.
Statistical theory (CLT, hypothesis tests) assumes SRS.

Other sampling methods (Edexcel S2 awareness only).

Method	Idea	Pros	Cons
Simple random	Pure random	Unbiased	May miss subgroups by chance
Stratified	Sample from each subgroup proportionally	Represents subgroups	More complex
Systematic	Every $k$ -th individual	Easy	Cyclic bias risk
Cluster	Sample whole groups	Cheap if groups exist	Less precise
Quota	Fill quotas non-randomly	Quick	Not random — biased
Opportunity	Whoever's available	Fast	Highly biased

S2 mainly tests SRS and (occasionally) stratified.

SRS: every subset size $n$ has equal probability.
Procedure: number, generate, discard, repeat.
Other methods exist but SRS is the S2 default.
Examiners want detailed procedural description.

Sampling distribution of $\bar X$

$E(\bar X) = \mu$ , $\text{Var}(\bar X) = \sigma^{2}/n$ . Standard error $\sigma/\sqrt n$ .

Setup. Take a random sample $X_1, X_2, \ldots, X_n$ from a population with mean $\mu$ and variance $\sigma^{2}$ . The $X_i$ are iid (independent, identically distributed).

Sample mean. $\bar X = (1/n)\sum X_i$ is itself a random variable.

Mean of $\bar X$ .

$E(\bar X) = \dfrac{1}{n}\sum E(X_i) = \dfrac{1}{n}(n\mu) = \mu.$

So $\bar X$ is an UNBIASED estimator of $\mu$ .

Variance of $\bar X$ . Using independence (so variances add):

$\text{Var}(\bar X) = \dfrac{1}{n^{2}}\sum \text{Var}(X_i) = \dfrac{1}{n^{2}}(n\sigma^{2}) = \dfrac{\sigma^{2}}{n}.$

Standard error.

$\text{SE}(\bar X) = \dfrac{\sigma}{\sqrt n}.$

Note. These hold REGARDLESS of the shape of the parent distribution. Only the SHAPE of $\bar X$ depends on the parent.

Effect of $n$ .

Doubling $n$ reduces variance by half, SE by $1/\sqrt 2 \approx 71\%$ .
To halve SE, $n$ must QUADRUPLE.
Precision $\propto 1/\sqrt n$ — slow improvement.

Numerical example. $\sigma = 10$ .

$n = 25$ : SE $= 2$ .
$n = 100$ : SE $= 1$ .
$n = 400$ : SE $= 0.5$ .

$E(\bar X) = \mu$ — unbiased.
$\text{Var}(\bar X) = \sigma^{2}/n$ .
SE $= \sigma/\sqrt n$ .
Quadruple $n$ to halve SE.
Holds for ANY parent shape.

Central Limit Theorem (CLT)

For large $n$ , $\bar X \approx N(\mu, \sigma^{2}/n)$ regardless of parent.

Statement. Let $X_1, \ldots, X_n$ be iid with mean $\mu$ and finite variance $\sigma^{2}$ . Then for large $n$ ,

$\bar X \approx N\left(\mu, \dfrac{\sigma^{2}}{n}\right).$

Why this is amazing. The parent distribution can be ANYTHING — uniform, skewed, multimodal, discrete — and $\bar X$ still becomes approximately normal for large $n$ .

Threshold. Edexcel uses $n \ge 30$ as a rough guide. In practice:

Parent close to normal: $n = 5$ - $10$ suffices.
Parent moderately skewed: $n = 30$ usually enough.
Parent heavily skewed: $n = 50$ + recommended.

When CLT NOT needed. If the parent is itself normal, $\bar X$ is EXACTLY normal — no approximation, no large $n$ required.

When CLT FAILS.

Parent variance is infinite (Cauchy, some heavy-tailed) — outside S2.
Observations are NOT independent.
Sample size is small AND parent is heavily skewed.

Application algorithm.

State $\mu, \sigma, n$ .
Write $\bar X \approx N(\mu, \sigma^{2}/n)$ , citing CLT or normal parent.
Standardise: $z = (\bar x - \mu)/(\sigma/\sqrt n)$ .
Look up $\Phi(z)$ and conclude.

Large $n$ : $\bar X \approx N(\mu, \sigma^{2}/n)$ .
Works for ANY parent (with finite variance).
$n \ge 30$ is the Edexcel threshold.
Normal parent: exact, no CLT needed.

Applications and interpretation

Compute probabilities for $\bar X$ , sample-size calculations, confidence intervals (S2 introduces this concept).

Compute $P(\bar X > a)$ etc.

Standard procedure:

Identify $\mu, \sigma, n$ .
Compute SE $= \sigma/\sqrt n$ .
Standardise the target value: $z = (a - \mu)/\text{SE}$ .
Look up probability.

Sample-size calculation. 'How large $n$ to ensure SE $\le k$ ?' Solve $\sigma/\sqrt n \le k$ to get $n \ge (\sigma/k)^{2}$ .

Interpretation.

A small SE means $\bar X$ is close to $\mu$ with high probability — the sample mean is a PRECISE estimate.
A large SE means $\bar X$ varies a lot — imprecise estimate.
Increasing $n$ improves precision at the rate $1/\sqrt n$ .

Quality control example. Bags of sugar with $\mu = 1000$ g, $\sigma = 12$ g. Sample $n = 36$ . $\bar X \approx N(1000, 4)$ , SE $= 2$ . About $95\%$ of sample means lie between $1000 - 4 = 996$ and $1000 + 4 = 1004$ . If a sample mean falls outside this, the process may be off-spec.

Contextual sentences. Always interpret the answer in context: 'There is approximately a $2.3\%$ chance that the sample mean weight is below $996$ g.'

Standardise: $z = (\bar x - \mu)/(\sigma/\sqrt n)$ .
Sample size: $n \ge (\sigma/k)^{2}$ for SE $\le k$ .
Precision $\propto 1/\sqrt n$ .
Always interpret in context.

How it’s examined

Sampling appears on every WST02/01 paper — typically a $6$ - $10$ mark question: (a) state population / sample / statistic ( $2$ - $3$ marks), (b) describe SRS procedure ( $3$ - $5$ marks), (c) compute a probability for $\bar X$ using CLT ( $4$ - $5$ marks). The key marks are dropped on (1) using $\sigma$ instead of $\sigma/\sqrt n$ in the $z$ -formula and (2) failing to cite CLT (or equivalent justification) when the parent is non-normal. Descriptions of SRS need procedural detail — vague 'choose at random' loses marks.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WST02/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

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Step-by-step worked examples — Sampling and Sampling Distributions

Step-by-step solutions to past-paper-style questions on sampling and sampling distributions, written exactly the way a tutor would explain them at the board.

1Population, sample, statistic (4 marks, S2)
Core• Adapted from WST02/01 January 2024 Q1• sampling, vocabulary
Question
A college has $1800$ students. To estimate the mean height, a random sample of $50$ students is selected and their heights measured. (a) State the population and the sample. (b) Give one reason why a sample is used instead of measuring everyone. (c) State a suitable statistic to estimate the population mean height.
Step-by-step solution
1. Step 1
  (a) Population: all $1800$ students at the college. Sample: the $50$ students chosen.
2. Step 2
  (b) Reasons (any one): cheaper, faster, less intrusive, less effort. Measuring every student would be impractical / time-consuming / expensive.
3. Step 3
  (c) Suitable statistic: the SAMPLE MEAN $\bar X = (1/n)\sum X_i$ — used as an UNBIASED ESTIMATOR of the population mean $\mu$ .
Answer
(a) Population: all 1800 students; sample: the 50 selected. (b) Cheaper / faster than measuring everyone. (c) Sample mean $\bar X$ .
Examiner tip
(a) B1 each for population and sample. (b) B1 for any sensible practical reason. (c) B1 for sample mean. 'Mean' alone is sometimes marked down — say 'sample mean' or ' $\bar X$ ' explicitly.
2Simple random sample procedure (5 marks, S2)
Core• Adapted from WST02/01 June 2024 Q1• sampling, method, random numbers
Question
Describe in detail how a simple random sample of $30$ employees can be selected from a workforce of $400$ .
Step-by-step solution
1. Step 1
  Number the employees $1, 2, \ldots, 400$ (or use existing IDs).
2. Step 2
  Use a random number generator (calculator, computer, or random number tables) to produce three-digit numbers in the range $001$ - $400$ .
3. Step 3
  Discard any number $> 400$ and any repeats.
4. Step 4
  Continue until $30$ distinct numbers are obtained.
5. Step 5
  The employees with the chosen numbers form the simple random sample.
Answer
Number employees $1$ - $400$ . Use random number generator. Discard $> 400$ and repeats. Continue until $30$ distinct numbers chosen. Selected employees form the sample.
Examiner tip
B1 for numbering. B1 for random number generator. B1 for handling out-of-range / duplicates. B1 for continuing until $n$ obtained. B1 for stating that the chosen IDs form the sample. The key feature of a SRS: every subset of size $n$ has equal probability of selection.
3Sampling distribution of $\bar X$ (6 marks, S2)
Extended• Adapted from WST02/01 January 2024 Q7• sampling, central limit, normal
Question
The weight of bags of sugar produced by a factory has mean $\mu = 1000$ g and standard deviation $\sigma = 12$ g. A quality controller takes a random sample of $36$ bags. Let $\bar X$ be the sample mean weight. (a) State the distribution of $\bar X$ , giving a reason. (b) Find $P(\bar X < 996)$ .
Step-by-step solution
1. Step 1
  (a) Sample size $n = 36$ is large. By the CLT,
  $\bar X \approx N\left(\mu, \dfrac{\sigma^{2}}{n}\right) = N\left(1000, \dfrac{144}{36}\right) = N(1000, 4)$
2. Step 2
  Reason: $n = 36 \ge 30$ is large enough for the CLT to apply (mark scheme accepts 'CLT' or ' $n$ large').
3. Step 3
  (b) Standardise. $\sigma_{\bar X} = \sqrt 4 = 2$ .
  $z = \dfrac{996 - 1000}{2} = -2$
4. Step 4
  From normal tables:
  $P(\bar X < 996) = \Phi(-2) = 1 - \Phi(2) = 1 - 0.9772 = 0.0228$
Answer
(a) $\bar X \approx N(1000, 4)$ by CLT. (b) $P(\bar X < 996) = 0.0228$ .
Examiner tip
(a) B1 for the distribution $N(1000, 4)$ . B1 for citing CLT. (b) M1 for standardising. A1 for $z = -2$ . A1 for $0.0228$ . Common error: using $\sigma = 12$ instead of $\sigma/\sqrt n = 2$ .
4Effect of sample size on SE (5 marks, S2)
Extended• Adapted from WST02/01 June 2024 Q8• sampling, standard error, sample size
Question
A random variable $X$ has $\mu = 50$ and $\sigma = 10$ . The mean $\bar X$ of a random sample of size $n$ is to be used to estimate $\mu$ . (a) Find the standard error of $\bar X$ when $n = 25$ . (b) How large must $n$ be to ensure SE $\le 1$ ? (c) What does this say about the trade-off between precision and sample size?
Step-by-step solution
1. Step 1
  (a) SE $= \sigma/\sqrt n = 10/\sqrt{25} = 10/5 = 2$ .
2. Step 2
  (b) Require $\sigma/\sqrt n \le 1$ :
  $\dfrac{10}{\sqrt n} \le 1 \Rightarrow \sqrt n \ge 10 \Rightarrow n \ge 100$
3. Step 3
  (c) SE shrinks as $1/\sqrt n$ . To halve the SE, $n$ must QUADRUPLE. So precision improves slowly with sample size; large samples are expensive.
Answer
(a) SE $= 2$ . (b) $n \ge 100$ . (c) SE $\propto 1/\sqrt n$ — halving SE requires quadrupling $n$ .
Examiner tip
(a) M1, A1. (b) M1 for inequality set-up, A1 for $n \ge 100$ . (c) B1 for the $1/\sqrt n$ insight. The quadratic relationship is a recurring theme — make sure students internalise it.
5CLT for non-normal parent (5 marks, S2)
Extended• Adapted from WST02/01 January 2024 Q8• sampling, clt, non-normal
Question
The time taken to complete a survey response is modelled by $X$ with mean $4$ minutes and SD $3$ minutes; the distribution of $X$ is heavily right-skewed (NOT normal). A random sample of $100$ responses is taken. Find the approximate probability that the SAMPLE MEAN response time exceeds $4.5$ minutes.
Step-by-step solution
1. Step 1
  Parent is non-normal but $n = 100$ is large. By the CLT,
  $\bar X \approx N\left(4, \dfrac{9}{100}\right) = N(4, 0.09)$
2. Step 2
  $\sigma_{\bar X} = \sqrt{0.09} = 0.3$ .
3. Step 3
  Standardise:
  $z = \dfrac{4.5 - 4}{0.3} = \dfrac{0.5}{0.3} \approx 1.667$
4. Step 4
  Probability:
  $P(\bar X > 4.5) \approx 1 - \Phi(1.67) \approx 1 - 0.9525 = 0.0475$
Answer
$P(\bar X > 4.5) \approx 0.0475$ .
Examiner tip
B1 for citing CLT. B1 for $\sigma_{\bar X} = 0.3$ . M1 for standardisation. A1 for $z \approx 1.67$ . A1 for $0.0475$ . The CLT works EVEN FOR NON-NORMAL parents — this is the key point. INDIVIDUAL values $X$ are skewed; the SAMPLE MEAN is approximately normal.

Key Formulae — Sampling and Sampling Distributions

The formulae you need to memorise for sampling and sampling distributions on the Pearson Edexcel IAL XMA01 / YMA01 paper, with every variable defined in plain English and a note on when to use it.

Sample mean
$\bar X = \dfrac{1}{n}\sum_{i=1}^n X_i$
$X_i$
$i$ -th observation in the sample
$n$
sample size
When to use
Standard estimator for population mean. NOT in formula booklet — definition.
Example
Sample of $30$ heights: $\bar X$ = arithmetic mean.
Central Limit Theorem
$X_1, \ldots, X_n \text{ iid, mean } \mu, \text{ variance } \sigma^{2}: \quad \bar X \approx N\left(\mu, \dfrac{\sigma^{2}}{n}\right) \text{ for large } n$
$\mu$
population mean
$\sigma^{2}$
population variance
$n$
sample size (large)
When to use
Whenever the question gives a sample mean and the sample size is moderate-to-large ( $n \ge 30$ , conservatively). NOT in the booklet — memorise.
Example
$X$ has $\mu = 1000, \sigma = 12$ ; $n = 36$ : $\bar X \approx N(1000, 4)$ .
Mean and variance of $\bar X$
$E(\bar X) = \mu, \quad \text{Var}(\bar X) = \dfrac{\sigma^{2}}{n}$
$\mu, \sigma^{2}$
population parameters
When to use
Holds for ANY sample size, NORMAL parent or not. The distribution shape may not be normal for small $n$ , but the mean / variance ARE these. IS in the IAL formula booklet.
Example
$X$ with $\mu = 50, \sigma = 10$ , $n = 25$ : $E(\bar X) = 50$ , $\text{Var}(\bar X) = 4$ .
Standard error
$\text{SE}(\bar X) = \dfrac{\sigma}{\sqrt n}$
$\text{SE}$
standard error
When to use
Measures precision of the sample mean as an estimator. To halve, quadruple $n$ .
Example
$\sigma = 10, n = 25$ : SE $= 2$ . $n = 100$ : SE $= 1$ .
Normal parent — exact distribution
$X_i \sim N(\mu, \sigma^{2}) \Rightarrow \bar X \sim N\left(\mu, \dfrac{\sigma^{2}}{n}\right) \text{ exactly}$
$X_i$
iid normal observations
When to use
If the parent is normal, the sample mean is EXACTLY normal — no need for CLT or large $n$ .
Example
$X \sim N(50, 100)$ , $n = 4$ : $\bar X \sim N(50, 25)$ exactly.

Key Definitions and Keywords — Sampling and Sampling Distributions

Definitions to memorise and the exact keywords mark schemes credit for sampling and sampling distributions answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IAL XMA01 / YMA01 sitting.

Population
Examiner keyword
The complete set of items / individuals about which a statistical question is being asked.
Sample
Examiner keyword
A subset of the population, chosen so that conclusions about the sample can be extended (with uncertainty) to the population.
Simple random sample (SRS)
Examiner keyword
A sample chosen so that every subset of size $n$ has equal probability of selection. Equivalently, every individual is equally likely to be chosen.
Statistic
Examiner keyword
A quantity computed from the sample (e.g. sample mean $\bar X$ , sample variance $s^{2}$ ). Treated as a random variable in its own right.
Sampling distribution
Examiner keyword
The probability distribution of a statistic across all possible samples of size $n$ .
Central Limit Theorem (CLT)
Examiner keyword
For iid $X_i$ with mean $\mu$ and variance $\sigma^{2}$ , the sample mean $\bar X$ is approximately $N(\mu, \sigma^{2}/n)$ for large $n$ , regardless of the parent distribution.
Standard error
Examiner keyword
SD of a statistic. For sample mean: SE $(\bar X) = \sigma/\sqrt n$ . Measures sampling variability.
Unbiased estimator
Examiner keyword
Statistic whose expected value equals the parameter being estimated. $\bar X$ is unbiased for $\mu$ because $E(\bar X) = \mu$ .

Common Mistakes and Misconceptions — Sampling and Sampling Distributions

The traps other students keep falling into on sampling and sampling distributions questions — taken from recent Pearson Edexcel IAL XMA01 / YMA01 examiner reports and mark schemes — and how to avoid them.

✕Using $\sigma$ instead of $\sigma/\sqrt n$ when standardising $\bar X$
WST02/01 examiner reports — top annual error
Why it happens
Students forget that $\bar X$ has a different SD from individual $X$ .
How to avoid it
Always write ' $\bar X$ has variance $\sigma^{2}/n$ ' as the first line. Use SE $= \sigma/\sqrt n$ in the $z$ -formula.
✕Stating 'by CLT' when the parent is normal
Why it happens
Students apply CLT habitually.
How to avoid it
If the parent is normal, $\bar X$ is EXACTLY normal — no approximation needed, NO 'by CLT'. Just write ' $X$ is normal, so $\bar X$ is normal'.
✕Invoking CLT for very small $n$ (e.g. $n = 4$ ) from a skewed parent
Why it happens
Misremembering the rule.
How to avoid it
CLT is reliable for $n \ge 30$ (rough rule). For smaller $n$ , only the EXACT distribution (or normal parent) works. Edexcel mark schemes use $n \ge 30$ as the threshold.
✕Reporting variance when SE was asked, or vice versa
Why it happens
Confusion between $\sigma^{2}/n$ and $\sigma/\sqrt n$ .
How to avoid it
VARIANCE of $\bar X$ : $\sigma^{2}/n$ (dimensions of $X^{2}$ ). STANDARD ERROR: $\sigma/\sqrt n$ (dimensions of $X$ ). Both are useful; know which the question asks for.
✕Vague description of simple random sampling
WST02/01 examiner reports
Why it happens
Students give the spirit, not the method.
How to avoid it
Include: numbering the population, using a random number generator, handling duplicates / out-of-range, continuing until $n$ chosen. Four explicit steps.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

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Sampling and Sampling Distributions — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Sampling and Sampling Distributions

Short Study Notes in the form of Slides

Detailed Notes

What you’ll learn

How it’s examined

1Population, sample, statistic (4 marks, S2)

2Simple random sample procedure (5 marks, S2)

3Sampling distribution of Xˉ\bar XXˉ (6 marks, S2)

4Effect of sample size on SE (5 marks, S2)

5CLT for non-normal parent (5 marks, S2)

Sample mean

Central Limit Theorem

Mean and variance of Xˉ\bar XXˉ

Standard error

Normal parent — exact distribution

Population

Sample

Simple random sample (SRS)

Statistic

Sampling distribution

Central Limit Theorem (CLT)

Standard error

Unbiased estimator

✕Using σ\sigmaσ instead of σ/n\sigma/\sqrt nσ/n​ when standardising Xˉ\bar XXˉ

✕Stating 'by CLT' when the parent is normal

✕Invoking CLT for very small nnn (e.g. n=4n = 4n=4) from a skewed parent

✕Reporting variance when SE was asked, or vice versa

✕Vague description of simple random sampling

Practice questions

Past paper style quiz

Sampling and Sampling Distributions — frequently asked questions

3Sampling distribution of $\bar X$ (6 marks, S2)

Mean and variance of $\bar X$

✕Using $\sigma$ instead of $\sigma/\sqrt n$ when standardising $\bar X$

✕Invoking CLT for very small $n$ (e.g. $n = 4$ ) from a skewed parent