Is there a formula sheet for H2 Maths sampling?

Yes - the "Formulas at a glance" section near the top of this page collects every result you need: the sample mean distribution, the Central Limit Theorem statement, the standard error formula, and both forms of the unbiased estimators (raw sums and shifted sums). Note that MF27 provides the unbiased variance estimator formula \$s^2 = \dfrac\{1\}\{n-1\}\left[\sum x^2 - \dfrac\{(\sum x)^2\}\{n\}\right]\$; however, the sample mean distribution \$\bar\{X\} \sim \mathcal\{N\}(\mu, \sigma^2/n)\$, the CLT approximation, and the standard error \$\sigma/\sqrt\{n\}\$ are not listed in MF27 and must be recalled from memory.

Is Topic 6.4 directly tested in the exam?

Topic 6.4 is labelled “for teaching and learning only” in the official syllabus, so standalone questions specifically on sampling theory are unlikely. However, the key ideas - \$E(\bar\{X\}) = \mu\$, \$\operatorname\{Var\}(\bar\{X\}) = \sigma^2/n\$, and unbiased estimation - appear inside Topic 6.5 Hypothesis Testing questions, so you must know them well.

Can I use the GC to compute \$\bar\{x\}\$ and \$s^2\$ from raw data?

Yes. Use 1-Var Stats on a TI or STAT mode on a Casio to obtain \$\bar\{x\}\$ and \$s_x\$ directly. For summarised data (given \$\sum x\$ and \$\sum x^2\$), show the manual substitution formula in working alongside the GC result.

When do I need to use the CLT versus the exact normal distribution?

If the original population is stated to be normal, \$\bar\{X\}\$ is exactly normal for any sample size \$n\$. Use the CLT only when the population distribution is unknown or non-normal, and only when \$n\$ is large. Always state which case applies. ---

H2 Maths Sampling Formula Sheet | CLT & Estimators

Study guide07 Oct 2025, 00:00 ZUpdated 21 Aug 2026

Reviewed by

Marcus Pang·Managing Director (Maths)

H2 Maths sampling formula sheet: sample mean distribution, Central Limit Theorem, unbiased estimators of population mean and variance, and standard error - every key result on o...

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Q: What does H2 Maths Notes (JC 1-2): 6.4) Sampling cover?
A: Sample mean distributions, Central Limit Theorem (CLT), and unbiased-estimation workflows aligned with the H2 Maths 2026 syllabus.

Download: Get the H2 Maths Sampling formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Before you revise
SEAB labels sub-topic 6.4 Sampling as "for teaching and learning only", but the core ideas (sample mean, $E(\bar{X})$ , $\operatorname{Var}(\bar{X})$

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Quantity	Formula
Expected value of $\bar{X}$	$E(\bar{X}) = \mu$
Variance of $\bar{X}$	$\operatorname{Var}(\bar{X}) = \dfrac{\sigma^2}{n}$
Standard error	$\dfrac{\sigma}{\sqrt{n}}$
Exact distribution (normal population)	$\bar{X} \sim \mathcal{N}!\left(\mu, \dfrac{\sigma^2}{n}\right)$

Condition	Result
Large $n$ (typically $n \geq 30$ ), any population	$\bar{X} \approx \mathcal{N}!\left(\mu, \dfrac{\sigma^2}{n}\right)$ approximately

Quantity	Formula
Sample mean (unbiased estimator of $\mu$ )	$\bar{x} = \dfrac{\sum x}{n}$
Unbiased sample variance (estimator of $\sigma^2$ )	$s^2 = \dfrac{1}{n-1}\left[\sum x^2 - \dfrac{(\sum x)^2}{n}\right]$
Mean from shifted sums	$\bar{x} = a + \dfrac{\sum (x - a)}{n}$
Variance from shifted sums	$s^2 = \dfrac{1}{n-1}\left[\sum (x-a)^2 - \dfrac{(\sum (x-a))^2}{n}\right]$

Question cue	Quantity to write	What it means	Common trap
"A random sample of size $n$ is taken. Find the distribution of $\bar{X}$ ."	$E(\bar{X}) = \mu$ , $\operatorname{Var}(\bar{X}) = \dfrac{\sigma^2}{n}$	The sample mean is a random variable before the sample is observed.	Using $s^2$ when the population variance $\sigma^2$ is supplied.
"The sample has $\sum x$ and $\sum x^2$ . Find an unbiased estimate of the variance."	$s^2 = \dfrac{1}{n - 1}\left[\sum x^2 - \dfrac{(\sum x)^2}{n}\right]$
"Use the sample to test a claim about $\mu$ ."	Start with $\bar{x}$ , then use the sampling model requested by the question.	The observed mean becomes evidence for a population claim.	Treating $\bar{x}$ as the true population mean.

Given in the question	Model for $\bar{X}$	What to write before calculating
Population is normal	Exactly normal for any $n$	$\bar{X} \sim \mathcal{N}(\mu, \sigma^2/n)$ .
Population is not stated normal, but $n$ is large	Approximately normal by CLT	By CLT, $\bar{X} \approx \mathcal{N}(\mu, \sigma^2/n)$ .
Population shape is unknown and $n$ is small	Normal model is not justified from the given information	Do not force a normal approximation unless the question supplies another reason.

H2 Maths Sampling Formula Sheet | CLT & Estimators

Formulas at a glance

Sampling Language

Statistic-choice checkpoint

Distribution of the Sample Mean

Sample-Mean Model Checkpoint

Probability-tail checkpoint

Unbiased Estimates from Summarised Data

Shifted-sum checkpoint

Calculator Workflows

Exam Watch Points

Practice Quiz

Quick Revision Checklist

Common exam mistakes

Frequently asked questions

Other H2 Maths formula sheets

Sources

Probability statement	Tail to mark	Cut-off form	Common trap
$P(\bar{X} \gt a)=0.05$	Right tail is 0.05, so left area is 0.95.	$a=\mu+z_{0.95}\dfrac{\sigma}{\sqrt{n}}$	Using $z_{0.05}$ , which gives the lower 5 percent point.
$P(\bar{X} \lt a)=0.05$	Left tail is 0.05.	$a=\mu+z_{0.05}\dfrac{\sigma}{\sqrt{n}}$
$P(\lvert\bar{X}-\mu\rvert\lt d)=0.95$	Middle 0.95 leaves 0.025 in each tail.	$d=z_{0.975}\dfrac{\sigma}{\sqrt{n}}$

Step	What to do	Why it works	Common trap
1	Identify the shift $a$ .	The question has centred the data around $a$ to keep sums smaller.	Treating $a$ as the sample mean.
2	Find the shifted mean: $\dfrac{\sum (x-a)}{n}$ .	This is the average distance from $a$ .	Reporting this shifted mean as $\bar{x}$ .
3	Add back the shift: $\bar{x}=a+\dfrac{\sum (x-a)}{n}$ .
4	Use the shifted variance formula with divisor $n-1$ .	Variance is unchanged by shifting all values by the same constant.	Dividing by $n$ , or expanding to raw $\sum x^2$ unnecessarily.