H2 Maths Sampling Formula Sheet | CLT & Estimators

Study guide07 Oct 2025, 00:00 ZUpdated 17 Jul 2026

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...

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, $E(\bar{X})$ , $\operatorname{Var}(\bar{X})$ , CLT) are exactly what you use inside 6.5 Hypothesis Testing. Recap mean/variance notation and get comfortable computing $\bar{x}$

A sample is a small window into a population: Identify the population parameter.
The sample mean varies less when sample size is larger: Divide the variance by n for the sample mean.
CLT lets large samples behave approximately normal: State whether normality is exact or approximate.

Concrete example: A class sample of 36 students gives a more stable average than a sample of 4 because random highs and lows cancel more strongly.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 6.4 is marked "for teaching and learning only" in the official syllabus; the sample-mean ideas feed Paper 2 Section B (Probability and Statistics, 60 marks) through Topic 6.5 Hypothesis Testing.

Formulas at a glance

Every result the 9758 syllabus expects you to use, on one screen. The unbiased variance estimator $s^2$ is provided in MF27; the sample mean distribution, CLT statement, and standard error formula are not, so commit them to memory. Worked examples for each appear in the sections below.

Sample mean distribution

Quantity	Formula
Expected value of $\bar{X}$	$E(\bar{X}) = \mu$
Variance of $\bar{X}$

Central Limit Theorem

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

Unbiased estimators

Quantity	Formula
Sample mean (unbiased estimator of $\mu$ )	$\bar{x} = \dfrac{\sum x}{n}$

Sampling Language

A population has true mean $\mu$ and variance $\sigma^2$ .
A simple random sample of size $n$ is one where every size- $n$ subset is equally likely.
From a sample $x_1, \dots, x_n$

Statistic-choice checkpoint

Before substituting a formula, decide whether the question is about a random sample mean or about estimating spread from the observed data. The symbols look similar, but they answer different questions.

Question cue	Quantity to write	What it means	Common trap
"A random sample of size $n$ is taken. Find the distribution of $\bar{X}$ ."	$E(\bar{X}) = \mu$

Worked check: if $\sigma^2 = 25$ and $n = 100$ , then $\operatorname{Var}(\bar{X}) = 25/100 = 0.25$

Misconception check: $s^2$ estimates population variance after data are collected; $\sigma^2/n$ is the variance of the sample mean before observing the sample.

Distribution of the Sample Mean

If $X_1, \dots, X_n$ are independent with mean $\mu$ and variance $\sigma^2$

Sample-Mean Model Checkpoint

Before using normalcdf or a z-score, decide why $\bar{X}$ can be treated as normal. This prevents the common error of using the right formula with an unsupported model.

Given in the question	Model for $\bar{X}$	What to write before calculating
Population is normal	Exactly normal for any $n$	$\bar{X} \sim \mathcal{N}(\mu, \sigma^2/n)$

Common trap: $\sigma/\sqrt{n}$ is the standard deviation of the sample mean, not the variance. The variance is $\sigma^2/n$ .

Example -- Average revision hours

JC students have revision hours with $\mu = 6.4$ , $\sigma = 1.8$ . For a random sample of $n = 36$ , find $P(\bar{X} > 7)$

$\bar{X} \sim \mathcal{N}(6.4, 1.8^2 / 36) = \mathcal{N}(6.4, 0.09)$ .
Standardise: $z = \frac{7 - 6.4}{0.3} = 2.0$

Example -- Sample size for desired precision (probability form)

How large should $n$ be so that $P(\lvert \bar{X} - \mu \rvert \lt 0.5) \geq 0.95$ when $\sigma = 2.4$

Require $P\left( \lvert Z \rvert \lt \dfrac{0.5}{\sigma / \sqrt{n}} \right) \geq 0.95 \Rightarrow \dfrac{0.5}{2.4 / \sqrt{n}} \geq z_{0.975} = 1.96$

Probability-tail checkpoint

When a sampling question gives a probability statement and asks for a cut-off, mark the tail before using invNorm or a z-value. Most wrong answers use the correct standard error but choose the wrong side of the distribution.

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.	$a=\mu+z_{0.95}\dfrac{\sigma}{\sqrt{n}}$

Worked check: if $\bar{X}\sim\mathcal{N}(50, 4/25)$ and $P(\bar{X}\gt a)=0.05$

Misconception check: a small probability does not always mean a negative z-value. First decide whether the small area is on the left tail, right tail, or split across two tails.

Unbiased Estimates from Summarised Data

SEAB explicitly allows questions where the data are summarised as $\sum x$ and $\sum x^2$ , or as $\sum (x - a)$ and $\sum (x - a)^2$

$\bar{x} = \dfrac{\sum x}{n}$ .
$s^2 = \dfrac{1}{n - 1} \left[\sum x^2 - \dfrac{(\sum x)^2}{n}\right]$

Shifted-sum checkpoint

When a question gives $\sum (x-a)$ and $\sum (x-a)^2$ , do not try to reconstruct every raw $x$ value. Treat the shifted values as the working data, then add the shift back only for the mean.

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: $\dfrac{\sum (x-a)}{n}$

Worked check: if $a=50$ , $n=20$ , and $\sum(x-a)=12$ , then the average shifted value is $12/20=0.6$

Misconception check: shifting is a calculation shortcut, not a new data set with a different spread. Add $a$ back for the mean; do not add $a$ to $s^2$ .

Example -- Using a shift

In a sample of $n = 40$ students, the data are summarised as $\sum (x - 8) = 16$ and $\sum (x - 8)^2 = 92$

$\bar{x} = 8 + \dfrac{16}{40} = 8.4$ .
$s^2 = \dfrac{1}{39} \left[92 - \dfrac{16^2}{40}\right] = \dfrac{85.6}{39} = 2.19$

Calculator Workflows

Use GC statistics mode (1-Var Stats) to obtain $\bar{x}$ , $s$ , $\sum x$ , and $\sum x^2$ (or the shifted sums if provided).

Exam Watch Points

Even though 6.4 is labelled “for teaching and learning only”, you are still expected to use $E(\bar{X})$ and $\operatorname{Var}(\bar{X})$ when doing 6.5 Hypothesis Testing.
Write the model line explicitly: “ $\bar{X} \sim \mathcal{N}(\mu, \sigma^2/n)$

Practice Quiz

Apply sample-mean modelling, CLT standardisation, and unbiased-estimation workflows under exam pacing.

Quick Revision Checklist

Distinguish population vs sample parameters quickly ( $\mu$ vs $\bar{X}$ ).
Compute $\operatorname{Var}(\bar{X})$ and standard errors without mixing up $S$

Want weekly guided practice on Sampling? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

Mixing up $\sigma$ and $s$ : Using the population standard deviation $\sigma$ in place of the unbiased sample standard deviation $s$ (or vice versa) is a common error. If $\sigma$ is unknown, use $s^2$

Frequently asked questions

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 $s^2 = \dfrac{1}{n-1}\left[\sum x^2 - \dfrac{(\sum x)^2}{n}\right]$

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 - $E(\bar{X}) = \mu$ , $\operatorname{Var}(\bar{X}) = \sigma^2/n$

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 $s_x$

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.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities
Sequences & series: Sequences & Series
Vectors: Vectors
Calculus: Differentiation · Maclaurin Series · Integration Techniques · Definite Integrals · Differential Equations
Statistics: Probability · Discrete Random Variables · Normal Distribution · Sampling (this page) · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 6 Probability and statistics sub-topic 6.4 Sampling (marked "for teaching and learning only"; $\bar{X}$ as a random variable, CLT for large $n$ , unbiased estimates from $\sum x$ , $\sum x^2$