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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.
Before you revise SEAB labels sub-topic 6.4 Sampling as “for teaching and learning only”, but the core ideas (sample mean, E(Xˉ), Var(Xˉ), CLT) are exactly what you use inside 6.5 Hypothesis Testing. Recap mean/variance notation and get comfortable computing xˉ
and
s2
from summarised data.
Status: SEAB H2 Mathematics (9758, first exam 2026) syllabus last checked 2026-01-13 (PDF last modified 2024-10-16). Topic 6.4 is marked “for teaching and learning only” in the official syllabus; the sample-mean ideas feed Paper 2 Section B (Probability & Statistics, 60 marks) through Topic 6.5 Hypothesis Testing.
Sampling Language
A population has true mean μ and variance σ2.
A simple random sample of size n is one where every size-n subset is equally likely.
From a sample x1,…,xn, the key statistics are:
Sample mean xˉ=n1∑xi.
Unbiased sample variance s2=n−11∑(xi−xˉ)2
Common data issues (good to mention even when not explicitly tested): undercoverage, non-response, and measurement bias.
Distribution of the Sample Mean
If X1,…,Xn are independent with mean μ and variance σ2, then Xˉ=n1∑i=1nXi has
E(Xˉ)=μ
Var(Xˉ)=nσ2
If the parent population is normal, then Xˉ is exactly normal for any n.
By the Central Limit Theorem (CLT), for sufficiently large n (often n≥30), Xˉ is approximately N(μ,σ2/n)
Example -- Average revision hours
JC students have revision hours with μ=6.4, σ=1.8. For a random sample of n=36, find P(Xˉ>7).
Xˉ∼N(6.4,1.82/36)=N(6.4,0.09).
Standardise: z=0.37−6.4=2.0.
Probability P(Xˉ>7)=1−Φ(2.0)≈0.0228.
Example -- Sample size for desired precision (probability form)
How large should n be so that P(∣Xˉ−μ∣<0.5)≥0.95 when σ=2.4 and sampling is i.i.d.?
Round up to the next whole number: take n=90 observations.
Unbiased Estimates from Summarised Data
SEAB explicitly allows questions where the data are summarised as ∑x and ∑x2, or as ∑(x−a) and ∑(x−a)2 (to reduce calculator rounding).
xˉ=n∑x.
s2=n−11[∑x2−n(∑x)2].
Shifted form (given a): xˉ=a+n∑(x−a)
Example -- Using a shift
In a sample of n=40 students, the data are summarised as ∑(x−8)=16 and ∑(x−8)2=92. Find xˉ and s2.
xˉ=8+4016=8.4.
s2=391[92−40162]=3985.6=2.19 (3 s.f.).
Calculator Workflows
Use GC statistics mode (1-Var Stats) to obtain xˉ, s, ∑x, and ∑x2 (or the shifted sums if provided).
When Xˉ is modelled as normal, compute probabilities using normalcdf after converting to z-scores.
If the question gives a probability and asks for a bound (e.g. find a such that P(Xˉ>a)=0.05), use invNorm on the Xˉ
Exam Watch Points
Even though 6.4 is labelled “for teaching and learning only”, you are still expected to use E(Xˉ) and Var(Xˉ) when doing 6.5 Hypothesis Testing.
Write the model line explicitly: “Xˉ∼N(μ,σ2/n)” (normal population) or “approximately normal by CLT” (large n).
For summarised data, use the unbiased variance with divisor n−1 and show the substitution cleanly.
Keep rounding until the end; avoid premature rounding of s when it appears inside s/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 (μ vs Xˉ).
Compute Var(Xˉ) and standard errors without mixing up S and σ.
Convert probability statements about Xˉ into z-scores (and back) cleanly.
Compute xˉ and s2 from ∑x, ∑x2
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 σ and s: Using the population standard deviation σ in place of the unbiased sample standard deviation s (or vice versa) is a common error. If σ is unknown, use s2 with divisor n−1.
Forgetting to divide variance by n: Writing Var(Xˉ)=σ2 instead of Var(Xˉ)=σ2/n
Invoking CLT without stating “large n”: The CLT approximation requires n to be sufficiently large (typically n≥30); always write “by CLT, since n is large” before using the approximate normal distribution.
Premature rounding of s: Rounding the sample standard deviation s too early before substituting into s/n introduces compounding errors. Keep full precision until the final answer.
Using the biased formula ∑(xi−xˉ)2/n: SEAB requires the unbiased estimate with divisor n−1
Frequently asked questions
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(Xˉ)=μ, Var(Xˉ)=σ2/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 xˉ and s2 from raw data? Yes. Use 1-Var Stats on a TI or STAT mode on a Casio to obtain xˉ and sx directly. For summarised data (given ∑x and ∑x2), 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, Xˉ 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.
