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
Reviewed by
Marcus Pang·Managing Director (Maths)
and
s2
from summarised data.
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.
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.
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 Xˉ."
E(Xˉ)=μ, Var(Xˉ)=nσ2
The sample mean is a random variable before the sample is observed.
Using s2 when the population variance σ2 is supplied.
"The sample has ∑x and ∑x2. Find an unbiased estimate of the variance."
s2=n−11[∑x2−n(∑x)2]
"Use the sample to test a claim about μ."
Start with xˉ, then use the sampling model requested by the question.
The observed mean becomes evidence for a population claim.
Treating xˉ as the true population mean.
Worked check: if σ2=25 and n=100, then Var(Xˉ)=25/100=0.25. Do not divide by 99, because this is not an unbiased-variance estimate from raw sample data.
Misconception check: s2 estimates population variance after data are collected; σ2/n is the variance of the sample mean before observing the sample.
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)
Sample-Mean Model Checkpoint
Before using normalcdf or a z-score, decide why Xˉ 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 Xˉ
What to write before calculating
Population is normal
Exactly normal for any n
Xˉ∼N(μ,σ2/n).
Population is not stated normal, but n is large
Approximately normal by CLT
By CLT, Xˉ≈N(μ,σ2/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.
Common trap: σ/n is the standard deviation of the sample mean, not the variance. The variance is 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.
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(Xˉ>a)=0.05
Right tail is 0.05, so left area is 0.95.
a=μ+z0.95nσ
Using z0.05, which gives the lower 5 percent point.
P(Xˉ<a)=0.05
Left tail is 0.05.
a=μ+z0.05nσ
P(∣Xˉ−μ∣<d)=0.95
Middle 0.95 leaves 0.025 in each tail.
d=z0.975nσ
Worked check: if Xˉ∼N(50,4/25) and P(Xˉ>a)=0.05, the standard error is 2/5=0.4. The right-tail probability 0.05 means z=1.645, so a=50+1.645(0.4)≈50.7.
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 ∑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)
Shifted-sum checkpoint
When a question gives ∑(x−a) and ∑(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: n∑(x−a).
This is the average distance from a.
Reporting this shifted mean as xˉ.
3
Add back the shift: xˉ=a+n∑(x−a).
Every observation was written as
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 ∑x2 unnecessarily.
Worked check: if a=50, n=20, and ∑(x−a)=12, then the average shifted value is 12/20=0.6, so xˉ=50.6. The shift changes the mean calculation, but the spread calculation still uses deviations and the divisor n−1.
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 s2.
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.
