IB Maths AI HLRandom VariablesPaper 1 & 2~6 min read
Unbiased Estimates
You rarely measure a whole population — you take a sample and use it to estimate the population’s mean and variance. The sample mean x̄ does this job perfectly: it’s an unbiased estimate of the population mean. The sample variance sn2 does not — it systematically undershoots, so you scale it up by nn − 1 to get the unbiased estimate sn−12. That one correction is the whole topic.
📘 What you need to know
Estimator = a random variable used to estimate a population parameter; the estimate is the value it gives from a sample.
Unbiased means the estimator’s expected value equals the population parameter.
Sample mean is unbiased: x̄ = ∑xn estimates the population mean μ.
Sample variance sn2 is biased — it underestimates the population variance σ2.
Unbiased variance (in the booklet): sn−12 = nn − 1sn2.
Dividing by n → biased; dividing by n − 1 → unbiased.
sn−1 is NOT an unbiased estimate of the standard deviation — you won’t be asked to find one.
Estimators & bias
An estimator is a recipe (a random variable) for guessing a population parameter from a sample. Run it on a sample and out comes an estimate. An estimator is unbiased if, on average across all possible samples, it lands exactly on the true parameter.
Sample mean
unbiased
E(X̄) = μ. On average it hits the population mean dead on.
Sample variance sₙ²
biased
E(Sn2) = n−1nσ2 < σ2 — it undershoots.
🤔 Why does the sample variance undershoot?
The sample variance measures spread around the sample mean x̄, not the true mean μ. The sample mean sits right in the middle of your data, so the data is always a little closer to x̄ than to μ. That makes sn2 a touch too small — and the bigger the sample, the smaller the gap.
The formulae
A sample of n values gives unbiased estimates of both the population mean and variance.
Unbiased estimate of the meanx̄ = ∑xnthe sample mean — already unbiased ✓
Unbiased estimate of the variancesn−12 = nn − 1sn2
=
∑(x − x̄)2n − 1in the formula booklet ✓
🧭 Recipe — finding the unbiased estimates
Mean: divide the data total by n → x̄ = ∑xn. That’s it — no correction needed.
Variance: if you’re given the sample variance sn2, multiply by nn − 1.
Read the wording first — make sure what you’re given is sn2, not σ2 or sn−12 already.
🧠 Memory aid — “n−1 on the bottom = unbiased”
Whenever the divisor is n − 1 you have the unbiased estimate; a plain n gives the biased one. The nn − 1 multiplier is always bigger than 1, which makes sense — you’re boosting the too-small sample variance up to size.
Which quantity has the question given you?
Always check the wording to see which of the three you’ve been handed.
Standard deviation & notation
One catch: even though sn−12 is an unbiased estimate of the variance, its square root sn−1 is not an unbiased estimate of the standard deviation. There’s no single formula that fixes this for all populations, so the exam never asks for it — always work with the unbiased variance.
Calculator notation warning: different calculators label the unbiased estimator differently — you may see sn−12, σn−12, s2 or ŝ2 (and their square roots). Know which button on your calculator gives the n − 1 version.
Two proofs (understanding only — not for memorising): using the linear-combination rules, E(X̄) = μ+μ+…+μn = μ, so the sample mean is unbiased. And E(Sn2) works out to n−1nσ2 ≠ σ2, which is exactly why multiplying by nn−1 repairs it.
Worked examples
WE 1
Unbiased mean and variance from a sample
The times, X minutes, of daily revision by 50 IB students are summarised by n = 50, ∑x = 6174, sn2 = 1384.3. Find unbiased estimates of the population mean and variance.
mean: x̄ = ∑x / n= 6174 / 50 = 123.48x̄ ≈ 123 minutes (3sf)variance: sₙ₋₁² = n/(n−1) × sₙ²= 50/49 × 1384.3 = 1412.55…sₙ₋₁² ≈ 1410 minutes² (3sf)mean needs no correction; variance is scaled up by 50/49.
WE 2
Correcting a sample variance
A sample of n = 20 has sample variance sn2 = 38. Find the unbiased estimate of the population variance.
sₙ₋₁² = n/(n−1) × sₙ²= 20/19 × 38= 1.0526… × 38 = 40sₙ₋₁² = 40the unbiased estimate is always a little larger than sₙ².
WE 3
Estimating from raw data
A sample of 5 values is 4, 7, 8, 10, 11. Find unbiased estimates of the population mean and variance.
A question states “the population variance is σ2 = 64.” Another states “the sample variance is sn2 = 64 for n = 16.” For each, give the unbiased estimate of the population variance.
case 1: already population varianceσ² = 64 → no estimate needed, it’s exact64case 2: sample variance → correct itsₙ₋₁² = 16/15 × 64 = 68.27…≈ 68.3 (3sf)only the SAMPLE variance needs the n/(n−1) boost.
WE 5
Working from ∑x and ∑x²
For a sample of n = 8, ∑x = 96 and ∑x2 = 1228. Find unbiased estimates of the mean and variance.
mean: x̄ = ∑x / n= 96 / 8 = 12x̄ = 12sₙ² = ∑x²/n − x̄²= 1228/8 − 12² = 153.5 − 144 = 9.5sₙ₋₁² = n/(n−1) × sₙ²= 8/7 × 9.5 = 10.857…sₙ₋₁² ≈ 10.9 (3sf)find the biased sₙ² first, then scale it up by 8/7.
💡 Top tips
Sample mean = unbiased straight away: x̄ = ∑xn, no correction.
Sample variance needs ×nn − 1 to become the unbiased estimate.
n − 1 on the bottom = unbiased; plain n = biased.
Read the wording: is it σ2, sn2 or sn−12? Only sn2 needs correcting.
Know your calculator’s button — there are several notations for the same thing.
Work with variance, not SD — the unbiased SD has no general formula.
⚠ Common mistakes
Reporting sn2 as the answer when an unbiased estimate was asked for — forgetting the nn−1 step.
Correcting σ2 — the population variance is already exact and needs no scaling.
Square-rootingsn−12 and calling it the unbiased standard deviation. It isn’t.
Dividing by n when finding the unbiased variance from raw data — divide by n − 1.
Multiplying by n−1n (upside down) — the unbiased estimate is larger, so use nn−1.
Mixing up the calculator outputs for sn and sn−1.
That wraps up the Random Variables unit! You can now transform and combine variables with the linear-combination rules, and use those same rules to produce unbiased estimates of a population’s mean and variance. Next this thinking feeds into hypothesis testing and confidence intervals, where the unbiased variance becomes the engine behind t-tests and interval estimates.
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