Recover unbiased estimates from coded sample data
Undo an affine coding for the estimated mean and square its scale for estimated variance.
Exam scope and assessment notes
Probability & Statistics 2 (Paper 6, A Level) · Mathematics 9709, 2026–2027, section 6.4. Paper 6 builds on Paper 5 and assumes Paper 3 calculus. Distinguish population parameters, sample statistics and repeated-sample random variables. Random selection needs a relevant frame and does not automatically remove undercoverage or non-response. Particular named sampling methods are not required. Sample-mean formulae here assume independent observations from one population with finite variance: E(X̄) = μ, Var(X̄) = σ²/n and SE = σ/√n. A mean of independent normal observations is normal; the CLT gives an approximate normal mean for an adequate large independent sample, not normal individual data. No universal sample-size cutoff guarantees that approximation. Unbiased sample variance uses n − 1, with n > 1. Mean intervals use the normal model with known variance or a large-sample approximation; a small sample with unknown variance does not justify the z method taught here. Proportion intervals use the large-sample normal approximation with adequate success and failure counts; extreme or sparse counts may invalidate it. Quoted two-sided critical values are 1.645 for 90%, 1.960 for 95% and 2.576 for 99%; retain other working unrounded. Confidence describes repeated-sampling coverage, not posterior probability for a fixed realised interval, individual observations or a guarantee. A narrower interval does not repair a biased sample. Hypothesis testing belongs to section 6.5. Written practice is self-checked; cumulative assessment and human subject-teacher review remain pending.
Before you begin
- Use summary-statistic variance estimates.
- Transform means and variances under fixed scaling.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
If U = (X − 100)/10, how is X recovered?
What you will learn
- Recover original-unit sample estimates.
- Distinguish coded sums from original measurements.
Compute in the coded units first
If U = (X − 100)/10, then X = 100 + 10U. For n = 5, Σu = 2 and Σu² = 6, ū = 0.4 and the centred squared sum is 6 − 4/5 = 5.2. The unbiased coded variance estimate is 5.2/4 = 1.3.
Recover the original mean as 100 + 10(0.4) = 104 and variance as 10²(1.3) = 130. Adding the fixed hundred affects the mean, while variance depends only on the squared scale.
Keep the coding equation explicit
For n = 6, Σu = −3 and Σu² = 15 with the same coding, ū = −0.5 and sᵤ² = (15 − 9/6)/5 = 2.7. Original mean is 95 and variance estimate 270.
If every original measurement is then corrected by adding five, the mean estimate rises by five and variance is unchanged. Random measurement error is a different situation; it cannot be treated as a fixed shift without an error model.
Pause and explain
What original variance follows from coded variance estimate 1.3?
Worked example
Recover mean and variance for the first coded summary.
Show the worked solution
- ū = 0.4 and sᵤ² = 1.3.
- Original mean = 100 + 10ū = 104.
- Original variance = 100sᵤ² = 130.
Answer Mean estimate 104; variance estimate 130.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For U = (X − 100)/10, n = 5, Σu = 2, Σu² = 6, recover unbiased mean and variance estimates.
Give me a hint
Calculate the coded estimates before reversing the scale.
Compare my reasoning
- ū = 0.4; sᵤ² = (6 − 4/5)/4 = 1.3.
- x̄ = 100 + 10(0.4) = 104.
- sₓ² = 100(1.3) = 130.
Mean 104; variance 130.
Look for these in your work
- Use n − 1 in coded variance.
- Square the reverse scale.
For the same coding, use n = 6, Σu = −3 and Σu² = 15.
Give me a hint
Keep the negative coded mean while computing a non-negative spread.
Compare my reasoning
- ū = −0.5.
- sᵤ² = (15 − 9/6)/5 = 2.7.
- Original estimates are 95 and 270.
Mean 95; variance 270.
Look for these in your work
- Retain the signed centre.
- Use squared scale for spread.
The first original sample is corrected by adding five to every value. What happens to its estimates?
Give me a hint
A fixed correction translates every deviation equally.
Compare my reasoning
- The mean rises from 104 to 109.
- Each centred deviation is unchanged.
- Variance estimate remains 130.
Mean 109; variance 130 unchanged.
Look for these in your work
- Distinguish fixed from random error.
- Translate centre without adding spread.
Common mistakes
- Undo the stated coding, not a guessed scale.
- Square the variance multiplier.
- A fixed shift adds no spread.
What can you explain now?
Does the coding shift hundred contribute variance?
Compare with the explanation
No.
Only the squared multiplicative scale changes variance.
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Make it stick
Recover coded mean with the affine equation and coded variance with its squared scale.
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