The standard error of the sample mean is:

σ/√n. The variance of X̄ is σ²/n, so the standard deviation is σ/√n. Using σ itself ignores that averaging reduces variability.

The central limit theorem allows you to assume X̄ is approximately normal when:

n is large. The theorem applies for large n whatever the population shape. If the population is already normal, X̄ is exactly normal and the theorem is not needed.

The unbiased estimate of population variance divides by:

n − 1. Dividing by n underestimates, because deviations are taken from the sample mean. n − 1 corrects that bias exactly.

A population has σ = 20. For a sample of 25, the standard error is:

4. 20/√25 = 20/5 = 4. Dividing by 25 rather than 5 gives 0.8.

Increasing the confidence level from 95% to 99% makes the interval:

wider. A larger z is needed to capture more of the distribution, so the interval widens. More confidence costs precision.

To halve the width of a confidence interval you should multiply the sample size by:

4. The width is proportional to 1/√n, so n must be quadrupled for √n to double.

A 95% confidence interval of (10, 14) means:

95% of such intervals would contain μ. μ is a fixed number, so it is either inside this interval or not. The 95% refers to the long-run success rate of the procedure.

When standardising a sample mean, the denominator should be:

σ/√n. A sample mean varies less than a single observation, so the standard error σ/√n is the correct denominator. Using σ makes z too small by a factor of √n.

Sampling and Estimation — Mathematics Notes, Worked Examples and Quiz

Why sample at all

Definition

Random sample — A sample in which every member of the population is equally likely to be chosen, and the choices are independent of one another.

Measuring an entire population is usually impossible, ruinously expensive, or destructive — you cannot test every match in a factory by striking it. So a sample is taken and used to say something about the whole.

For that inference to be sound the sample must be random: every member of the population must have an equal chance of selection, and selections must be independent. A sample that is convenient rather than random is biased, and no amount of statistical machinery afterwards can repair it.

This is the reason opinion polls conducted only among people who volunteer answers are unreliable: those who choose to respond differ systematically from those who do not.

The distribution of the sample mean

Take a sample, compute its mean, and you get a number. Take another sample and you get a slightly different number. So the sample mean X̄ is itself a random variable, and it has its own distribution — the key object of the whole topic.

Two facts describe it. Its mean equals the population mean, so it is centred on the right answer. Its variance is σ²/n, so it is less spread out than the population, and increasingly so as the sample grows.

The central limit theorem supplies the third fact and is what makes the topic usable: for a large enough sample, X̄ is approximately normal regardless of the shape of the population. The population may be skewed, bimodal, or discrete; the distribution of its sample means still tends to a bell.

E(X̄) = μVar(X̄) = σ²/nso s.d.(X̄) = σ/√nfor large n:X̄ ~ N(μ, σ²/n)approximately(central limit theorem)the √n is why quadrupling the sample only halves the uncertainty

X̄: the sample meana random variable, different for each sample
σ/√n: the standard errorthe spread of the sample mean, not of the population
n large: usually n > 30not needed at all if the population is already normal

The sample mean is normally distributed about μ with standard deviation σ/√n. Because that spread shrinks as n grows, the same z-score corresponds to a much narrower band of actual values for a large sample than a small one.

σ/√n, not σ

When standardising a sample mean the denominator is the standard error σ/√n, not the population standard deviation. Using σ by itself treats one sample mean as though it were a single observation and produces a z-score that is too small by a factor of √n — which typically turns a significant result into a non-significant one.

Estimating what you do not know

The population mean is estimated by the sample mean, which needs no adjustment. The population variance is different: the obvious formula computed about the sample mean comes out systematically too small, because the sample mean sits, by construction, in the middle of its own data.

Dividing by n − 1 instead of n corrects exactly this. The result is the unbiased estimate s², meaning that over many samples its average equals the true population variance.

unbiased estimate of μ:x̄ = Σx / nunbiased estimate of σ²:s² = ( Σx² − (Σx)²/n ) / (n − 1)= ( Σ(x − x̄)² ) / (n − 1)divide by n − 1, never by n, when estimating a population variance

n − 1: the degrees of freedomone is used up by estimating the mean
s²: the unbiased estimateslightly larger than the sample variance
Σx²: the sum of squaressquare each value, then add — not the square of the sum

Confidence intervals

A single estimate is a point with no indication of how much to trust it. A confidence interval gives a range instead, built by taking the sample mean and reaching out a fixed number of standard errors either side.

The interpretation must be stated carefully, and examiners are strict about it. A 95% confidence interval does not mean there is a 95% probability that μ lies inside this particular interval — μ is a fixed number and is either in it or not. It means that if the procedure were repeated many times, 95% of the intervals produced would contain μ.

confidence interval for μ:x̄ ± z · σ/√n(σ known)x̄ ± z · s/√n(σ estimated, n large)z = 1.645for 90%z = 1.960for 95%z = 2.576for 99%higher confidence means a wider interval — certainty costs precision

z: the critical valuefrom the normal table for the stated confidence
σ/√n: the standard errorhalved by quadrupling the sample size
width: 2z·σ/√nnarrower with a larger sample, wider with more confidence

Worked example

A sample of 100 components has mean length 25.4 mm. The population standard deviation is known to be 2 mm. Find a 95% confidence interval for the population mean.

Standard error = σ/√n = 2/√100 = 0.2 mm.The spread of the sample mean, not of the individual components.
For 95% confidence, z = 1.960.Leaving 2.5% in each tail, so the table value is for 0.975.
Interval = 25.4 ± 1.960 × 0.2 = 25.4 ± 0.392.Reaching out just under two standard errors either side of the estimate.
= (25.0, 25.8) mm.Quoted as an interval, which is what the question asks for.
Interpretation: if this procedure were repeated many times, 95% of the intervals produced would contain the true mean.The correct phrasing. Saying "there is a 95% chance μ is in this interval" is the version that loses the mark.

(25.0, 25.8) mm

Points examiners return to

The standard error is σ/√n, and it shrinks only with the square root of the sample size.
Use n − 1 when estimating a population variance from a sample.
The central limit theorem is not needed if the population is already normal.
A wider interval means more confidence, not more accuracy.
Quadrupling n halves the width of the interval.
State the interpretation in terms of repeated sampling, never as a probability about this one interval.

How samples are actually taken

Questions ask you to name and evaluate a sampling method, so the common ones are worth distinguishing — particularly on the point of whether each produces a genuinely random sample.

In simple random sampling every member has an equal chance, usually via numbered lists and random numbers. Systematic sampling takes every kth member from a random start — quick, but it fails badly if the list has a repeating pattern of period k. Stratified sampling divides the population into groups and samples each in proportion, which improves accuracy when the groups genuinely differ. Opportunity sampling takes whoever is available, and is not random at all.

Method	How	Main weakness
Simple random	random numbers on a numbered list	needs a full list of the population
Systematic	every kth from a random start	breaks if the list has a period of k
Stratified	proportional shares of each group	group sizes must be known
Opportunity	whoever is to hand	not random — biased by who was there
Self-selected	volunteers respond	volunteers differ from non-volunteers

Larger is not the same as better

A biased sample does not improve by being bigger — it just measures the wrong thing more precisely. A survey of ten thousand volunteers is less trustworthy than a genuinely random sample of two hundred, because increasing n narrows the confidence interval around an answer that was systematically wrong to begin with.

Sampling and Estimation