Sample means and sampling distributions: VCE Specialist Mathematics Units 3 & 4

1. One sample produces one mean

A population variable X describes a single randomly selected observation. A sample of size n contains observations X₁, X₂, …, Xₙ, and its sample mean is X̄ = (X₁ + ··· + Xₙ)/n. Before sampling, each observation and therefore their mean is random. After sampling, a particular numerical mean x̄ is observed. Uppercase X̄ denotes the random variable; lowercase x̄ denotes its realised value.

The sampling distribution of X̄ is the distribution of means that would arise from repeated samples of the same size, drawn by the same procedure from the same population. It is not a histogram of the observations in one sample. If each of 500 students draws 25 observations and reports one mean, a histogram of their 500 reported means estimates a sampling distribution for n = 25.

Keep three objects separate: an individual observation X, a total T, and a mean X̄. Their units may be similar, but their typical values and spreads differ. Questions about an average should not use the population standard deviation unchanged. Questions about a total should not divide the mean by n a second time. Defining the requested variable first prevents many later mistakes.

2. Derive the centre and spread

Assume the sample observations are independent and identically distributed, with population mean μ and finite variance σ². From linear combinations, the total has mean nμ and variance nσ². Multiplying the total by 1/n gives the following exact moment relationships:

The standard deviation of the sample mean is often called its standard error. It describes the variability of an estimator across repeated samples. The word “error” does not mean that the sample was collected incorrectly. Even perfectly collected random samples give different means because different observations are selected.

The sample mean is centred on the population mean: averaging does not systematically shift the centre. Increasing n concentrates its distribution. Multiplying the sample size by four divides the standard error by two; multiplying it by nine divides the standard error by three. This square-root relationship explains why a large gain in precision can require substantially more data.

These mean and variance formulas do not require a normal population. They do require the stated sampling model. Samples selected because they are convenient, or repeated measurements that share a source of dependence, may not behave like iid observations. In a finite population, sampling without replacement is not literally independent. Use the model supplied in the question rather than applying formulas indiscriminately to every list of numbers.

3. Separate exact normality from approximate normality

If the population is normal, X ∼ N(μ, σ²), every sample mean from n independent observations is exactly normal: X̄ ∼ N(μ, σ²/n). There is no minimum sample size for this exact result. It follows from the normal linear-combination rule, including the factor 1/n.

If the population is not normal, the central limit idea says that, for sufficiently large n under the usual iid finite-variance conditions, the sampling distribution of the mean becomes approximately normal. Its mean and variance remain exactly μ and σ²/n under that model; it is the normal shape that is approximate.

There is no universal sample size at which all populations suddenly produce a good normal approximation. Moderate asymmetry may settle quickly, while a population with rare extreme values can require much larger samples. A guideline such as n ≥ 30 is useful in many practical contexts, but it is not a mathematical guarantee. Examine the information provided about shape and use approximation language when warranted.

The original observations do not become normal as more are collected. Increasing the sample size changes the distribution of their average. A skewed population can stay skewed while its averages look nearly symmetric. This distinction is particularly important when explaining a simulation: the plotted variable must be labelled clearly.

4. Work backwards to a sample size

For a normal population with known σ, a target probability can be converted into a condition on n. Standardise the event, identify the required standard normal quantile, and solve the inequality. The sample size is a positive integer, so a lower bound must usually be rounded upward. Ordinary rounding to the nearest integer can leave the requested probability just below its target.

Use full precision for the quantile until the final integer is selected. After rounding upward, substitute both the chosen integer and the preceding integer into the original probability. This verifies that the answer works and, when required, that it is the smallest allowed sample size. A calculation about precision does not establish that a real sampling procedure is unbiased.

5. Simulate the distribution, not just one sample

To investigate sampling distributions, choose a population model, sample size n and repetition count R. Each repetition must generate a fresh sample, reset the running total, and store exactly one mean. A histogram of the R stored means can then be compared with the proposed normal distribution. Raising R makes the simulated picture more stable; raising n changes the distribution being investigated. These two roles must not be confused.

input n, R
means ← empty list
for r from 1 to R
    total ← 0
    for j from 1 to n
        x ← fresh independent draw from population
        total ← total + x
    end for
    append total / n to means
end for
output histogram, mean and standard deviation of means

Run the algorithm for a normal population, a uniform population on [0, 12], and a two-point population that equals 0 with probability 0.98 and 50 with probability 0.02. Compare n = 4, 30 and 100, keeping R large. The uniform model has mean 6 and standard deviation √12; the rare-event model has mean 1 and standard deviation 7. The formulas predict the centre and standard error even when the shape is visibly nonnormal.

The rare-event example explains the limitation of the “30 is enough” shortcut. At n = 30, the probability that every observation is zero is 0.98³⁰ ≈ 0.5455. More than half the sample means therefore equal exactly zero. A continuous symmetric bell curve is a poor description of that pronounced spike. Simulation complements, rather than replaces, this mathematical explanation.

6. Read evidence carefully

A simulated mean close to μ and a simulated standard deviation close to σ/√n support a correctly implemented experiment. They should not match exactly: simulation has random variation. Use a fixed seed when reproducibility matters, but do not choose a seed merely because it gives a prettier result. Increasing R reduces simulation noise without repairing a wrong sampling model.

When writing an exam solution, state which variable is distributed normally, include its parameters, identify whether the result is exact or approximate, and show the probability event before reporting a decimal. Keep units on the mean and standard error. For probabilities, four decimal places are often useful unless a different accuracy is requested; never round intermediate bounds so early that the final decision changes.