Actuarial Statistics · Random sampling and sampling distributions

Population, Sample and Random Sampling in Actuarial Statistics

Updated 11 October 2026 · Fact-checked

A population is the full set of items you want to study. A sample is the part you observe. A parameter describes the population; a statistic is computed from the sample. A random sample X₁,…,Xₙ is iid: independent and identically distributed. Inference uses the statistic to learn about the parameter.

Understand Population, Sample and Random Sampling

A population is the whole group you care about. It can be real, such as all motor policies of an insurer. It can also be a model, such as the distribution of claim sizes that could arise. You rarely see all of it.

A sample is the part of the population you actually observe. You use it to learn about the population. In actuarial statistics, a sample of size n is written as random variables X₁, X₂, …, Xₙ before data is collected. After collection you get the observed values x₁, …, xₙ. Capital letters are random; small letters are fixed numbers.

A parameter is a fixed, usually unknown number describing the population, such as the mean μ, variance σ² or a rate λ. A statistic is any function of the sample that does not contain unknown parameters, such as the sample mean X̄. A statistic is a random variable, because it changes from sample to sample. Its distribution is called a sampling distribution.

A simple random sample gives every possible sample of the same size an equal chance of selection. Sampling with replacement means an item can be picked more than once, so draws are independent. Sampling without replacement means each item can be picked once, so draws are dependent, though they are identically distributed.

A random sample in the IAI sense means X₁,…,Xₙ are iid: independent, and each has the same distribution as the population variable X. This is the basis for almost all inference in CS1. If the population is very large compared with n, sampling without replacement is close to iid, so the iid model is a good approximation.

Key rules to remember

iid random sample: X₁, …, Xₙ iid with the distribution of X; Independent and identically distributed. Each Xᵢ has the same mean μ and variance σ² as the population.
Joint density of an iid sample: f(x₁, …, xₙ) = f(x₁) × f(x₂) × … × f(xₙ); Uses independence. It is the starting point for the likelihood function.
Sample mean: X̄ = (1/n) Σ Xᵢ; A statistic. Its observed value is x̄.
Sample variance: S² = Σ (Xᵢ − X̄)² ÷ (n − 1); A statistic with divisor n − 1. It is an unbiased estimator of σ² for an iid sample.
Mean and variance of X̄ for an iid sample: E(X̄) = μ and Var(X̄) = σ² ÷ n; Needs independence and finite variance. Standard error of X̄ is σ ÷ √n.
Number of samples of size n: With replacement: Nⁿ ordered samples. Without replacement: N!/(n!(N − n)!) unordered samples; N is the population size. Each is equally likely under simple random sampling.

How to solve Population, Sample and Random Sampling questions

Use this method for any question on populations, samples, statistics and random sampling.

1Identify the population and the random variable X that describes it. Say what the parameters are.
2Write down the sample as X₁,…,Xₙ and state whether it is random (iid) or not. Check the sampling method: with or without replacement.
3Decide whether each quantity is a parameter or a statistic. A parameter is fixed and unknown. A statistic is computed from the sample alone.
4If independence holds, write the joint distribution as the product of the individual densities or probabilities.
5Use E(X̄) = μ and Var(X̄) = σ²/n where needed. Check that the conditions hold.
6For small finite populations, list or count the possible samples and give each equal probability.
7Compute the answer. Show the formula, the substitution and the result.
8State your assumptions and give a one-line conclusion in context.

Quickest way: Parameter or statistic, and iid check

When to use it: Use this for multiple-choice questions that ask you to classify a quantity or judge whether a sample is random.

Ask: does it use sample data only and have no unknown parameters in it? If yes, it is a statistic.
Ask: is it a fixed property of the whole population? If yes, it is a parameter.
For iid, check two things: same distribution for all draws, and no draw affecting another.
With replacement means iid. Without replacement means not independent, unless N is very large compared with n.
For counting samples, decide ordered or unordered before you calculate.

Common mistakes in Population, Sample and Random Sampling

Calling X̄ a parameter or μ a statistic.
Both are averages, so they look alike.
Fix: μ belongs to the population and is fixed. X̄ comes from the sample and varies. Ask which one changes if you redraw the sample.
Treating draws without replacement as independent.
Students memorise iid and apply it everywhere.
Fix: Without replacement, the draws are dependent. They are identically distributed but not independent. The iid model is only an approximation when N is much larger than n.
Using a statistic that contains an unknown parameter, such as Σ(Xᵢ − μ)² with μ unknown.
Students forget the definition of a statistic.
Fix: A statistic must be calculable from the data alone. If μ is unknown, the quantity is not a statistic.
Mixing X and x, so that X̄ is treated as a fixed number.
Notation is ignored or written carelessly.
Fix: Use capitals for random variables and small letters for observed values. Only x̄ is a number.
Dividing the sample variance by n and calling it the standard S².
It copies the population variance formula.
Fix: The IAI sample variance uses n − 1. Divisor n gives a biased estimator of σ².
Assuming a convenience sample is a random sample.
Students think any sample that is large is random.
Fix: Randomness comes from the selection method, not the size. Selection must give equal chances and avoid bias.

Worked examples

Example 1

A population has N = 4 values: 2, 4, 6, 8. A simple random sample of size 2 is drawn without replacement. (a) How many different unordered samples are there? (b) Find the probability that the sample mean is 5. (c) Find the mean of the population, which is a parameter.

Show the solution

(a) Number of unordered samples = 4!/(2! × 2!) = 6.
List them: (2,4), (2,6), (2,8), (4,6), (4,8), (6,8). Each has probability 1/6.
(b) Sample means: 3, 4, 5, 5, 6, 7.
The mean 5 arises from (2,8) and (4,6). So the probability is 2/6 = 1/3.
(c) Population mean μ = (2 + 4 + 6 + 8) ÷ 4 = 20 ÷ 4 = 5.
Check: the average of the six sample means is (3 + 4 + 5 + 5 + 6 + 7) ÷ 6 = 30 ÷ 6 = 5, which equals μ.

Answer: (a) 6 samples. (b) 1/3. (c) μ = 5.

Example 2

Claim sizes X are modelled as an iid sample X₁,…,X₁₀₀ with mean μ = ₹40,000 and standard deviation σ = ₹12,000. Find E(X̄) and the standard error of X̄. Also write the joint density of the sample if X ~ Exponential with rate λ.

Show the solution

The sample is iid, so E(X̄) = μ = ₹40,000.
Var(X̄) = σ² ÷ n = 12,000² ÷ 100 = 144,000,000 ÷ 100 = 1,440,000.
Standard error = √1,440,000 = ₹1,200. This equals σ ÷ √n = 12,000 ÷ 10.
For an Exponential(λ) variable, f(x) = λe^(−λx) for x > 0.
By independence, f(x₁,…,x₁₀₀) = Π λe^(−λxᵢ) = λ¹⁰⁰ e^(−λ Σxᵢ), for all xᵢ > 0.

Answer: E(X̄) = ₹40,000. Standard error = ₹1,200. Joint density = λ¹⁰⁰ e^(−λΣxᵢ) for xᵢ > 0.

Exam tips

In MCQs, classify each quantity as parameter or statistic first. Check whether any unknown parameter appears in it.
Always state the iid assumption in written answers. Marks are often given for stating it.
Keep capital and small letters correct. Examiners notice X̄ versus x̄.
For finite populations, decide ordered or unordered and with or without replacement before counting.
In Paper B, a sample in R is a vector of draws. Say clearly what the population distribution is and what statistic you compute.

Practice questions from Random sampling and sampling distributions

Population, Sample and Random Sampling in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

ACCA Applied KnowledgeRandom Sampling Methods

Population, Sample and Random Sampling: frequently asked questions

What is the difference between a parameter and a statistic?

A parameter is a fixed, usually unknown number that describes the population, such as μ or σ². A statistic is a function of the sample data, such as X̄ or S². A statistic is random before data is collected and has its own sampling distribution.

What is the difference between sampling with and without replacement?

With replacement, each item can be selected again, so the draws are independent and identically distributed. Without replacement, each item is selected at most once, so the draws are dependent. When the population is much larger than the sample, the difference is small.

What does iid mean in actuarial statistics?

It means independent and identically distributed. Each observation has the same distribution as the population variable, and no observation affects another. This lets you write the joint density as a product.

Why is the sample variance divided by n − 1?

Dividing by n − 1 makes S² an unbiased estimator of σ² for an iid sample. The sample mean is used in the calculation, which uses up one degree of freedom.