Actuarial Statistics · Estimators and their properties

Unbiasedness and Bias of Estimators: Definition and How to Prove It

Updated 11 October 2026 · Fact-checked

The bias of an estimator θ̂ is E(θ̂) − θ. If the bias is zero for every possible value of θ, the estimator is unbiased. To check it, find E(θ̂) using linearity of expectation, subtract θ, and simplify. If the bias tends to zero as n grows, the estimator is asymptotically unbiased.

Understand Unbiasedness and Bias of Estimators

An estimator is a rule, written as a function of the sample, that gives a guess for an unknown parameter θ. Because the sample is random, the estimator is itself a random variable. It has its own mean and variance.

The bias measures whether the estimator is right on average. It is defined as bias(θ̂) = E(θ̂) − θ. A positive bias means the estimator tends to overshoot. A negative bias means it tends to undershoot. If the bias is 0 for all θ, the estimator is unbiased.

Unbiased does not mean accurate in one sample. A single value can be far from θ. Unbiasedness only says that over many repeated samples the estimates average out to θ. Spread is measured separately by the variance.

Some estimators are biased for finite n but the bias shrinks to zero as n → ∞. These are asymptotically unbiased. The maximum likelihood estimator of σ² for a normal sample, with divisor n, is the standard example.

The classic results are these. The sample mean X̄ is unbiased for μ whenever the mean exists. The sample variance S² with divisor n − 1 is unbiased for σ² for independent identically distributed observations with finite variance. The divisor n version is biased downwards. Bias is also not preserved by non-linear functions: if θ̂ is unbiased for θ, then θ̂² is generally not unbiased for θ².

Key rules to remember

Bias: bias(θ̂) = E(θ̂) − θ; The expectation is taken over the sampling distribution of θ̂. θ is treated as fixed.
Unbiased estimator: E(θ̂) = θ for all θ in the parameter space; It must hold for every possible value of θ, not just one.
Asymptotically unbiased: E(θ̂ₙ) → θ as n → ∞; Equivalently, the bias tends to 0 as n → ∞.
Expectation of sample mean: E(X̄) = μ, Var(X̄) = σ² ÷ n; For i.i.d. observations with mean μ and variance σ². The variance result also needs independence.
Sample variance: S² = Σ(Xᵢ − X̄)² ÷ (n − 1), E(S²) = σ²; Uses i.i.d. observations with finite variance.
Divisor n version: E[Σ(Xᵢ − X̄)² ÷ n] = (n − 1)σ² ÷ n; Bias is −σ² ÷ n, which tends to 0, so it is asymptotically unbiased.
Key identity: Σ(Xᵢ − X̄)² = ΣXᵢ² − nX̄²; Used in the proof that S² is unbiased.
Variance and second moment: E(Y²) = Var(Y) + [E(Y)]²; Used to find E(Xᵢ²) and E(X̄²).

How to solve Unbiasedness and Bias of Estimators questions

Use this method for any question that asks you to show an estimator is unbiased or to find its bias.

1Write down the estimator clearly in terms of the sample X₁, ..., Xₙ, and state the assumptions (for example i.i.d. with mean μ and variance σ²).
2Apply linearity of expectation to find E(θ̂). This needs no independence.
3If squares appear, use E(Y²) = Var(Y) + [E(Y)]² for each term. Find E(Xᵢ²) = σ² + μ² and E(X̄²) = σ²÷n + μ².
4Simplify E(θ̂) fully as an expression in θ and n.
5Subtract θ to get the bias. If it is zero for all θ, say the estimator is unbiased.
6If the bias is not zero, state whether it depends on n and take the limit as n → ∞ to decide on asymptotic unbiasedness.
7If asked for a corrected estimator, multiply or adjust by a constant to remove the bias, and state it.

Quickest way: Compute E(θ̂) and compare with θ

When to use it: Use in multiple-choice questions or when the estimator is a linear combination of the observations.

For a linear estimator Σaᵢ Xᵢ with E(Xᵢ) = μ, the expectation is μ Σaᵢ. It is unbiased for μ only if Σaᵢ = 1.
For a variance estimator, remember that divisor n − 1 gives an unbiased result and divisor n gives a mean of (n − 1)σ² ÷ n.
For bias of a divisor n estimator, use −σ² ÷ n directly.
For a non-linear function such as θ̂², compute E(θ̂²) = Var(θ̂) + θ² and expect a bias equal to Var(θ̂) when θ̂ is unbiased.

Common mistakes in Unbiasedness and Bias of Estimators

Writing E(X̄²) = μ² when finding E(S²).
Students treat the expectation of a square as the square of the expectation.
Fix: Use E(X̄²) = Var(X̄) + μ² = σ²÷n + μ². The extra σ²÷n term is what produces the n − 1.
Saying an unbiased estimator is always close to the true value.
Unbiased sounds like correct.
Fix: Unbiased means correct on average only. A large variance can still give poor single estimates. Mention variance or mean squared error when comparing estimators.
Assuming the square root of an unbiased estimator is unbiased.
Students think unbiasedness carries through any function.
Fix: It holds only for linear functions. S² is unbiased for σ², but S is generally biased for σ. Use Jensen's inequality or direct calculation.
Getting the sign of the bias wrong.
Students compute θ − E(θ̂) instead.
Fix: Always write bias = E(θ̂) − θ. For the divisor n variance, the bias is −σ²÷n, so it underestimates.
Using only one value of θ to claim unbiasedness.
Students check E(θ̂) = θ at an example value.
Fix: Show E(θ̂) = θ as an algebraic identity in θ, valid for all permitted θ.
Confusing unbiased with asymptotically unbiased or consistent.
All three ideas involve behaviour near the true value.
Fix: Unbiased: E(θ̂) = θ for every n. Asymptotically unbiased: bias → 0. Consistent: θ̂ converges to θ in probability. They are separate properties.

Worked examples

Example 1

X₁, ..., Xₙ are independent with mean μ and variance σ². Show that S² = Σ(Xᵢ − X̄)² ÷ (n − 1) is an unbiased estimator of σ².

Show the solution

Use the identity Σ(Xᵢ − X̄)² = ΣXᵢ² − nX̄².
Find E(Xᵢ²) = Var(Xᵢ) + μ² = σ² + μ². So E(ΣXᵢ²) = n(σ² + μ²).
Find E(X̄²) = Var(X̄) + μ² = σ²÷n + μ². So E(nX̄²) = σ² + nμ².
Then E[Σ(Xᵢ − X̄)²] = nσ² + nμ² − σ² − nμ² = (n − 1)σ².
Divide by n − 1: E(S²) = (n − 1)σ² ÷ (n − 1) = σ².

Answer: E(S²) = σ² for all σ², so S² is unbiased for σ².

Example 2

A random sample of size 10 is taken from a distribution with variance σ². The estimator T = Σ(Xᵢ − X̄)² ÷ 10 is used for σ². Find the bias of T, and say whether T is asymptotically unbiased. If σ² = 50, find the numerical bias.

Show the solution

From the previous result, E[Σ(Xᵢ − X̄)²] = (n − 1)σ².
So E(T) = (n − 1)σ² ÷ n = 9σ² ÷ 10 for n = 10.
Bias = E(T) − σ² = 0.9σ² − σ² = −0.1σ² = −σ² ÷ n.
As n → ∞, −σ² ÷ n → 0, so T is asymptotically unbiased.
With σ² = 50, bias = −50 ÷ 10 = −5.

Answer: Bias = −σ²÷n, which is −5 when σ² = 50. T underestimates σ² on average and is asymptotically unbiased but not unbiased.

Exam tips

Show the identity Σ(Xᵢ − X̄)² = ΣXᵢ² − nX̄² and each expectation line by line. Marks are given for the method.
State assumptions at the start, for example independent and identically distributed with finite variance.
In MCQs on linear estimators, check whether the weights sum to 1 before anything else.
If an estimator is non-linear, such as a square or a square root, do not assume unbiasedness. Compute it or use the variance link.
Link bias to mean squared error when asked to compare estimators: MSE = Var + bias².

Practice questions from Estimators and their properties

Unbiasedness and Bias of Estimators: frequently asked questions

How do I prove that the sample variance is an unbiased estimator?

Expand Σ(Xᵢ − X̄)² as ΣXᵢ² − nX̄². Take expectations using E(Xᵢ²) = σ² + μ² and E(X̄²) = σ²÷n + μ². You get (n − 1)σ², so dividing by n − 1 gives σ².

What is the difference between a biased and an unbiased estimator?

An unbiased estimator has expected value equal to the parameter for every parameter value. A biased estimator has a non-zero difference E(θ̂) − θ. A biased estimator can still be useful if its variance is small.

Can you give an example of an asymptotically unbiased estimator?

The divisor n sample variance has expectation (n − 1)σ²÷n, so its bias is −σ²÷n. This tends to 0 as n grows. It is therefore asymptotically unbiased but not unbiased.

Is a biased estimator always worse than an unbiased one?

No. Mean squared error combines variance and squared bias. A slightly biased estimator with much smaller variance can have a lower MSE than an unbiased one.