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Estimator — Definition, Formula & Examples

An estimator is a rule or formula applied to sample data to produce a numerical guess (called an estimate) for an unknown population parameter. For example, the sample mean xˉ\bar{x} is an estimator of the population mean μ\mu.

An estimator is a statistic T=g(X1,X2,…,Xn)T = g(X_1, X_2, \dots, X_n)—a function of random variables drawn from a population—whose realized value serves as an estimate of a target parameter θ\theta. Key properties used to evaluate estimators include unbiasedness (E[T]=θE[T] = \theta), consistency, and efficiency.

Key Formula

θ^=g(X1,X2,…,Xn)\hat{\theta} = g(X_1, X_2, \dots, X_n)
Where:
  • θ^\hat{\theta} = The estimator (and its computed value, the estimate) of the parameter θ
  • X1,…,XnX_1, \dots, X_n = Random sample of size n from the population
  • gg = A function that maps sample data to a single numerical value

How It Works

You collect a random sample from a population whose parameter (such as μ\mu or pp) is unknown. You then apply the estimator's formula to the sample values to compute an estimate. Different estimators can target the same parameter; statisticians prefer estimators that are unbiased (their expected value equals the parameter) and have low variance. When an estimator is both unbiased and has the smallest possible variance among all unbiased estimators, it is called the minimum-variance unbiased estimator (MVUE).

Worked Example

Problem: A researcher measures the heights (in cm) of 5 randomly selected students: 162, 170, 168, 175, 165. Use the sample mean as an estimator of the population mean height.
Identify the estimator: The sample mean is the estimator for the population mean μ.
μ^=xˉ=1n∑i=1nxi\hat{\mu} = \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i
Compute the estimate: Substitute the observed values and the sample size n = 5.
xˉ=162+170+168+175+1655=8405=168\bar{x} = \frac{162 + 170 + 168 + 175 + 165}{5} = \frac{840}{5} = 168
Answer: The estimate of the population mean height is 168 cm.

Why It Matters

Every confidence interval and hypothesis test in AP Statistics relies on an estimator—such as xˉ\bar{x}, p^\hat{p}, or s2s^2—to draw conclusions about a population. Understanding what makes one estimator better than another is foundational in experimental design, quality control, and data science.

Common Mistakes

Mistake: Confusing the estimator with the estimate.
Correction: The estimator is the formula or rule (e.g., Xˉ\bar{X}, a random variable). The estimate is the specific number you get after plugging in observed data (e.g., xˉ=168\bar{x} = 168). The estimator varies from sample to sample; a single estimate does not.