Estimators

AP Statistics· 12 min read

1. Introduction to Point Estimators★★☆☆☆⏱ 3 min

A point estimator is any sample statistic used to approximate an unknown fixed population parameter. For inference for proportions, the most common estimator is the sample proportion $\overset{p}{^}$ , which estimates the true population proportion $p$ . A single calculated value of an estimator from one specific sample is called a point estimate.

📘 Definition

Point Estimator

$\hat{θ}$ (general form)

A function of sample data that produces a single value intended to estimate an unknown fixed population parameter $θ$

Example:

Sample proportion $\overset{p}{^} = \frac{number of successes in sample}{sample size n}$ is the standard estimator for population proportion $p$ .

📐 Worked Example

Identify which of the following qualify as estimators for population proportion p: (A) Fixed sample size n, (B) Raw count of successes X in sample, (C) Sample proportion X/n

1
First, eliminate any values that are not functions of random sample outcomes designed to estimate p
2
Sample size n is fixed by study design, not a function of sample results, so it is not an estimator
3
Raw count X does not normalize by sample size, so it cannot estimate p directly, so it is not an estimator
4
Sample proportion X/n is a function of sample data designed to approximate p, so it is the only valid estimator here

✓ Quick check

Test your basic understanding of estimator definitions

Which of the following is a valid estimator for population mean $μ$ ?
- A) Population standard deviation $σ$
- B) Sample mean $\overset{x}{ˉ}$
- C) Fixed sample size n
Reveal answer
B —
Sample mean is calculated from sample data to estimate the unknown population mean, making it a valid point estimator.

2. Unbiasedness of Estimators★★★☆☆⏱ 4 min

An estimator is defined as unbiased if the expected value of the sampling distribution of the estimator equals the true value of the population parameter it is estimating. This means that the average value of the estimator across all possible random samples of the same size will exactly match the population parameter, with no systematic over- or under-estimation.

E[\hat{\theta}] = \theta

🔬 Derivation

Goal:

Prove that sample proportion $\overset{p}{^}$ is an unbiased estimator of population proportion p

Starting from:

We know the count of successes X in a binomial sample has expectation E[X] = np

1
The sample proportion is defined as $\overset{p}{^} = X / n$
2
Apply linearity of expectation: $E [\overset{p}{^}] = E [X / n] = \frac{1}{n} E [X]$
3
Substitute E[X] = np: $E [\overset{p}{^}] = \frac{1}{n} \times n p = p$

Result:

The expected value of $\overset{p}{^}$ is exactly equal to p, so sample proportion is an unbiased estimator of population proportion.

📐 Worked Example

A student proposes a new estimator for population proportion p: $\overset{p}{^}_{a d j} = \frac{X + 1}{n + 2}$ . If the true p is 0.3 and n=10, calculate the bias of this new estimator.

1
Bias is defined as $E [\hat{θ}] - θ$ , so first calculate $E [\overset{p}{^}_{a d j}]$
2
$E [\overset{p}{^}_{a d j}] = E [\frac{X + 1}{n + 2}] = \frac{E [ X ] + 1}{n + 2} = \frac{n p + 1}{n + 2}$
3
Substitute n=10, p=0.3: $E [\overset{p}{^}_{a d j}] = \frac{10 * 0.3 + 1}{10 + 2} = \frac{4}{12} \approx 0.333$
4
Calculate bias: $0.333 - 0.3 = 0.033$ , so the estimator has a positive bias of ~0.033

3. Minimum Variance and Consistency★★★☆☆⏱ 3 min

Unbiasedness alone is not enough to make an estimator optimal. You also want the estimator to have low sampling variability, measured by the variance of its sampling distribution. An estimator that has the smallest possible variance among all unbiased estimators for a parameter is called a minimum variance unbiased estimator (MVUE).

An estimator is consistent if as the sample size n grows to infinity, the probability that the estimator is arbitrarily close to the true parameter approaches 1. All unbiased estimators for proportions are consistent, as increasing sample size reduces their standard error.

📐 Worked Example

Two unbiased estimators for p are proposed: Estimator A has variance $\frac{p ( 1 - p )}{n}$ , Estimator B has variance $\frac{2 p ( 1 - p )}{n}$ . Which estimator is better, and why?

1
Both estimators are unbiased, so we compare their variances to select the optimal one
2
Estimator A has half the variance of Estimator B for any fixed sample size n
3
This means Estimator A will produce estimates that are, on average, closer to the true p than Estimator B
4
Therefore Estimator A is the better minimum variance unbiased estimator

4. Estimator Selection for Proportion Inference★★☆☆☆⏱ 2 min

All confidence interval and hypothesis test procedures for population proportion on the AP exam use the sample proportion $\overset{p}{^}$ as the point estimator. You will never be asked to use an alternative estimator for standard inference questions, but you may be tested on why $\overset{p}{^}$ is chosen over other possible estimators.

📐 Worked Example

A researcher surveys 200 randomly selected US adults and finds 120 support a new public health policy. What is the correct point estimator for the true population proportion of support?

1
Identify the number of successes X = 120, sample size n = 200
2
Calculate the sample proportion: $\overset{p}{^} = 120/200 = 0.6$
3
This value is the unbiased minimum variance estimator for the true population proportion of support

5. Common Pitfalls

Wrong move:

Claiming all sample statistics are unbiased estimators

Why:

Many common statistics (e.g., sample range, sample standard deviation s) are biased estimators for their corresponding population parameters

Correct move:

Only confirm an estimator is unbiased if you can prove its expected value equals the target parameter

Wrong move:

Prioritizing low variance over unbiasedness when selecting an estimator

Why:

A biased estimator with very low variance will systematically produce wrong estimates, even if they are consistent

Correct move:

First confirm the estimator is unbiased, then select the one with the smallest variance

Wrong move:

Confusing a point estimate (single calculated value from a sample) with a point estimator (the general formula)

Why:

AP graders will deduct points if you refer to a specific 0.6 sample value as the estimator, rather than the formula X/n

Correct move:

Label the formula as the estimator, and the single calculated number as the point estimate

Wrong move:

Calculating bias as the difference between your sample point estimate and the true parameter

Why:

Bias is a property of the estimator's sampling distribution, not a single sample result

Correct move:

Bias is defined as the difference between the expected value of the estimator across all samples and the true parameter

6. Quick Reference Cheatsheet

Property	Definition	Requirement for $\overset{p}{^}$ (estimator of p)
Unbiasedness	$E [\hat{θ}] = θ$	Satisfied, $E [\overset{p}{^}] = p$
Minimum Variance	Smallest variance among all unbiased estimators	Satisfied for simple random samples
Consistency	Estimator converges to $θ$ as $n \to \infty$	Satisfied, standard error shrinks as n increases
Bias	$E [\hat{θ}] - θ$	Bias of $\overset{p}{^}$ = 0

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · MCQ Section
Identify unbiased estimator for proportion
2022 · Free Response Q2
Calculate bias of adjusted proportion estimator
2021 · MCQ Section
Compare variance of two unbiased estimators

What's Next

Mastering the properties of estimators is the critical foundation you need to build for all subsequent inference for proportions topics. The unbiasedness of $\overset{p}{^}$ is the core justification for why one-proportion z-confidence intervals and z-hypothesis tests produce valid, reliable results that do not systematically over- or under-estimate the true population proportion. You will see estimator properties tested directly in 1-2 multiple choice questions on almost every AP exam, and they are frequently required as justification steps in free response inference questions to earn full reasoning points. Next, you will apply these estimator properties to calculate the standard error of the sample proportion, then move on to constructing formal confidence intervals for population proportions.