Sampling Distributions for Sample Proportions

AP Statistics· Unit 6: Inference for Categorical Data: Proportions, Topic 6.2· 12 min read

1. Defining the Sampling Distribution of $\hat{p}$★★☆☆☆⏱ 10 min

It is critical to distinguish three distinct distribution types for categorical data: the population distribution (all individual outcomes for the full population), the sample distribution (outcomes from one single collected sample), and the sampling distribution of $\overset{p}{^}$ , which aggregates the sample proportion statistic across every possible random sample of the same fixed size.

📘 Definition

Sampling Distribution of Sample Proportion $\hat{p}$

$\hat{p} = rac{ ext{Number of successes in sample}}{n}$

The full probability distribution of the sample proportion values calculated from all possible simple random samples of identical size n drawn from the same defined population.

Example:

For a population of 2000 college students where 55% live on campus, all SRS of size 100 will produce $\overset{p}{^}$ values that form this sampling distribution.

📐 Worked Example

A small town has 1500 registered voters, 62% of whom support a local ballot measure. You take repeated SRS of size 75 voters. Explain what the sampling distribution of $\overset{p}{^}$ (proportion of measure supporters in samples of size 75) represents.

1
First, identify the population: all 1500 registered town voters, with true population proportion p = 0.62.
2
The sampling distribution describes every unique $\overset{p}{^}$ value you could get if you selected every possible SRS of size 75 from this 1500-person population.
3
It also quantifies the relative frequency (probability) of each possible $\overset{p}{^}$ value occurring across all those samples.

✓ Quick check

Confirm you can tell the three distribution types apart

Which distribution describes the values of a categorical variable for every individual member of the full population?
- Sampling distribution of $\overset{p}{^}$
- Population distribution
- Single sample distribution
Reveal answer
Population distribution —
The population distribution includes raw individual outcomes, not aggregated sample statistics.

2. Mean and Standard Deviation of the Sampling Distribution★★★☆☆⏱ 12 min

The sampling distribution of $\overset{p}{^}$ has two fixed, easily calculated parameters when the sample is selected randomly. The mean of the distribution is exactly equal to the true population proportion p, meaning $\overset{p}{^}$ is an unbiased estimator with no systematic tendency to over or underestimate p.

\mu_{\hat{p}} = p

\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}

🔬 Derivation

Goal:

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

Starting from:

X is a binomial random variable counting successes in n independent trials, with E[X] = np

1
Define the sample proportion as $\overset{p}{^} = \frac{X}{n}$
2
Apply linearity of expectation: $E [\overset{p}{^}] = E [\frac{X}{n}] = \frac{1}{n} E [X]$
3
Substitute the known binomial expectation: $\frac{1}{n} (n p) = p$

Result:

The expected value of $\overset{p}{^}$ is exactly equal to the population proportion p, confirming no estimation bias.

📐 Worked Example

A factory produces lightbulbs, 3% of which are defective. If you select an SRS of 400 bulbs, calculate the mean and standard deviation of the sampling distribution of $\overset{p}{^}$ (proportion of defective bulbs in the sample).

1
Identify given values: population proportion p = 0.03, sample size n = 400
2
Calculate the mean: $μ_{\overset{p}{^}} = p = 0.03$
3
Calculate standard deviation: $σ_{\overset{p}{^}} = \frac{0.03 ( 0.97 )}{400} \approx 0.0085$

3. Conditions for Normal Approximation★★★☆☆⏱ 10 min

To use the normal distribution to approximate the shape of the sampling distribution of $\overset{p}{^}$ , you must verify three non-negotiable conditions, all of which are explicitly graded on AP exam FRQs.

📘 Definition

Large Counts Condition

$n p \geq 10$ and $n (1 - p) \geq 10$

The requirement that the expected number of successes and failures in the sample are both at least 10, to ensure the discrete binomial distribution is well-approximated by a continuous normal curve.

📐 Worked Example

Determine if the sampling distribution of $\overset{p}{^}$ is approximately normal for a sample of size 25 drawn from a population where p=0.3.

1
Check Random: Assume the sample is a valid SRS as specified in the problem
2
Check 10% Condition: Assume population size is at least 250, so n=25 is less than 10% of the population
3
Check Large Counts: np = 25*0.3 = 7.5 < 10, n(1-p) = 17.5 ≥ 10
4
Conclusion: The large counts condition fails, so the sampling distribution is not approximately normal.

4. Calculating Probabilities for Sample Proportions★★★★☆⏱ 12 min

Once all three conditions are verified, you can treat the sampling distribution of $\overset{p}{^}$ as exactly normal, and use standard normal CDF calculations to find the probability of observing any range of $\overset{p}{^}$ values.

z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}

📐 Worked Example

Public health data shows 18% of US adults smoke. We take an SRS of 300 adults. Find the probability that the sample proportion of smokers is less than 14%.

1
Verify conditions: Random (SRS), 10% (300 < 10% of all US adults), Large Counts: np=54 ≥10, n(1-p)=246 ≥10. All conditions met.
2
Calculate parameters: $μ_{\overset{p}{^}} = 0.18$ , $σ_{\overset{p}{^}} = \frac{0.18 * 0.82}{300} \approx 0.0222$
3
Calculate z-score: $z = \frac{0.14 - 0.18}{0.0222} \approx - 1.80$
4
Find P(Z < -1.80) = 0.0359. The probability of observing a sample proportion below 14% is ~3.6%.

5. Common Pitfalls

Wrong move:

Using the sample $\overset{p}{^}$ instead of the true population p to calculate $σ_{\overset{p}{^}}$

Why:

The theoretical standard deviation of the sampling distribution is defined using the known population proportion, not an estimate from one sample.

Correct move:

Always use the given population p value when calculating the exact spread of the sampling distribution.

Wrong move:

Skipping condition checks before applying normal approximation

Why:

AP graders deduct 1-2 full points for proportion inference problems if you do not explicitly verify all three conditions.

Correct move:

State and confirm Random, 10%, and Large Counts conditions for every problem that uses normal approximation for $\overset{p}{^}$ .

Wrong move:

Using n instead of $n$ in the denominator for $σ_{\overset{p}{^}}$

Why:

This gives the standard deviation of the binomial count X, not the sample proportion $\overset{p}{^}$ , leading to drastically incorrect spread values.

Correct move:

Use the formula $p (1 - p) / n$ to get the correct standard deviation for the proportion distribution.

Wrong move:

Checking the 10% condition when sampling with replacement

Why:

Observations are fully independent when sampling with replacement, so the 10% rule is unnecessary and irrelevant.

Correct move:

Only verify the 10% condition when sampling without replacement from a finite population.

Wrong move:

Requiring observed sample successes and failures to be ≥10 instead of expected np and n(1-p)

Why:

The large counts condition uses expected values calculated from the population p, not observed counts from your specific sample.

Correct move:

Calculate np and n(1-p) using the given population proportion p, not your sample $\overset{p}{^}$ .

6. Quick Reference Cheatsheet

Parameter	Formula	Required Condition
Mean of $\overset{p}{^}$	$μ_{\overset{p}{^}} = p$	Sample is randomly selected
Standard Deviation of $\overset{p}{^}$	$σ_{\overset{p}{^}} = \frac{p ( 1 - p )}{n}$	n ≤ 10% of total population
Normal Approximation	$\overset{p}{^} \sim N (μ_{\overset{p}{^}}, σ_{\overset{p}{^}})$	$n p \geq 10$ , $n (1 - p) \geq 10$
Z-score for $\overset{p}{^}$	$z = \frac{p ^ - p}{p ( 1 - p ) / n}$	All 3 conditions verified

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 · Paper 1
MCQ on normality condition checks
2022 · Paper 2
FRQ sample proportion probability calculation
2021 · Paper 1
Standard deviation of $\overset{p}{^}$ calculation

What's Next

Mastering sampling distributions for sample proportions is the non-negotiable foundation for all subsequent categorical inference content, which makes up roughly 15% of your total AP Statistics exam score. You will reuse the exact conditions and parameter formulas you learned here to construct confidence intervals that estimate unknown population proportions, and run formal hypothesis tests to evaluate claims about population proportions. This topic also directly connects to sampling distributions for sample means, which you will apply for quantitative inference later in the course. Explicit condition verification is one of the most commonly missed grading points on AP FRQs, so prioritize practicing that step for every problem. Move on to the next topics to build your full proportion inference toolkit.