Sampling Distribution
In short: Sampling distribution shows the pattern of values a statistic, such as an average or a proportion, takes across every possible sample of one fixed size from the same population.
AP StatisticsStatisticsAP Statistics Unit 2
What is Sampling Distribution?
A sampling distribution answers: “if I ran this survey over and over, which values of the statistic would show up?” It is not the histogram of one sample. It is the pattern of the statistic across all possible samples of size \(n\).
For a sample proportion, the center is \(\mu_{\hat{p}} = p\) and the spread is \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\); the shape is approximately normal when \(np \ge 10\) and \(n(1-p) \ge 10\). For a sample mean, the center is \(\mu\) and the spread is \(\frac{\sigma}{\sqrt{n}}\). Larger samples give narrower distributions.
How to spot it
- A question talks about all possible samples, or about repeating a survey many times.
- The task asks for the center and spread of p-hat or x-bar.
- A simulation drops dots on a plot, each dot being one sample’s statistic.
- The problem asks how likely it is to see a sample proportion or mean beyond a certain value.
The method
- Name the statistic (\(\hat{p}\) or \(\bar{x}\)) and the sample size \(n\).
- Compute its center: \(p\) for a proportion or \(\mu\) for a mean.
- Compute its spread: \(\sqrt{\frac{p(1-p)}{n}}\) or \(\frac{\sigma}{\sqrt{n}}\), after checking the conditions for shape.
- Use the normal curve, with \(z\)-scores, to find probabilities about the statistic.
Sampling Distribution: solved example
Problem
In a large school, 60% of students bike to campus. A club surveys random samples of 50 students. Describe the sampling distribution of the sample proportion \(\hat{p}\).
Solution
Center: \(\mu_{\hat{p}} = 0.60\).
Spread: \(\sigma_{\hat{p}} = \sqrt{\frac{0.60 \times 0.40}{50}} = \sqrt{0.0048} \approx 0.069\).
Shape: \(np = 30\) and \(n(1-p) = 20\) are both at least 10, so the distribution is approximately normal with mean 0.60 and standard deviation about 0.069.
Try it
Question 1
Test scores in a district have \(\mu = 72\) and \(\sigma = 12\). For samples of \(n = 36\) students, what are the center and spread of \(\bar{x}\)?
Show the answer
Mean 72; standard deviation \(\frac{12}{\sqrt{36}} = 2\).
Question 2
Suppose you quadruple the sample size. How does the spread of \(\bar{x}\) change?
Show the answer
It is cut in half, because \(\frac{\sigma}{\sqrt{4n}} = \frac{1}{2}\cdot\frac{\sigma}{\sqrt{n}}\).
Sampling Distribution vs. Standard Error
The sampling distribution is the whole pattern of values a statistic takes. The standard error is a number that measures its spread, the standard deviation of that pattern estimated from one sample. Think of the first as the picture and the second as the width of the picture.
Keep going
Lessons
Worksheets and quizzes
- Sampling Distributions and the Central Limit Theorem – Worksheet
- Sampling Distributions and the Central Limit Theorem – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics