Sampling Distributions

A **parameter** is a number that describes a whole **population**, for example the population mean $\mu$, the population standard deviation $\sigma$ or the population proportion $p$. A **statistic** is a number computed from a **sample**, for example the sample mean $\bar{x}$ or the sample proportion $\hat{p}$. The parameter is fixed (but usually unknown); the statistic changes from sample to sample. The **sampling distribution** of a statistic is the distribution of its values over all possible random samples of the same size $n$. ## Sampling distribution of the sample mean $\bar{x}$ - **Center:** $\mu_{\bar{x}} = \mu$ (sample means are centered on the population mean) - **Spread:** $\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}$, called the **standard error** - **Shape (central limit theorem):** if the population is normal, $\bar{x}$ is normal for any $n$. If not, $\bar{x}$ is still approximately normal when $n$ is large, usually $n \geq 30$. So a probability about a sample mean uses the z-score $$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$$ ## Sampling distribution of the sample proportion $\hat{p}$ - **Center:** $\mu_{\hat{p}} = p$ - **Spread:** $\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}$ - **Shape:** approximately normal when $np \geq 10$ and $n(1-p) \geq 10$ These formulas assume a random sample of independent values (in practice: the sample is less than about 10% of the population).

Statistic or Parameter?

A school has 1200 students. Their mean height is 165 cm. A class project measures a random sample of 50 students and finds a mean height of 163.2 cm. Which number is a parameter and which is a statistic?

Find the number that describes the whole population: 165 cm is the mean of all 1200 students = Parameter: $\mu = 165$ cm

Find the number that comes from the sample: 163.2 cm is the mean of the 50 measured students = Statistic: $\bar{x} = 163.2$ cm

Think about a second sample: Another random sample of 50 students would give a slightly different mean, for example 166.0 cm = $\bar{x}$ varies, $\mu$ stays fixed

Answer: $\mu = 165$ cm is the parameter, $\bar{x} = 163.2$ cm is a statistic. The statistic changes from sample to sample, which is exactly what a sampling distribution describes.

Center and Standard Error of the Sample Mean

Exam scores have mean $\mu = 70$ and standard deviation $\sigma = 15$. You take random samples of $n = 36$ students. Find the mean and the standard error of $\bar{x}$. What happens with $n = 100$?

Center of the sampling distribution: $\mu_{\bar{x}} = \mu$ = $\mu_{\bar{x}} = 70$

Standard error for $n = 36$: $\sigma_{\bar{x}} = \dfrac{15}{\sqrt{36}} = \dfrac{15}{6}$ = $\sigma_{\bar{x}} = 2.5$

Standard error for $n = 100$: $\sigma_{\bar{x}} = \dfrac{15}{\sqrt{100}} = \dfrac{15}{10}$ = $\sigma_{\bar{x}} = 1.5$

Answer: The sample means are centered at 70. Their standard error is 2.5 for $n = 36$ and only 1.5 for $n = 100$: bigger samples give means that stay closer to $\mu$.

When Is the Sample Mean Normal?

Waiting times at a call center are strongly skewed to the right, with $\mu = 8$ minutes and $\sigma = 6$ minutes. Describe the sampling distribution of $\bar{x}$ for samples of $n = 4$ calls and for samples of $n = 40$ calls.

Center for both sample sizes: $\mu_{\bar{x}} = \mu$ no matter what $n$ is = Center 8 minutes

Standard error for $n = 4$: $\dfrac{6}{\sqrt{4}} = \dfrac{6}{2}$ = 3 minutes

Shape for $n = 4$: The population is not normal and $n = 4$ is small, so the central limit theorem does not help yet = Still skewed to the right

Standard error and shape for $n = 40$: $\dfrac{6}{\sqrt{40}} \approx \dfrac{6}{6.32}$, and $n = 40 \geq 30$ = About 0.95 minutes, approximately normal

Answer: Both distributions are centered at 8 minutes. For $n = 4$ the means are spread out (standard error 3) and still skewed; for $n = 40$ they are tightly packed (standard error about 0.95) and approximately normal by the central limit theorem.

Probability About a Sample Mean

Phone batteries last $\mu = 50$ hours on average with $\sigma = 12$ hours. A shop tests a random sample of 36 batteries. What is the probability that the sample mean is below 47 hours? Compare with the probability that ONE battery lasts less than 47 hours.

Check the shape: $n = 36 \geq 30$, so by the central limit theorem $\bar{x}$ is approximately normal = We may use z-scores

Find the standard error: $\sigma_{\bar{x}} = \dfrac{12}{\sqrt{36}} = \dfrac{12}{6}$ = 2 hours

Compute the z-score of $\bar{x} = 47$: $z = \dfrac{47 - 50}{2}$ = $z = -1.5$

Read the z-table: $P(Z < -1.5) = 0.0668$ = About 6.7%

Compare with one battery (if battery life is normal): $z = \dfrac{47 - 50}{12} = -0.25$, $P(Z < -0.25) = 0.4013$ = About 40%

Answer: $P(\bar{x} < 47) \approx 0.0668$, only about 6.7%. One battery below 47 hours is common (about 40%), but the average of 36 batteries that low is rare, because averages vary much less than single values.

Sampling Distribution of a Sample Proportion

In a town, 40% of adults support a new park ($p = 0.4$). A newspaper polls a random sample of $n = 100$ adults. Describe the sampling distribution of $\hat{p}$ and find the probability that more than half of the sample supports the park.

Check the conditions: $np = 100 \cdot 0.4 = 40 \geq 10$ and $n(1-p) = 100 \cdot 0.6 = 60 \geq 10$ = Approximately normal

Center and standard error: $\mu_{\hat{p}} = 0.4$, $\sigma_{\hat{p}} = \sqrt{\dfrac{0.4 \cdot 0.6}{100}} = \sqrt{0.0024}$ = $\sigma_{\hat{p}} \approx 0.049$

Compute the z-score of $\hat{p} = 0.5$: $z = \dfrac{0.5 - 0.4}{0.049}$ = $z \approx 2.04$

Find the right-tail area: $P(Z > 2.04) = 1 - 0.9793$ = $\approx 0.0207$

Answer: $\hat{p}$ is approximately normal with mean 0.4 and standard error about 0.049. The chance that the poll shows a majority (more than 50%) is only about 2%.

Almost every number in the news that comes from a survey is a statistic, not a parameter: - **Polls**: a poll of 1000 voters estimates the share of all voters. The standard error tells you how far the poll is likely to be from the truth. - **Quality control**: a factory checks the mean weight of a sample of packets instead of weighing every packet. - **Medicine**: a clinical trial measures the mean effect on a few hundred patients and draws conclusions about everyone. - **Science**: averaging repeated measurements reduces random error. The key idea: the standard error $\sigma / \sqrt{n}$ shrinks as the sample grows. Four times as many people halve the error, which is why larger polls are more precise. The central limit theorem then lets you use the normal distribution, z-scores and the z-table you already know, even when the population itself is not normal.

Mistake: Using $\sigma$ instead of $\sigma / \sqrt{n}$ in a question about a sample mean

Why: $\sigma$ is the spread of single values. Sample means vary much less, so using $\sigma$ makes unusual averages look far too likely.

Correct: For $\bar{x}$ always divide by the standard error: $z = \dfrac{\bar{x} - \mu}{\sigma / \sqrt{n}}$. For one value use $z = \dfrac{x - \mu}{\sigma}$.

Mistake: Dividing by $n$ instead of $\sqrt{n}$

Why: With $\sigma = 12$ and $n = 36$, dividing by $n$ gives $0.33$ instead of the correct standard error $2$, so every z-score comes out 6 times too large.

Correct: Take the square root of $n$ first: $\dfrac{12}{\sqrt{36}} = \dfrac{12}{6} = 2$.

Mistake: Thinking the central limit theorem makes the population or the data normal

Why: A large sample from a skewed population is still skewed: its histogram looks like the population. Only the distribution of the sample MEANS becomes normal.

Correct: The central limit theorem is about the sampling distribution of $\bar{x}$, not about individual values.

Mistake: Using the normal model for $\hat{p}$ without checking $np \geq 10$ and $n(1-p) \geq 10$

Why: With $p = 0.05$ and $n = 40$ we get $np = 2$: most samples have 0, 1, 2 or 3 successes, and the distribution of $\hat{p}$ is strongly skewed.

Correct: Check both conditions first. If one fails, take a bigger sample or use the binomial distribution.

Election Polls

Polling companies ask about 1000 people and report a margin of error. That margin comes straight from the standard error of $\hat{p}$.

If the true support is $p = 0.5$, a poll of 1000 people has standard error $\sqrt{0.5 \cdot 0.5 / 1000} \approx 0.016$, so about 95% of such polls land within roughly 3 percentage points of the truth.

Quality Control in a Coffee Shop

Machines that fill cups are checked by averaging small samples. An average that is too far from the target, measured in standard errors, sets off an alarm.

A machine fills cups with $\mu = 250$ mL and $\sigma = 8$ mL. An inspector measures 16 cups, so the standard error is $8 / \sqrt{16} = 2$ mL.

Planning a Medical Study

Before a study starts, researchers choose the sample size so that the standard error is small enough. Because of the square root, halving the error needs four times as many patients.

Blood pressure has $\sigma = 20$ mmHg. With 25 patients the standard error is $20 / 5 = 4$ mmHg; with 100 patients it is $20 / 10 = 2$ mmHg.

A parameter ($\mu$, $\sigma$, $p$) describes the population and is fixed; a statistic ($\bar{x}$, $\hat{p}$) comes from a sample and varies from sample to sample

The sampling distribution of $\bar{x}$ has mean $\mu$ and standard error $\sigma / \sqrt{n}$

Central limit theorem: for $n \geq 30$ (about) the sample mean is approximately normal whatever the population shape; if the population is normal, for any $n$

For a probability about a sample mean use $z = \dfrac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ and the z-table

The sample proportion $\hat{p}$ has mean $p$ and standard error $\sqrt{p(1-p)/n}$; it is approximately normal when $np \geq 10$ and $n(1-p) \geq 10$

Bigger samples give more precise estimates: four times the sample size halves the standard error

Q: Why does the standard error use $\sqrt{n}$ and not $n$?

A: The variance of a mean of $n$ independent values is $\sigma^2 / n$. The standard error is its square root: $\sqrt{\sigma^2 / n} = \sigma / \sqrt{n}$. That is why you need four times as many values to halve the error.

Q: Is 30 a magic number?

A: No, it is a rule of thumb. For a population that is already close to normal, much smaller samples are fine; for a very skewed population (like incomes) you may need more than 30. Try it in the simulator: the skewed population needs a bigger $n$ than the die.

Q: What is the difference between standard deviation and standard error?

A: The standard deviation $\sigma$ measures how much single values vary. The standard error $\sigma / \sqrt{n}$ measures how much a statistic, like the sample mean, varies from sample to sample. The standard error is always smaller when $n > 1$.

Q: Does the size of the population matter?

A: Hardly, as long as the sample is a small part of the population (less than about 10%). A random sample of 1000 people is about equally precise for a city or for a whole country. What matters is the sample size $n$.

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Sampling Distributions

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Probability / Probability Distributions

Sampling Distributions

Learn how sample means and sample proportions vary from sample to sample, what the standard error is, and why the central limit theorem makes them approximately normal.

Advanced30 min

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