Is 30 a magic number?

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.

What is the difference between standard deviation and standard error?

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 $n > 1$.

Does the size of the population matter?

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

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

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.

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.

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Definition

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).

Try it now

A survey of 200 randomly chosen voters finds that 54% support a new law. In this situation, the 54% is a...

Worked Examples

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

Find the number that comes from the sample

163.2 cm is the mean of the 50 measured students → Statistic:

\bar{x} = 163.2

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.

Common Mistakes

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

Why it's wrong: $\sigma$ 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 $z = \dfrac{x - \mu}{\sigma}$ .

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

Why it's wrong: 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: $\dfrac{12}{\sqrt{36}} = \dfrac{12}{6} = 2$ .

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

Why it's wrong: 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.

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

Why it's wrong: 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.

Interactive Visual

Central limit theorem: sample means from a skewed population

Sample size n

Population mean: μ = 2.6
Mean of the sample means: ?
SD of the sample means: ?
Standard error: σ/√n = 2.285/√1 ≈ 2.285

Try n = 1 first, then a bigger n: the sample means pile up closer to μ, their SD shrinks like σ/√n, and the shape turns normal even when the population is not.

Average of n die rolls (μ = 3.5, σ ≈ 1.71)

Sample size n

Population mean: μ = 3.5
Mean of the sample means: ?
SD of the sample means: ?
Standard error: σ/√n = 1.708/√2 ≈ 1.208

Try n = 1 first, then a bigger n: the sample means pile up closer to μ, their SD shrinks like σ/√n, and the shape turns normal even when the population is not.

Poll: sample proportion when p = 0.4

Sample size n

Population proportion: p = 0.4
Mean of the sample proportions: ?
SD of the sample proportions: ?
Standard error: √(p(1 − p)/n) = √(0.4 · 0.6/100) ≈ 0.049

Try a small n, then a bigger one: the sample proportions pile up closer to p, their SD shrinks like √(p(1 − p)/n), and the shape turns normal.

P(x̄ < 47) when μ = 50, σ = 12, n = 36 (standard error 2)

μ = 50 h, σ = 2 h · Drag a handle or type x or z.

z = (x − μ)/σ = (47 − 50)/2 = -1.5

P(X ≤ 47) = P(Z ≤ -1.5) ≈ 0.0668 = 6.68%

About 6.68% of all values fall in the shaded region.

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Practice Problems

18 problems

Problem 1 of 18

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A survey of 200 randomly chosen voters finds that 54% support a new law. In this situation, the 54% is a...

Practice Problems: Sampling Distributions

1. A survey of 200 randomly chosen voters finds that 54% support a new law. In this situation, the 54% is a...
2. A population has standard deviation \sigma = 18. What is the standard error of the sample mean for samples of size n = 36?
3. A population has mean \mu = 40 and standard deviation \sigma = 8. What is the mean of the sampling distribution of \bar{x} for samples of size n = 25?
4. Bolts from a machine have mean length \mu = 50 mm and \sigma = 2 mm. A worker measures random samples of 16 bolts. Find the mean and the standard error of \bar{x}.
5. IQ scores have \mu = 100 and \sigma = 10 in a test group. What is the standard error of the mean IQ of random samples of n = 100 people?
6. 20% of the apps in a store are games (p = 0.2). You pick random samples of n = 100 apps. Check the conditions and find the standard error of \hat{p}.
7. House prices in a region are strongly skewed to the right. For which sample size is the sampling distribution of the mean price closest to normal?
8. A population has \mu = 80 and \sigma = 12. A random sample of n = 16 has mean \bar{x} = 86. What is the z-score of this sample mean?
9. A population has \sigma = 20. Compare the standard error of \bar{x} for n = 25 and for n = 100.
10. A coin is fair (p = 0.5). You flip it n = 100 times and record the proportion of heads \hat{p}. What is the standard error of \hat{p}?
11. Exam scores are normally distributed with \mu = 70 and \sigma = 15. For a random class of n = 25 students, find P(\bar{x} < 67). Use P(Z < -1) = 0.1587.
12. Phone batteries last \mu = 50 hours with \sigma = 12 hours. For a random sample of n = 36 batteries, find P(\bar{x} > 54). Use P(Z < 2) = 0.9772 and give the answer as a decimal.
13. About 5% of a factory's light bulbs are faulty (p = 0.05). An inspector takes random samples of n = 120 bulbs. Can she use the normal model for \hat{p}? If not, how large must n be?
14. Apple weights are normally distributed with \mu = 180 g and \sigma = 20 g. Find the probability that ONE apple weighs more than 185 g, and that the mean of a bag of 16 apples is more than 185 g. Use P(Z < 0.25) = 0.5987 and P(Z < 1) = 0.8413.
15. Which statement about the central limit theorem is correct?
16. Delivery times have \sigma = 15 minutes. A company wants the standard error of the mean delivery time to be 1.5 minutes. How many deliveries must it sample? How many for a standard error of 0.75 minutes?
17. 40% of a city's households have a pet (p = 0.4). In a random sample of n = 150 households, find P(\hat{p} > 0.46). Use P(Z < 1.5) = 0.9332 and give the answer as a decimal.
18. LED lamps last \mu = 1000 hours with \sigma = 80 hours. For a random sample of n = 64 lamps, find P(990 < \bar{x} < 1020). Use P(Z < -1) = 0.1587 and P(Z < 2) = 0.9772.

Why It Matters

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.

Real World Applications

Election Polls

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

Example:

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.

1Try It Yourself

A smaller poll asks $n = 400$ randomly chosen voters. The true support for a candidate is $p = 0.5$ .

What is the standard error of the sample proportion $\hat{p}$ ?

Step 1: Write the mathematical expression

Compute $\sqrt{\dfrac{p(1-p)}{n}}$ :

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.

Example:

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.

2Try It Yourself

The 16 cups have a mean of $\bar{x} = 246$ mL. The machine is set to $\mu = 250$ mL with $\sigma = 8$ mL.

What is the z-score of this sample mean?

Step 1: Write the mathematical expression

Compute $z = \dfrac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ :

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.

Example:

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.

3Try It Yourself

Researchers want the standard error of the mean blood pressure to be 2 mmHg. The population standard deviation is $\sigma = 20$ mmHg.

How many patients do they need?

Step 1: Write the mathematical expression

From $\dfrac{\sigma}{\sqrt{n}} = 2$ we get $n = \left(\dfrac{\sigma}{2}\right)^2$ :

Key Takeaways

1A 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
2The sampling distribution of $\bar{x}$ has mean $\mu$ and standard error $\sigma / \sqrt{n}$
3Central limit theorem: for $n \geq 30$ (about) the sample mean is approximately normal whatever the population shape; if the population is normal, for any $n$
4For a probability about a sample mean use $z = \dfrac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ and the z-table
5The sample proportion $\hat{p}$ has mean $p$ and standard error $\sqrt{p(1-p)/n}$ ; it is approximately normal when $np \geq 10$ and $n(1-p) \geq 10$
6Bigger samples give more precise estimates: four times the sample size halves the standard error

Frequently Asked Questions

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.

Glossary

Parameter: A number that describes a whole population, such as $\mu$ , $\sigma$ or $p$ ; usually unknown
Statistic: A number computed from a sample, such as $\bar{x}$ or $\hat{p}$ ; it changes from sample to sample
Sampling distribution: The distribution of a statistic over all possible random samples of the same size $n$
Standard error: The standard deviation of a sampling distribution, for example $\sigma / \sqrt{n}$ for the sample mean
Central limit theorem: For large $n$ the sampling distribution of the sample mean is approximately normal, whatever the shape of the population
Sample proportion: The share of successes in a sample, $\hat{p} = \dfrac{\text{number of successes}}{n}$

Formula Card

Mean of the sample mean

\mu_{\bar{x}} = \mu

Sample means are centered on the population mean

Standard error of the mean

\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}

Spread of the sample means; shrinks as $n$ grows

z-score of a sample mean

z = \dfrac{\bar{x} - \mu}{\sigma / \sqrt{n}}

Use with the z-table when $n \geq 30$ or the population is normal

Standard error of a proportion

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

Normal model if $np \geq 10$ and $n(1-p) \geq 10$

Today's challengeOne problem, three tries

Sampling Distributions

Definition

Sampling distribution of the sample mean $\bar{x}$

Sampling distribution of the sample proportion $\hat{p}$

Worked Examples

Common Mistakes

Interactive Visual

Central limit theorem: sample means from a skewed population

Average of n die rolls (μ = 3.5, σ ≈ 1.71)

Poll: sample proportion when p = 0.4

P(x̄ < 47) when μ = 50, σ = 12, n = 36 (standard error 2)

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Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Election Polls

Quality Control in a Coffee Shop

Planning a Medical Study

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Mean of the sample mean

Standard error of the mean

z-score of a sample mean

Standard error of a proportion

More in This Topic

Introduction to Probability Distributions

Binomial Probability

Discrete vs Continuous Distributions

Expected Value

Definition

Sampling distribution of the sample mean xˉ\bar{x}xˉ

Sampling distribution of the sample proportion p^\hat{p}p^​

Worked Examples

Common Mistakes

Interactive Visual

Central limit theorem: sample means from a skewed population

Average of n die rolls (μ = 3.5, σ ≈ 1.71)

Poll: sample proportion when p = 0.4

P(x̄ < 47) when μ = 50, σ = 12, n = 36 (standard error 2)

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Election Polls

Quality Control in a Coffee Shop

Planning a Medical Study

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Mean of the sample mean

Standard error of the mean

z-score of a sample mean

Standard error of a proportion

More in This Topic

Introduction to Probability Distributions

Binomial Probability

Discrete vs Continuous Distributions

Expected Value

Sampling distribution of the sample mean $\bar{x}$

Sampling distribution of the sample proportion $\hat{p}$