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.

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Definition

A parameter is a number that describes a whole population, for example the population mean μ\mu, the population standard deviation σ\sigma or the population proportion pp. A statistic is a number computed from a sample, for example the sample mean xˉ\bar{x} or the sample proportion p^\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 nn.

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

  • Center: μxˉ=μ\mu_{\bar{x}} = \mu (sample means are centered on the population mean)
  • Spread: σxˉ=σn\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}, called the standard error
  • Shape (central limit theorem): if the population is normal, xˉ\bar{x} is normal for any nn. If not, xˉ\bar{x} is still approximately normal when nn is large, usually n≥30n \geq 30.
So a probability about a sample mean uses the z-score
z=xˉ−μσ/nz = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}

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

  • Center: μp^=p\mu_{\hat{p}} = p
  • Spread: σp^=p(1−p)n\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}
  • Shape: approximately normal when np≥10np \geq 10 and n(1−p)≥10n(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).

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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...

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?
1

Find the number that describes the whole population

165 cm is the mean of all 1200 students → Parameter: μ=165\mu = 165 cm
2

Find the number that comes from the sample

163.2 cm is the mean of the 50 measured students → Statistic: xˉ=163.2\bar{x} = 163.2 cm
3

Think about a second sample

Another random sample of 50 students would give a slightly different mean, for example 166.0 cm → xˉ\bar{x} varies, μ\mu stays fixed

Common Mistakes

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

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

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

Dividing by nn instead of n\sqrt{n}

Why it's wrong: With σ=12\sigma = 12 and n=36n = 36, dividing by nn gives 0.330.33 instead of the correct standard error 22, so every z-score comes out 6 times too large.

Correct: Take the square root of nn first: 1236=126=2\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 xˉ\bar{x}, not about individual values.

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

Why it's wrong: With p=0.05p = 0.05 and n=40n = 40 we get np=2np = 2: most samples have 0, 1, 2 or 3 successes, and the distribution of p^\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
Populationμ = 2.6 · σ = 2.28012345678910
Sample means (n = 1)0 samples012345678910Draw samples: each sample mean lands here.Value
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
Populationμ = 3.5 · σ = 1.71123456
Sample means (n = 2)0 samples123456Draw samples: each sample mean lands here.Die roll
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
Populationp = 0.4 · σ = 0.49Does not support 60%Supports the plan 40%00.250.50.751
Sample proportions (n = 100)0 samples00.250.50.751Draw samples: each sample proportion lands here.
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)

47 · z = -1.5xz44-346-248-1500521542563Mean battery life x̄ (n = 36) (h)

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...

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 σ/n\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=0.5p = 0.5, a poll of 1000 people has standard error 0.5⋅0.5/1000≈0.016\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=400n = 400 randomly chosen voters. The true support for a candidate is p=0.5p = 0.5.

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

Step 1: Write the mathematical expression

Compute p(1−p)n\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 μ=250\mu = 250 mL and σ=8\sigma = 8 mL. An inspector measures 16 cups, so the standard error is 8/16=28 / \sqrt{16} = 2 mL.

2Try It Yourself

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

What is the z-score of this sample mean?

Step 1: Write the mathematical expression

Compute z=xˉ−μσ/nz = \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 σ=20\sigma = 20 mmHg. With 25 patients the standard error is 20/5=420 / 5 = 4 mmHg; with 100 patients it is 20/10=220 / 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 σ=20\sigma = 20 mmHg.

How many patients do they need?

Step 1: Write the mathematical expression

From σn=2\dfrac{\sigma}{\sqrt{n}} = 2 we get n=(σ2)2n = \left(\dfrac{\sigma}{2}\right)^2:

Key Takeaways

  • 1A parameter (μ\mu, σ\sigma, pp) describes the population and is fixed; a statistic (xˉ\bar{x}, p^\hat{p}) comes from a sample and varies from sample to sample
  • 2The sampling distribution of xˉ\bar{x} has mean μ\mu and standard error σ/n\sigma / \sqrt{n}
  • 3Central limit theorem: for n≥30n \geq 30 (about) the sample mean is approximately normal whatever the population shape; if the population is normal, for any nn
  • 4For a probability about a sample mean use z=xˉ−μσ/nz = \dfrac{\bar{x} - \mu}{\sigma / \sqrt{n}} and the z-table
  • 5The sample proportion p^\hat{p} has mean pp and standard error p(1−p)/n\sqrt{p(1-p)/n}; it is approximately normal when np≥10np \geq 10 and n(1−p)≥10n(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 nn independent values is σ2/n\sigma^2 / n. The standard error is its square root: σ2/n=σ/n\sqrt{\sigma^2 / n} = \sigma / \sqrt{n}. That is why you need four times as many values to halve the error.
The variance of a mean of nn independent values is σ2/n\sigma^2 / n. The standard error is its square root: σ2/n=σ/n\sqrt{\sigma^2 / n} = \sigma / \sqrt{n}. That is why you need four times as many values to halve the error.
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 nn than the die.
The standard deviation σ\sigma measures how much single values vary. The standard error σ/n\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>1n > 1.
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 nn.

Glossary

Parameter
A number that describes a whole population, such as μ\mu, σ\sigma or pp; usually unknown
Statistic
A number computed from a sample, such as xˉ\bar{x} or p^\hat{p}; it changes from sample to sample
Sampling distribution
The distribution of a statistic over all possible random samples of the same size nn
Standard error
The standard deviation of a sampling distribution, for example σ/n\sigma / \sqrt{n} for the sample mean
Central limit theorem
For large nn 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, p^=number of successesn\hat{p} = \dfrac{\text{number of successes}}{n}

Formula Card

Mean of the sample mean

μxˉ=μ\mu_{\bar{x}} = \mu

Sample means are centered on the population mean

Standard error of the mean

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

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

z-score of a sample mean

z=xˉ−μσ/nz = \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

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

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

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