Sampling Distribution Calculator

Results

With n = 40, the central limit theorem makes x̄ approximately normal even if the population is not, unless the population is extremely skewed or has heavy outliers.

How to Use the Sampling Distribution Calculator

1. 1. Choose the statistic. The sample mean x̄ for measurements, or the sample proportion p̂ for yes-or-no outcomes.
2. 2. Enter the population values and the sample size. For a mean, the population mean μ and standard deviation σ; for a proportion, the population proportion p. Then n, the size of each sample.
3. 3. Describe the event. Below a value, above it, or between two values of the sample statistic.
4. 4. Read the results. The probability, the standard error and z-scores, the working, a check of the conditions behind the normal approximation, and a sketch of the sampling distribution.

What a Sampling Distribution Is

Take every possible random sample of size n from a population and compute the same statistic — the mean, say — for each one. The distribution of those values is the sampling distribution. Its centre is the population value, and its spread, the standard error, shrinks as n grows: averages of many observations vary less than single observations do.

The central limit theorem is what makes this work: whatever the shape of the population, the mean of a large enough random sample is approximately normally distributed. "Large enough" depends on the population — n = 30 is a rough guide that suits moderately skewed data, while heavily skewed populations or ones with outliers need more. If the population itself is normal, x̄ is exactly normal for every n.

The probability then comes from the normal curve: z = (value − centre) / SE, and the area follows from Φ(z), as in the Normal Distribution Calculator. The formulas assume a random sample of independent observations — and, when sampling without replacement, a sample under about 10% of the population.

Worked Examples

A sample mean. Commute times in a city have a mean of 28 minutes and a standard deviation of 9, and are skewed to the right. For random samples of 40 commuters the sample mean has a standard error of 9 / √40 = 1.423, so the probability that a sample averages more than 30 minutes is P(x̄ > 30) = 0.0799, at z = 1.4055. With n = 40, the central limit theorem takes care of the skew.

With only 10 commuters per sample, the standard error rises to 2.846 and the same probability to 0.2411 — but for a skewed population a sample that small is not covered by the central limit theorem, and the calculator says so.

A sample proportion. If 40% of voters support a policy, a poll of 100 has a standard error of √(0.4 × 0.6 / 100) = 0.049. The chance that the poll shows support below 35% is P(p̂ < 0.35) = 0.1537. With 400 respondents the standard error halves to 0.0245 and that chance falls to 0.0206. The success–failure check passes in both cases: with n = 100, np = 40 and n(1 − p) = 60.

Common mistakes

• Using σ instead of the standard error. A question about a sample mean needs σ / √n; plugging in σ answers a question about one individual instead.
• Assuming any n makes x̄ normal. The central limit theorem is about large samples. With a small sample from a skewed population, the normal answer can be badly wrong.
• Skipping the success–failure condition. With p = 0.02 and n = 100, np = 2, and the normal approximation to p̂ is poor. Use the binomial distribution instead.

Frequently Asked Questions

What is the standard error?

The standard deviation of a sampling distribution: how much a statistic varies from one sample to the next. For a mean it is σ / √n, so quadrupling the sample size halves it. Confidence intervals and test statistics are built from it.

What is the difference between standard deviation and standard error?

The standard deviation describes the spread of individual observations; the standard error describes the spread of a statistic, such as the mean, across repeated samples. With σ = 9 and n = 40, the standard error of the mean is 1.423 — single commutes vary far more than averages of 40 commutes.

When does the central limit theorem apply?

When the observations are independent and the sample is large enough for the population's shape. n ≥ 30 is a common rule of thumb for moderately skewed populations; symmetric ones need fewer, and very skewed or heavy-tailed ones need more. For a normal population, no minimum is needed.

Why must np and n(1 − p) both be at least 10?

p̂ is a count of successes divided by n, and that count is binomial. The binomial looks normal only when the expected numbers of successes and failures are both reasonably large; 10 is the threshold most current textbooks use, and some older ones use 5. Below it, compute exact binomial probabilities instead.

Does the population size matter?

Not while the sample is a small part of it — under about 10% when sampling without replacement. For a larger share of a finite population, multiply the standard error by the finite population correction √((N − n) / (N − 1)).

How is a sampling distribution different from a confidence interval?

A sampling distribution starts from known population values and asks where sample statistics will fall. A confidence interval runs the other way: it starts from one sample and estimates the unknown population value. Both rest on the same standard error.