Central Limit Theorem Calculator
Result
Standard error
- Mean of sample means
- 100.0000
- Variance of sample means
- 6.2500
A central limit theorem calculator describes the sampling distribution of a sample mean: where it is centred, how wide it is, and how both respond to the sample size. The centre is the surprising part — the average of every sample mean equals the population mean at any sample size, so collecting more data does not move it. The width is what changes: the standard error is the population standard deviation divided by √n, so quadrupling the sample halves it, while the variance falls fourfold instead. The theorem's own claim — that the shape approaches normal as n grows — is about form rather than width, and no arithmetic on this page can show it.
How the standard error falls as the sample grows
| Sample size, n | √n | Standard error ÷ σ |
|---|---|---|
| 1 | 1 | 1 |
| 4 | 2 | 0.5 |
| 16 | 4 | 0.25 |
| 25 | 5 | 0.2 |
| 100 | 10 | 0.1 |
| 400 | 20 | 0.05 |
| 2500 | 50 | 0.02 |
| 10000 | 100 | 0.01 |
The third column is the factor to multiply σ by — a population standard deviation of 15 with a sample of 100 gives a standard error of 0.1 × 15 = 1.5. Nothing in the table depends on your mean or your standard deviation, which is why it cannot contradict the panel above it: it is the function 1/√n, and the panel evaluates that function at one point. Read down the rows and the square-root law is visible as a pattern rather than as an assertion — every time the sample size multiplies by four, the standard error halves, and the eight rows are chosen in four-fold steps so that the pattern repeats rather than drifts. The sample sizes are the ones whose square roots are whole numbers, so each row can be checked by hand.
Formula
E[X̄] = μ Var(X̄) = σ² / n SE = σ / √n
- μ
- The population mean — the centre of the distribution the samples are drawn from. It comes back out unchanged as the mean of the sampling distribution, which is the first of the three results and the one that is true at every sample size
- σ
- The population standard deviation. Its units are the units of the data, and so are the units of the standard error, which is why σ and SE can be compared directly — a standard error of 2.5 next to a σ of 15 is a statement about the sample size, not about the data
- n
- How many observations are in one sample — the size of each draw, not the number of draws. It appears as √n under a division and as n under another, and that difference is the whole practical message: variance falls in proportion to n, the standard error only to its square root
- X̄
- The sample mean, treated as a random quantity rather than as a number. One sample gives one X̄; repeating the sampling gives a distribution of them, and the three outputs describe that distribution rather than any single sample
- SE
- The standard error — the standard deviation of the sample means, and the page's main result. It is what a confidence interval is built out of and what a test statistic divides by, which is why it is the number worth reading rather than the variance it is the square root of
Use it when you need to know how much a sample mean can be expected to move from sample to sample — before designing a study, when reading a standard error off a report and wanting to see where it came from, or when checking that an intuition about sample size is right. The intuition it usually corrects is that a bigger sample gives a proportionally better estimate: it does not, because the standard error falls with the square root. Going from 100 observations to 400 halves the spread of the estimate, and getting another halving costs 1600. The two results at the top hold for any population shape, which is why they can be used before knowing anything about the distribution. What the page cannot tell you is whether a given n is large enough for the sample mean to be roughly normal — that depends on how skewed the underlying distribution is, and it is a judgement rather than a calculation.
Worked examples
The default: a population with mean 100 and standard deviation 15, sampled 36 at a time
- Variance of the sample means: σ² / n = 225 / 36 = 6.25
- Standard error: √6.25 = 2.5
- Mean of the sample means: 100, the population mean unchanged
- Compare with σ = 15: sampling 36 at a time shrinks the spread by a factor of 6, which is √36
Both 15 and 36 were chosen so the arithmetic can be done in the head — 225 over 36 is 6.25, and its square root is a round 2.5. The third row is worth a second look precisely because it looks like nothing happened: the mean of all sample means is the population mean, at any sample size, and that is a fact about expectation rather than an approximation. Seeing 100 = 100 is the point of the row, and it is why it is printed even though it echoes the input.
The same population sampled 9 at a time instead of 36
- The sample is a quarter of the size, so the variance is four times as large: 225 / 9 = 25
- Standard error: √25 = 5
- The mean is still 100 — cutting the sample changed the spread and nothing else
- Compared with the first example, n fell to a quarter and the standard error only doubled
This pair is the square-root law made concrete: divide the sample by four and the standard error only doubles, because the variance is what scales with n. Read in the other direction it is the reason precision is expensive — the four-fold sample buys a two-fold improvement, and the next two-fold improvement costs sixteen times the original sample. Every sample size decision is this trade, and the reference table below shows it across a wider range of n.
Sampling one observation at a time leaves the spread untouched
- With n = 1 the sample mean is the single observation, so the sampling distribution is the population itself
- Variance: 225 / 1 = 225, which is σ²
- Standard error: √225 = 15, which is σ
- The mean of the sample means is still 100
The n = 1 end of the range is worth seeing because it anchors the whole idea: a sample of one is a draw from the population, so nothing has been averaged away and the standard error is the population standard deviation itself. Every other sample size is this starting point divided by √n. The page accepts n = 1 rather than requiring at least two, because it is a legitimate and instructive input — the sampling distribution simply has not been narrowed by the sample.
Limitations
The two identities at the top of the formula hold for any population and any n, but they describe only the first two moments of the sampling distribution. The theorem's third claim — approximate normality — is what most of the practical use rests on, and it is the part this page cannot check, because how large n has to be depends on how skewed the underlying distribution is: for a roughly symmetric population 10 is often plenty, while a distribution with a long tail can need hundreds before the sample mean settles down. Treat the widely repeated threshold of 30 as a rule of thumb rather than as a result. The formula also assumes the observations are independent, which sampling without replacement from a small population violates — the sample size page applies that correction and will ask for fewer people than this one implies. Nothing here is estimated from data: both inputs are population values, and if you only have a sample standard deviation, the confidence interval page is the one that knows the difference.
Frequently asked questions
- What is the difference between the standard deviation and the standard error?
- The standard deviation describes how far individual observations sit from the mean; the standard error describes how far a sample mean sits from the population mean. The second is the first divided by √n, and the division is the entire content of the difference: a population with σ = 15 is just as spread out whether you sample 9 of it or 900, but the mean of 900 observations lands three times closer to the truth than the mean of 100. That is why σ appears in the data's own units while the standard error shrinks as the study grows — and why the standard error is the number that goes into a confidence interval or a test statistic.
- Does the mean of the sample means change if I take bigger samples?
- No, and that is the result people find hardest to believe. The mean of the sampling distribution equals the population mean at every sample size, from n = 1 upwards, because averaging does not push the estimate in either direction — it only makes it less variable. The distribution of sample means becomes narrower and taller around the same centre. Sample size buys precision, never accuracy: it shrinks the spread of the estimate without moving where the estimate is centred. A biased sampling method, by contrast, moves the centre and no sample size will bring it back, which is why the two failure modes are worth keeping apart.
- How large does my sample have to be for the central limit theorem to apply?
- It depends on the shape of the population, and there is no threshold that works for all of them. A symmetric population is well behaved at n = 10, while a strongly skewed one may need several hundred observations before the sample mean is close to normal. The familiar rule of thumb of 30 comes from simulation studies of mildly skewed data and is a reasonable default, not a theorem. What the page does give you without any assumption about shape is the centre and the spread — the mean and the standard error are exact at any n. Only the normal approximation needs the large sample, and only the confidence intervals and p-values built on it inherit that requirement.
- Why is the variance of the sample means so much smaller than the variance of the data?
- Because averaging cancels out the independent errors of individual observations, and the cancellation runs in proportion to the sample size — while the standard error, which is what people usually compare, runs in proportion to its square root. With σ = 15 and n = 36 the variance drops from 225 to 6.25, a factor of 36, while the standard deviation drops from 15 to 2.5, a factor of only 6. Both rows are printed for that reason. The variance is how the theorem is normally stated, and the standard error is the form that gets used, and the gap between the two factors is the square root.
- Is this the same as the standard error calculator?
- They compute the same quantity from different inputs, and which one you want depends on what you have. This page takes the population standard deviation and the sample size — the values that exist before any data is collected — and is the right tool for planning a study or checking an intuition about how n drives precision. The standard error calculator takes the sample itself and computes the standard deviation from it, which is what you use once the data is in hand and the population value is unknown. The two differ by exactly the estimation step in the middle, and the confidence interval page is where that difference has consequences.
References
- 1.3.6.1. What is a Probability Distribution — e-Handbook of Statistical Methods (a sample statistic as a quantity with its own distribution, which is the object this page describes) — National Institute of Standards and Technology (NIST)
- 1.3.6.7.1. Cumulative Distribution Function of the Standard Normal Distribution — e-Handbook of Statistical Methods (the curve the sampling distribution approaches as the sample grows, and the table it is read from) — National Institute of Standards and Technology (NIST)
- 3.1 Terminology — Introductory Statistics 2e (the law of large numbers: sample averages settling towards the population mean as the sample grows, which is the behaviour the mean of sample means states exactly) — OpenStax (Rice University)