Central Limit Theorem Calculator
The central limit theorem says the average of a large enough sample behaves like a normal distribution, centered on the population mean with spread sigma / sqrt(n). This tool turns that promise into numbers: sampling probabilities, the sample mean that cuts off any target probability, confidence intervals, the sample size you need for a target precision, and a live simulation that shows the theorem happening.
What the four modes do
Probability answers P(x-bar below, above, or between values): it computes the standard error, converts to a z-score, and reads the normal CDF, with 95% and 99% confidence intervals shown alongside. Quantile runs the inverse: give it a target probability and it returns the sample-mean value that cuts it off. Sample size solves n = (sigma / SE)^2 for a target standard error or CI half-width, always rounding up. Simulation draws thousands of real random samples from a uniform, exponential, or normal population so you can watch the means pile into a bell curve.
Worked example
A population has mean 70 and standard deviation 7. With n = 35, the standard error is 7 / sqrt(35) = 1.1832. The probability that the sample mean falls between 68 and 72: z1 = (68 - 70) / 1.1832 = -1.6903, z2 = 1.6903, so P = 90.9031%. The 95% confidence interval for the sample mean is 70 +/- 1.96 x 1.1832, i.e. 67.6809 to 72.3191. In Quantile mode, the value with 95% of sample means below it is 70 + 1.6449 x 1.1832 = 71.9462. In Sample size mode, a target SE of 1 needs n = (7/1)^2 = 49.
The formulas, stated plainly
Mean of the sampling distribution: mu_x-bar = mu. Standard error: SE = sigma / sqrt(n). Z-score for a sample mean: z = (x-bar - mu) / SE. The 30 rule of thumb: n = 30 is the usual minimum for the normal approximation when the population is not normal, though skewed populations may need more. The sample-size solver inverts the SE formula, which is why it rounds up: n = 48.2 means 49, because 48 would miss your precision target.
What competing calculators miss
The typical rival computes the standard error and one probability, then stops. One competitor adds a quantile mode but no confidence intervals and no sample-size solver; another shows a 95% interval but cannot do between-value probabilities. None combine probability, quantile, sample-size solving, and a live simulation in one tool, and none show both 95% and 99% intervals next to every probability result.
Central limit theorem FAQs
What is the standard error of the mean?
It is the standard deviation of the sampling distribution: SE = sigma / sqrt(n). It measures how much sample means vary around the population mean, and it shrinks as n grows.
How large does n need to be for the CLT to apply?
The common rule of thumb is n = 30, but it depends on the population shape. Symmetric populations converge fast; heavily skewed ones need larger samples. The simulation mode above lets you see the difference directly.
What is the difference between standard deviation and standard error?
Standard deviation (sigma) describes the spread of individual observations. Standard error (sigma / sqrt(n)) describes the spread of sample means. Confusing the two is the most common CLT mistake.
How do I find the sample size for a target margin of error?
Use the Sample size tab: enter sigma and your target SE (or CI half-width with a confidence level). The tool computes n = (sigma / SE)^2 and rounds up.
Can I use the CLT when the population is not normal?
That is exactly what the theorem is for: for large n, the sampling distribution of the mean is approximately normal regardless of the population shape. Try the exponential population in simulation mode with n = 5 versus n = 50.
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