Calculadora do Teorema Central do Limite
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
SE = σ / √n
About this calculator
The Central Limit Theorem Calculator computes the standard error of the sample mean, given the population standard deviation (σ) and sample size (n). The central limit theorem states that for large samples, the distribution of sample means approaches a normal distribution, with mean equal to the population mean and standard deviation equal to σ/√n. This tool is essential for estimating the precision of a sample mean and constructing confidence intervals.
The calculation is straightforward: enter the population standard deviation (σ) and the sample size (n). The formula used is SE = σ / √n, where SE is the standard error. The larger the n, the smaller the standard error, indicating greater precision of the sample mean. The calculator assumes random and independent samples, and that σ is known. For small samples (n < 30), the normal approximation may not be adequate if the population is not normal.
Use this calculator in opinion polls, quality control, scientific studies, and any situation where you need to know how close the sample mean is to the population mean. For example, when measuring the average height of a population, the standard error indicates the expected margin of error. It is also useful for determining the sample size needed for a given level of precision.
Cautions: The population standard deviation (σ) is rarely known in practice; often the sample standard deviation (s) is used as an estimate, but this introduces additional uncertainty. Also, the central limit theorem requires independent and identically distributed samples. If the population has a very skewed distribution or outliers, larger samples (n > 30 or more) are needed for a good approximation.
Frequently asked questions
What does the calculated standard error mean?
The standard error measures the variability of the sample mean around the population mean. A smaller standard error indicates that the sample mean tends to be closer to the true mean.
Can I use this calculator if I don't know the population standard deviation?
Not directly, because the formula requires σ. If you have the sample standard deviation (s), you can use it as an estimate, but the result will be approximate, and for small samples it is better to use the t-distribution.
What is the minimum sample size for the central limit theorem to apply?
There is no fixed value, but generally n ≥ 30 is considered sufficient for the normal approximation, provided the population is not too skewed. For normal populations, any n works.
How does the standard error relate to the confidence interval?
The confidence interval for the mean is constructed as sample mean ± (critical value × standard error). For example, for 95% confidence, approximately 1.96 × standard error is used.
Can this calculator be used for hypothesis testing?
Yes, the standard error is used in z-tests for the mean when σ is known. The test statistic is (sample mean - hypothesized mean) / standard error.