Compute the standard error and confidence intervals for sampling distributions using the central limit theorem.
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About this calculator
The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population distribution shape. This calculator derives the standard error and confidence intervals for the sampling distribution.
Sampling distribution
Confidence intervals
Interpretation
With a sample size of 30, the standard error of the mean is 2.74. This means repeated samples of this size will produce sample means that typically fall within about ±5.37 of the true population mean at 95% confidence.
Also in Statistics
Statistics
A central limit theorem calculator computes the standard error of the sampling distribution and confidence intervals at common levels (90%, 95%, 99%). The CLT states that the distribution of sample means approaches a normal distribution as sample size increases, regardless of the population distribution.
The sampling distribution of the mean has a mean equal to the population mean (μ) and a standard error equal to σ/√n, where σ is the population standard deviation and n is the sample size.
Confidence intervals are constructed as μ ± z × SE, where z is the critical value for the desired confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%).
SE = σ / √n
Standard error of the mean.
CI = μ ± z × SE
Confidence interval formula.
Frequently asked questions
A common rule of thumb is n ≥ 30 for the CLT to provide a good approximation, though this depends on how non-normal the population distribution is.
No — that is the power of the CLT. Regardless of the population shape, the sampling distribution of the mean approaches normal as n increases.
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