Statistics & Probability

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Central Limit Theorem Calculator.

Calculate the sampling-distribution standard error and normal interval.

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01

Set your values

Results update as you type.

standardError: 2

standardError

2.000000
lower: 46.08

lower

46.080000
upper: 53.92

upper

53.920000
marginOfError: 3.92

marginOfError

3.920000

Results update automatically as you type.

Use Cases

Estimate the precision of a sample mean

Use this calculator to determine how much the sample mean is expected to vary from the true population mean, which helps in assessing the reliability of sample estimates.

Example: If the population mean is 100, standard deviation is 15, and sample size is 30, the standard error is about 2.74.

Construct a confidence interval for the population mean

Given the population mean and standard deviation, you can calculate a range within which the sample mean is likely to fall, aiding in hypothesis testing and decision making.

Example: With the same inputs, a 95% confidence interval would be approximately 94.6 to 105.4.

Frequently Asked Questions

What does the Central Limit Theorem Calculator do?
This calculator uses the Central Limit Theorem to compute the standard error of the sample mean (population standard deviation divided by the square root of the sample size) and a confidence interval around the population mean based on your inputs.
What inputs do I need to provide?
You need to enter the population mean, the population standard deviation, and the sample size. The calculator then uses these to compute the standard error and the confidence interval.
What is the standard error and why is it important?
The standard error measures the variability of the sample mean from the true population mean. It is calculated as the population standard deviation divided by the square root of the sample size. A smaller standard error indicates a more precise estimate of the population mean.

Tips & Common Mistakes

Tips

  • Ensure the sample size is sufficiently large (typically n ≥ 30) for the Central Limit Theorem to hold, so the sampling distribution of the mean is approximately normal.
  • The population standard deviation must be known; if you only have a sample standard deviation, use a t-distribution instead.
  • The confidence interval provided is based on the normal distribution; for small samples, consider using a t-distribution for more accurate intervals.
  • Remember that the standard error decreases as the sample size increases, so larger samples yield more precise estimates.

Common Mistakes to Avoid

  • Using the sample standard deviation instead of the population standard deviation, which can lead to incorrect standard error and interval calculations.
  • Entering the sample size as the number of samples rather than the number of observations in each sample.
  • Assuming the confidence interval is for the sample mean rather than the population mean; the interval estimates where the population mean lies based on the sample.

Last updated: August 13, 2026