Central Limit Theorem Calculator

Your details

Stats mode returns the standard error. Probability mode converts a sample mean to a tail probability. Quantile mode is the reverse: enter a probability and get the cutoff sample mean.
The average of the full population you are sampling from. This equals the mean of the sampling distribution.
The spread of individual values in the population. Must be greater than zero.
How many observations are in each sample. The Central Limit Theorem typically requires n >= 30 for a non-normal population.
Choose which region of the sampling distribution you want to compute. Left tail is the most common for hypothesis tests and confidence intervals.
The specific sample mean you want to find a probability for. Used for left-tail and right-tail modes.
Lower bound for the "between" or "outside" probability types.
Upper bound for the "between" or "outside" probability types.
Standard Error (SE)
1.8257

Standard deviation of the sampling distribution: SE = σ / sqrt(n)

Mean of sampling distribution (μ_x-bar)50
Z-score2.1909
Probability98.5770%
2.190998.6% below · z

P(X-bar <= 54) = 98.5770%

  • The sampling distribution of the mean has a standard error of 1.8257, which is 10 / sqrt(30). With a larger sample the curve narrows proportionally.
  • With n = 30 (>= 30) the CLT guarantees the sampling distribution is approximately normal regardless of the population shape.
  • The sample mean 54 corresponds to a z-score of 2.1909, which is 2.19 standard errors above the population mean.
  • Doubling the sample size reduces the standard error by a factor of sqrt(2) (about 29%), so collecting more data is a powerful way to tighten the sampling distribution.

Next stepUse these results to construct a confidence interval (add/subtract z * SE from the sample mean) or to set a decision threshold for a hypothesis test.

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