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Statistics · Sampling & confidence

Central Limit Theorem Calculator

Explore the normal approximation for a sample mean using population mean, spread, and sample size.

Statistics · Sampling & confidence

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Private in-browser calculation · explicit assumptions
Means from repeated samples vary less than individual observations: standard error = SD ÷ √n.
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
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Enter plain numbers without measurement units. Datasets accept commas, spaces, semicolons, or line breaks and are limited to 10,000 values. Results stay in this browser.

Statistics result

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Understand sampling uncertainty

Why estimates vary from sample to sample

Central Limit Theorem. A sampling distribution describes the values an estimator would take across repeated samples drawn under the same design.

Method used

Z=(x̄−μ)/(σ/√n).

Increasing n often improves the normal approximation, but no universal cutoff guarantees adequacy for every population.

What the result cannot establish

The central limit theorem is an approximation whose adequacy depends on sample size, dependence, skew, tails, and the statistic being studied.

Assumption: Independence and finite variance are required; severe skew, heavy tails, dependence, or selection bias can invalidate a routine approximation.

Quick guide

How to use this calculator

  1. Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
  2. Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
  3. Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.

Calculation method

How the central limit theorem calculator works

Z=(x̄−μ)/(σ/√n).

Increasing n often improves the normal approximation, but no universal cutoff guarantees adequacy for every population.

Worked example

Central Limit Theorem example

With μ=50, σ=10, n=100, the sample-mean standard error is 1.

Z=(x̄−μ)/(σ/√n).

Supported inputs

Precision and limits

Model and design

Independence and finite variance are required; severe skew, heavy tails, dependence, or selection bias can invalidate a routine approximation.

Numerical scope

Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.

Interpretation

Increasing n often improves the normal approximation, but no universal cutoff guarantees adequacy for every population.

Decision boundary

The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.

Privacy

Entered values and calculated results stay in this browser and are not sent to an analytics service.

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