Central Limit Theorem Simulator | CalidadSA

Interactive central limit theorem simulation. See how sample means become approximately normal even when the starting distribution is skewed, flat, or bimodal.

8 min readIntermediate

How can averages look normal when the raw data does not?

The central limit theorem explains it. Repeated sample averages tend to form a bell-shaped distribution, even when the original data starts skewed, flat, discrete, or split into two peaks.

The core idea

In this simulation, the left chart shows the raw distribution you sample from. The right chart shows the averages of many repeated samples. As subgroup size grows, those averages usually become smoother, narrower, and more normal-looking.

Step 1: Start anywhere

Choose a uniform, skewed, bimodal, or discrete source distribution.

Step 2: Average small samples

Each simulated subgroup produces one sample mean.

Step 3: Watch the shape change

As n increases, the distribution of sample means often becomes more bell-shaped and less skewed.

Run the simulation

Pick a starting distribution, choose how many observations go into each sample mean, and run repeated samples.

Starting distribution

Uniform Flat and evenly spread. No bell curve to begin with.

Right-skewed Many small values with a long right tail.

Bimodal Two peaks, like two different process settings mixed together.

Discrete 0/1 A simple pass/fail style source distribution.

Sample size (n)

How many observations are averaged together in each subgroup.

Repeated samples

How many subgroup means to simulate.

Quick presets

Source shape

right-skewed

The raw data you start from.

Source skewness

1.84

How asymmetric the starting distribution is.

Sample-mean skewness

0.766

Skewness after averaging samples of size 5.

Average of sample means

0.001

It stays near the source mean.

Source distribution

Raw values drawn from the chosen starting shape.

Distribution of sample means (n = 5)

Histogram of 2000 repeated sample averages.

What this means

The sample means are moving toward a bell shape, but with n = 5 the right-skewed source still leaves visible asymmetry. Try a larger subgroup size.

Source spread is about 0.951, while the spread of the sample means is about 0.44. Averaging reduces variation.

With n = 5, the averages from a right-skewed source now have skewness around 0.766. That is the central limit theorem in action.

Why quality engineers care

Subgroup averages

X-bar style reasoning depends on how subgroup averages behave, not just on the raw measurements.

Stable center estimates

Sample means stay centered near the true process mean, which helps when comparing process shifts.

Important caveat

Very skewed or highly discrete sources may need larger subgroup sizes before the bell shape becomes obvious.

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Try It Yourself

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