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Premium experiment • sampling distributions

STATISTICS • PROBABILITY DISTRIBUTIONS

Central Limit Theorem Explorer

Repeatedly sample non-normal parent distributions and watch the distribution of sample means concentrate as sample size increases.

● SAMPLING DISTRIBUTIONCentral Limit Theorem
Mean of means
Measured SD
Theoretical SE
Trials

Reasoning inspector

live model state
Sample size n—
Trials—
Measured SD—
Theoretical SE—
Measured / theory—

From parent population to sample means

The theorem concerns the distribution of averages across repeated independent samples. As n grows, that sampling distribution narrows according to σ/√n and often becomes approximately normal.

LEARN WITH THE SIMULATION

Understand Central Limit Theorem Explorer by experimenting.

START HERE

The Central Limit Theorem describes the behavior of sampling distributions, not a transformation of raw data into a normal distribution. This lab repeatedly draws independent samples from uniform, exponential and Bernoulli parent populations, builds the distribution of sample means, and compares the measured spread with the theoretical standard error σ/√n. Changing sample size and trial count separately makes the difference between statistical precision and Monte Carlo smoothness visible.

BUILD THE IDEA

What you should understand first

  • Distinguish the parent distribution from the sampling distribution of the sample mean.
  • Explain why the standard deviation of sample means shrinks approximately as 1/√n.
  • Compare the measured spread of simulated sample means with the theoretical standard error.
  • Recognize that a sampling distribution can become approximately normal even when the parent distribution is skewed or discrete.

TRY IT YOURSELF

How quickly does the sampling distribution forget the shape of a non-normal parent population?

  1. Choose the exponential parent and start with n = 1 so the sample-mean distribution still resembles the skewed parent.
  2. Increase n while holding the trial count fixed and compare histogram width and symmetry.
  3. Turn on the theoretical comparison and compare its center and spread with the simulated histogram.
  4. Reseed several times to separate persistent CLT behavior from random histogram roughness.
Read the visual:

As n grows, the sample means concentrate around the parent mean and their spread approaches σ/√n. The histogram also becomes more symmetric under the model conditions, while a finite number of trials still creates visible random variation.

WHY IT WORKS 01

Sample size and trial count do different jobs

Sample size n changes each mean that is being generated. A larger n reduces the theoretical sampling variation of the mean. Trial count changes only how many simulated means are used to draw the histogram. More trials reduce Monte Carlo roughness but do not alter the underlying sampling distribution, so the two controls should never be interpreted as interchangeable.

WHY IT WORKS 02

Why the square root appears

For independent observations, variances add. A sum of n observations has variance nσ², while dividing that sum by n to form a mean divides variance by n². The result is variance σ²/n and therefore standard deviation σ/√n. This square-root law is the reason precision improves with diminishing returns as sample size grows.

DO NOT MIX THESE UP

Common misconception

The Central Limit Theorem does not say that original observations become normal, and it does not guarantee a perfect normal approximation for every distribution and every sample size. Increasing the number of simulation trials only makes the displayed histogram more stable; increasing n changes the sampling distribution itself.

PRACTICE CHALLENGE

Can you reach a target standard error with the smallest sample size possible?

  • Choose a parent distribution and set a target standard error in the premium controls.
  • Increase n until the theoretical standard error first falls below the target.
  • Decrease n one step and explain why the target fails again using the 1/√n relationship.
Reveal the reasoning

The theoretical standard error is σ/√n, so the boundary is the smallest integer n for which σ/√n is no larger than the target. Because n sits under a square root, halving standard error requires roughly four times as many observations.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • Samples are independent draws from a fixed uniform, exponential or Bernoulli parent distribution.
  • A deterministic pseudo-random generator is used so identical seeds reproduce identical histograms.
  • Theoretical standard error uses the known variance of the selected parent distribution.

Model limitations

  • Finite trial counts can produce visibly noisy histograms, especially for small samples.
  • Heavy-tailed distributions with infinite variance, dependence and finite-population corrections are outside this model.

CONNECT IT

Where this matters

  • Confidence intervals and standard errors
  • Survey and experiment sampling
  • Approximate inference for averages and totals
  • Monte Carlo reasoning about repeated samples

WORDS THAT MATTER

Key vocabulary

Parent distribution
The probability distribution from which individual observations are drawn before averaging.
Sampling distribution
The distribution of a statistic, such as the sample mean, across repeated samples of the same size.
Standard error
The standard deviation of a sampling distribution; for an independent sample mean it is σ/√n in the ideal model.
Monte Carlo noise
Random variation in a simulated result caused by using a finite number of repeated trials.