Central Limit Theorem (CLT)

Maths: Statistics for machine learning

2 min read

Published Oct 22 2025, updated Oct 23 2025

Machine LearningMathsNumPyPandasPythonStatistics

The Central Limit Theorem (CLT) is one of the most important concepts in statistics.

It states that the sampling distribution of the sample mean will approach a normal (bell-shaped) distribution, regardless of the shape of the original population, as long as the sample size is sufficiently large (typically n ≥ 30).

In simple terms:

“No matter what your data looks like, if you take enough random samples and average them, the distribution of those averages will look normal.”

Key Points

Works for any population distribution — whether it’s skewed, uniform, or irregular.
As sample size (n) increases:
- The sampling distribution of the mean becomes more symmetric and normal.
- The standard error (spread of sample means) gets smaller.
The mean of the sampling distribution equals the population mean (μ).
The standard deviation of the sampling distribution (the standard error) is:

Why It Matters

Enables use of Normal Distribution methods (Z-scores, confidence intervals, hypothesis tests) even when population data are not normal.
Forms the foundation for statistical inference and many machine learning techniques that assume normally distributed errors or averages.

Example

Even if your population is heavily skewed (like income data), if you repeatedly: