Central limit theorem states that sample means converge to normal distribution as sample size increases
Central limit theorem
Central limit theorem states that sample means converge to normal distribution as sample size increases
The central limit theorem (CLT) is a fundamental concept in probability theory, asserting that the distribution of sample means approaches a normal distribution as the sample size grows, regardless of the original population's distribution.
The theorem's significance lies in its broad applicability; it allows the use of statistical methods designed for normal distributions in various scenarios involving different types of distributions. This universality simplifies many probabilistic and statistical analyses.
Historically, the CLT has evolved over time. Initial formulations date back to 1811, but it was only precisely defined in the 1920s. This evolution reflects the growing understanding and formalization of probability theory.
Example
Suppose we repeatedly sample 50 students' test scores from a non-normally distributed population. According to the CLT, the distribution of the sample means will approach a normal distribution as we increase the number of samples.
Remember this
Understanding the CLT is crucial for applying statistical methods correctly and confidently across diverse datasets.
Text adapted from Wikipedia, licensed under CC BY-SA 4.0.
Related concepts
Nyquist–Shannon sampling theorem
Sample at ≥ 2× the highest frequency to avoid aliasing
Glossary of probability and statistics
Ever wonder why your average score on quizzes improves with more quizzes?
importance sampling does: reweights samples from proposal to estimate target expectation
Why can't we always use the same samples to figure out what's happening in a complex system?
Law of large numbers
Law of large numbers: X̄_ n → μ as n → ∞ with probability 1
Langevin dynamics does: adds noise to gradient descent to sample from a distribution
How can we simulate randomness in physics?
CAP theorem states: you can have at most 2 of consistency, availability, partition tolerance
Ever wondered why you can't always get the latest news instantly?
Swipe through 100 ML concepts daily
Open Pocket Polymath