Central Limit Theorem
Sums of many variables approach a normal distribution.
Key idea
Sums of many variables approach a normal distribution.
In depth
The distribution of the sample mean of n independent observations is approximately normal with mean μ and SD σ/√n, for large n, whatever the shape of the original distribution. It justifies normal-based inference.
Worked example
Averages of 30 skewed incomes already look nearly normal, centred on the true mean.
Formula
Real-world uses
- Justifying the use of the normal distribution for averages in polls and quality control
Common mistake
Applying it to tiny samples from heavily skewed data.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Central Limit Theorem?
Sums of many variables approach a normal distribution. It is a topic at the expert level in Probability & Statistics.
What is the formula for Central Limit Theorem?
The key formula is x̄ ~ N(μ, σ²/n).
Where is Central Limit Theorem used in real life?
Real-world examples: Justifying the use of the normal distribution for averages in polls and quality control.
What should I learn before Central Limit Theorem?
Make sure you are comfortable with Law of Large Numbers, the previous topic in Probability & Statistics.
What is a common mistake with Central Limit Theorem?
Applying it to tiny samples from heavily skewed data.
How can I practise Central Limit Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Central Limit Theorem?
Move on to Sampling Distributions next.
Related expert topics in Probability & Statistics
- Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
- Residuals and R² – Error of a fit and proportion of variance explained.
- Simpson’s Paradox – A trend reverses when groups are combined.
- Law of Total Probability – Split an event over a partition.
- Bayes’ Theorem – Update beliefs with evidence.
- Medical Tests and Base Rates – Why positive results can still be probably false.