Central limit theorem
The central limit theorem says that sample means from almost any population are close to normal when n is large enough (often 30 or more). The mean of the sample means is mu and their SD is sigma / square root of n.
Last updated
Analogy
Even if single dice rolls are flat, the average of many rolls forms a bell shape.
Example
Population mean 50, SD 10, n = 25SE = 10 / 5 = 2z = (53 - 50) / 2 = 1.5P(Z > 1.5) = 1 - 0.9332 = 0.0668
In practice
It is why z and t methods work for data that is not itself normal.
Practice
Flashcards
- What does the central limit theorem say?Sample means are approximately normal for large n, whatever the population shape.
Interview questions
- Explain the central limit theorem and why it justifies using normal methods for sample means.
- Sample means become approximately normal for large n
- Holds for non-normal populations
- Mean is mu, SD is sigma / square root of n
- Does not change the shape of the raw data
Questions people ask
How large must n be for the central limit theorem?
A common rule is 30 or more, but it depends on the shape. Near-symmetric data may need only 10 or so, while heavily skewed data may need hundreds. If the population is already normal, any sample size gives a normal sample mean.
Does the central limit theorem make the data normal?
No. It says the distribution of the sample mean becomes normal. The individual values keep their original shape, skewed or not. A histogram of raw data stays lopsided, while a histogram of many sample means looks like a bell, which is the part the theorem describes.