The chi-square distribution
Square k independent standard normal values and add them up: the total follows a chi-square distribution with k degrees of freedom. It is never negative and is skewed to the right. It measures how far a sample variance strays, and it is the piece of Student's t that carries the estimated σ. Move k, then ask a question: the area it counts is shaded.
Parameters
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Compare with
P(X ≤ 4) = 0.5940
P(X ≤ 4) = F(4) = 0.5940
Summary
| Measure | χ²(k = 4) |
|---|---|
| mean | 4.0000 |
| variance | 8.0000 |
| standard deviation | 2.8284 |
| skewness | 1.4142 |
The same in code
- Python (scipy.stats)
chi2.cdf(4, 4)- Excel
=CHISQ.DIST(4, 4, TRUE)
Formulas
Try this
Z² and the normal
Open the preset for one squared normal.
Look for: P(Z² ≤ 3.841) = 0.9500, because 3.841 is 1.96² rounded: the chi-square with k = 1 is the square of the standard normal.
Towards the normal
Turn on the normal approximation and raise k from 2 to 100.
Look for: The skewness √(8/k) falls from 2 to 0.28 and the curve becomes a bell centred on k: a sum of many independent pieces is close to normal.
Mean k, variance 2k
Move k and watch the mean marker and the band.
Look for: The mean marker sits at k and the ±1σ band reaches √(2k) either side of it.
The piece of t that varies
Look at k = 9, then open Student's t with ν = 9.
Look for: T = Z / √(V/ν) divides a standard normal by this chi-square scaled by ν. Because V/ν varies from sample to sample, the t has heavier tails than the normal.