How the χ² (chi-square) distribution is built

χ² Builder

Draw values one at a time from the standard normal distribution, square them, and stack them up. The distribution of that running total is the χ² distribution. Pick a degrees of freedom (df) below and start drawing.

Samples completed: 0 Current df: 2

1. Draw a Z

Draw a value from the standard normal distribution (mean 0, SD 1)

2. Square it and stack

Squaring a Z always gives a value ≥ 0; stacking the squares gives the χ² statistic

χ² = —

3. Growing histogram

Completed samples pile up and trace out the χ² distribution (solid line)

The math

This is the real-data version of the Z's you've been squaring and stacking above.

Expected count for row i, column j

E_{i j} = \frac{R_{i} \times C_{j}}{N}

R = row total, C = column total, N = grand total

Pearson residual

r_{i j} = \frac{O_{i j} - E_{i j}}{\sqrt{E_{i j}}}

This is the piece that behaves like one of the Z's above, once N is reasonably large.

χ² statistic

χ^{2} = \sum_{i, j} \frac{{(O_{i j} - E_{i j})}^{2}}{E_{i j}} df = (r - 1) (c - 1)

Square and sum the residuals above, cell by cell — exactly the sum of squared Z's this page builds up one draw at a time.