Chi-Square ( $\chi^2$ ) Distribution

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The Chi-Square distribution is primarily used in inferential statistics, specifically for hypothesis testing. You rarely use it to model real-world events directly; instead, you use it to evaluate how well your theoretical models fit your actual data.

Core Concepts

The entire Chi-Square distribution is built upon the Standard Normal Distribution (Z).

If you take a Standard Normal variable ( $Z \sim \mathcal{N}(0,1)$ ), draw a random sample, and square it, the resulting value follows a Chi-Square distribution with 1 degree of freedom.

If you take $k$ independent Standard Normal variables, square them all, and add them together, the sum follows a Chi-Square distribution with $k$ degrees of freedom.

Q = \sum_{i=1}^{k} Z_i^2 \sim \chi^2(k)

Why square them?

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By squaring the Z-scores, we ensure that all negative and positive deviations from the mean become positive.

This makes the Chi-Square distribution perfect for measuring total error or total variance—because an error of -2 is just as bad as an error of +2.

Probability Density Function (PDF)

(Note: The PDF formula is mathematically dense and involves the Gamma function $\Gamma$ . In practice, statisticians rely on software or lookup tables rather than computing this by hand).

f(x; k) = \frac{x^{(k/2)-1} e^{-x/2}}{2^{k/2} \Gamma(k/2)} \quad \text{for } x > 0

Key Metrics

Expected Value (Mean): $E[X] = k$
Variance: $\text{Var}(X) = 2k$

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Notice how beautifully simple the mean and variance are! They depend entirely on the degrees of freedom ( $k$ ).

Real-World Applications

Goodness of Fit Test: Checking if a six-sided die is actually fair by comparing your observed rolls against the expected uniform rolls.
Test of Independence: Determining if two categorical variables (e.g., "Gender" and "Voting Preference") are mathematically related or completely independent.

Interactive Visualization

Use the slider below to adjust the Degrees of Freedom ( $k$ ). Notice two critical things:

The distribution is defined only for positive numbers ( $x > 0$ ) because squares cannot be negative.
When $k$ is small, the distribution is heavily skewed to the right. As $k$ grows larger, the Central Limit Theorem takes over, and the Chi-Square distribution slowly begins to look like a symmetrical Normal Distribution!

Chi-Square Distribution

Common in variance and goodness-of-fit tests. Adjust k (degrees of freedom).

Degrees of Freedom (k)

Test Your Knowledge

Example: Chi-Square Test Statistic

You flip a coin 100 times. You expect 50 Heads and 50 Tails. You actually get 60 Heads and 40 Tails. Calculate the Chi-Square test statistic $\chi^2$ .

View Step-by-Step Solution

Formula: $\chi^2 = \sum \frac{(O - E)^2}{E}$

For Heads: $\frac{(60 - 50)^2}{50} = \frac{10^2}{50} = \frac{100}{50} = 2$

For Tails: $\frac{(40 - 50)^2}{50} = \frac{(-10)^2}{50} = \frac{100}{50} = 2$

$\chi^2 = 2 + 2 = 4.0$

(To finish a real test, you would compare $4.0$ to a Chi-Square table with 1 degree of freedom).

Correlation Student's t-Distribution

Chi-Square (χ2\chi^2χ2) Distribution