Chi-Squared Goodness of Fit Tests

The chi-squared ($\chi^2$χ2) goodness of fit test is used to determine whether observed data follow a particular theoretical distribution. This is a key hypothesis testing technique in Further Statistics.

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The Idea Behind the Test

You have observed frequencies $O_i$Oi​ from an experiment or survey, and expected frequencies $E_i$Ei​ from a theoretical model (e.g., uniform, binomial, Poisson, normal). The test asks: are the differences between observed and expected frequencies larger than what we would expect from random variation alone?

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The Test Statistic

$X^2 = \sum_i \frac{(O_i - E_i)^2}{E_i}$X2=∑i​Ei​(Oi​−Ei​)2​

This statistic measures the overall discrepancy between observed and expected. Under the null hypothesis, $X^2$X2 approximately follows a chi-squared distribution with $\nu$ν degrees of freedom.

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Degrees of Freedom

$\nu = (\text{number of classes}) - 1 - (\text{number of estimated parameters})$ν=(number of classes)−1−(number of estimated parameters)

Scenario	Parameters estimated	$\nu$ν
Testing a uniform distribution	0	$k - 1$k−1
Testing a Poisson distribution with known $\lambda$λ	0	$k - 1$k−1
Testing a Poisson distribution with $\lambda$λ estimated from data	1	$k - 2$k−2
Testing a binomial with known $p$p	0	$k - 1$k−1
Testing a binomial with $p$p estimated from data	1	$k - 2$k−2
Testing a normal with both $\mu$μ and $\sigma^2$σ2 estimated	2	$k - 3$k−3

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Procedure for a Goodness of Fit Test

Step 1: State the hypotheses.

$H_0$H0​: The data follow the specified distribution. $H_1$H1​: The data do not follow the specified distribution.

Step 2: Calculate expected frequencies $E_i$Ei​ from the theoretical model.

Step 3: Combine classes if any $E_i < 5$Ei​<5 (this ensures the chi-squared approximation is valid).

Step 4: Calculate the test statistic $X^2 = \sum \frac{(O_i - E_i)^2}{E_i}$X2=∑Ei​(Oi​−Ei​)2​.

Step 5: Find the degrees of freedom $\nu$ν.

Step 6: Compare $X^2$X2 with the critical value from $\chi^2_\nu$χν2​ tables at the given significance level.

Step 7: Conclude: if $X^2 > \text{critical value}$X2>critical value, reject $H_0$H0​; otherwise, do not reject $H_0$H0​.

Exam Tip: You must combine classes when expected frequencies are less than 5. Show this clearly in your working — examiners look for it. After combining, recalculate the number of classes and adjust degrees of freedom.

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Worked Example: Testing a Uniform Distribution

A die is rolled 120 times with the following results:

Face	1	2	3	4	5	6
Observed	15	23	18	25	17	22

$H_0$H0​: The die is fair (uniform distribution). Each face has probability $1/6$1/6.

Expected: $E = 120/6 = 20$E=120/6=20 for each face.

Face	$O$O	$E$E	$(O-E)^2/E$(O−E)2/E
1	15	20	1.25
2	23	20	0.45
3	18	20	0.20
4	25	20	1.25
5	17	20	0.45
6	22	20	0.20