Maths › Further Statistics 1 › Goodness-of-fit tests
Goodness-of-fit tests
Put a model's predicted frequencies beside the observed ones, square the gaps, and one statistic decides whether the difference is more than chance would produce.
Builds on Hypothesis tests for Poisson and geometric models and Representing and interpreting data.
IN THIS TOPIC
- Compute expected frequencies from a proposed model and form the χ² statistic.
- Count degrees of freedom, subtracting one for each estimated parameter.
- Pool classes with expected frequency below 5 and state the conclusion.
WHAT YOU PROBABLY THINK
A large chi-squared statistic proves the model is wrong; a small one proves it is right.
Measuring the gaps
Multiply each class's model probability by the total to get its expected frequency, then accumulate:
Squaring makes over- and under-shoots count alike; dividing by E keeps a gap of 3 in a class expecting 5 from being drowned by the same gap in a class expecting 500. A large value means the model fits badly, but the test never proves anything either way: it compares the statistic with a critical value and reports evidence, which is what the opening claim overstates in both directions.
WORKED EXAMPLE
Is the die fair?
A die is thrown 60 times, giving 8, 10, 14, 7, 13, 8. Test at the 5% level whether it is fair.
H₀: the die is fair, so each expected frequency is 10.
χ² = (4 + 0 + 16 + 9 + 9 + 4)/10 = 4.2.
Degrees of freedom: 6 classes − 1 = 5, and the 5% critical value is 11.07.
4.2 < 11.07, so do not reject H₀: there is insufficient evidence at the 5% level that the die is unfair.
Counting the degrees of freedom
Start with the number of classes, subtract 1 because the frequencies must total the sample size, and subtract 1 more for every parameter estimated from the data. Fitting a Poisson model with λ taken from the sample mean costs a degree of freedom; fitting one with λ specified in advance does not.
Small expected frequencies distort the statistic, so classes with Ei below 5 are pooled with their neighbours before anything is computed, and the class count used for the degrees of freedom is the pooled one.
YOUR TURN
Degrees of freedom under two regimes
A Poisson model is fitted to data in 7 classes after pooling. Give the degrees of freedom when λ is specified in advance, and when λ is estimated from the sample mean.
Show the working
Specified in advance: 7 − 1 = 6 degrees of freedom.
Estimated from the data: 7 − 1 − 1 = 5.
Estimating a parameter lets the model bend towards the data, so one degree of freedom is surrendered to pay for the flexibility.
THE EXAM BIT
- Compute expected frequencies to at least one decimal place before squaring anything.
- Pool classes with expected frequency under 5 first, then count classes for the degrees of freedom.
- Subtract one degree of freedom for every parameter estimated from the data, and say which.
- Conclude with evidence language: 'insufficient evidence to reject', never 'the model is correct'.
CHECK YOURSELF
A goodness-of-fit test uses 5 classes after pooling, with the distribution's single parameter estimated from the data. State the degrees of freedom.
Show a hint
Classes minus one, minus one per estimated parameter.
Show the answer
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χ² sums (O − E)²/E: squared gaps, each weighted down by how large its class was expected to be.
Degrees of freedom are classes minus 1, minus one more for each parameter estimated from the data.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Compute expected frequencies from a proposed model and form the χ² statistic.
- Count degrees of freedom, subtracting one for each estimated parameter.
- Pool classes with expected frequency below 5 and state the conclusion.
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