Goodness of fit — Edexcel A-Level Statistics
Test yourself on Goodness of fit with PEARSON EDEXCEL A-Level practice questions.
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Goodness of fit explained
A goodness of fit test checks whether observed frequencies agree with a proposed distribution.
Read the full explanation
State H₀ that the data follow the named distribution and H₁ that they do not. Estimate any unknown parameters from the data, then calculate expected frequencies E = n × p for each class, combining classes so every E is at least 5. Compute X² = ∑ (O − E)² / E, where O is observed and E is expected. The statistic is compared with a χ² distribution whose degrees of freedom are the number of classes after combining minus 1, minus one further degree for each parameter estimated from the data. For a binomial with p estimated, or a Poisson with λ estimated, that is one estimated parameter; for a normal with μ and σ estimated, that is two. For example, testing Poisson fit across five classes with λ estimated gives ν = 5 − 1 − 1 = 3. A large X² relative to the critical value gives evidence against H₀.
Your focus
- Set up H₀ and H₁ for a goodness of fit test for binomial, Poisson, normal, exponential or a specified discrete distribution.
- Estimate unknown parameters and calculate expected frequencies E = n × p, combining classes so every E is at least 5.
- Calculate X² = ∑ (O − E)² / E and determine degrees of freedom allowing for estimated parameters.
Show all 4 objectives
- Compare the statistic with the appropriate critical value and interpret the outcome in context.
Goodness of fit exam tips
Marking Points
- State H₀ that the data follow the specified distribution and H₁ that they do not.
- Estimate unknown parameters from the sample data before calculating expected frequencies.
- Calculate expected frequencies as E = n × p for each class and combine classes so that every E is at least 5.
- Compute the test statistic X² = ∑ (O − E)² / E, summing over all classes after any combining.
- Obtain degrees of freedom as (number of classes after combining) − 1 − (number of parameters estimated from the data).
- Compare X² with the critical value at the stated significance level and interpret the result in context.
Examiner Tips
- 💡Write the hypotheses and the degrees of freedom before doing any arithmetic so the structure of the test is clear.
- 💡Combine adjacent classes until every expected frequency is at least 5, and record how many classes remain.
- 💡Keep a table of O, E, O − E and (O − E)² / E, and check that the sum of the O values equals n.
- 💡State the conclusion in the context of the original data, referring to the distribution being tested.
Common Mistakes
- Using the raw number of classes before combining: the error is counting classes with E below 5, whereas degrees of freedom use the number of classes after combining.
- Forgetting to subtract a degree of freedom for each estimated parameter: the error is using ν = k − 1 only, whereas a Poisson with λ estimated needs ν = k − 2.
- Calculating E as an observed proportion rather than n × p: the error is omitting the sample size, whereas each expected frequency must be n multiplied by the model probability.
- Comparing X² with the wrong tail or wrong critical value: the error is rejecting H₀ for a small X², whereas a large X² relative to the critical value supports rejection.