Start a short session
Check your answer after each question. Use a calculator where needed.
Observed counts · expected ratios · goodness of fit
Chi-squared calculations
Calculate chi-squared contributions and total goodness-of-fit statistics from original count tables, then choose degrees of freedom for a fully specified ratio.
Method and worked examples ↓Learn the method
How to approach these questions
- Use counts in independent, mutually exclusive categories. Calculate expected counts from the specified model and total sample size.
- Subtract expected from observed in each category, square the difference and divide by expected.
- Add the unrounded contributions. Do not square the sum of signed differences.
- For a fully specified goodness-of-fit model with no fitted parameters, subtract one from the number of categories.
- Compare the statistic with an appropriate supplied critical value before drawing a conclusion. These practice tables have expected counts of at least five; other settings need additional care.
Worked examples
Observed counts are 82 and 18; a 3:1 model predicts 75 and 25. Calculate χ².
χ² = (82 − 75)²/75 + (18 − 25)²/25 = 49/75 + 49/25 = 2.61 to 3 significant figures. Retain the unrounded terms when adding.
A fully specified 9:3:3:1 ratio has four phenotype categories. No parameters are fitted. Find the degrees of freedom.
Degrees of freedom = 4 − 1 = 3. This is the goodness-of-fit setting described; a contingency table or fitted model needs the appropriate different calculation.
Keep practising
Topic map and study list · Practical data guides · Active recall decks
Course resources
AQA A-level Biology · OCR A A-level Biology · Textbook answers
Skill coverage follows the common A-level science subject content. Check your board’s current requirements: AQA 7402 and OCR A H420.