Apply chi-squared to inheritance data
Compare observed and expected offspring counts and interpret chi-squared evidence using degrees of freedom.
A Level; Cambridge International Biology 9700, 2028–2030. Written answers are self-checked; this is not practical-skills certification.
Before you begin
- Genes are DNA sequences that can be inherited; meiosis and fertilisation link parental genomes.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does a large chi-squared value indicate relative to the expected counts?
What you will learn
- Calculate chi-squared from observed and expected values
- Use degrees of freedom and a critical value to evaluate a null hypothesis
Original teaching model. Pedigrees and records are fictional; allele and probability assumptions are stated.
| Feature | Meaning |
|---|---|
| 3:1 expected total 80 | 60 present, 20 absent |
| Invented observed total 80 | 62 present, 18 absent |
| Observed percentages | 77.5% present; 22.5% absent |
| Departure | 2.5 percentage points; not proof of a causal mechanism |
Build the genetic explanation
For each category, calculate (observed−expected)²/expected and sum the values to obtain χ². The expected counts come from the genetic hypothesis and total sample size. Degrees of freedom are the number of categories minus one when the expected ratio is specified. Compare the calculated value with the supplied critical value at the stated probability level: if χ² exceeds the critical value, reject the null hypothesis; otherwise there is insufficient evidence to reject it.
Check the evidence and limits
Chi-squared tests whether deviations could reasonably arise from chance under a hypothesis; it does not prove a genetic model true. Expected counts must be suitable for the test and should be calculated from the total number observed.
Pause and explain
A student uses the total observed count in every denominator. Explain the correction.
Worked example
Observed counts are 32 and 8; expected counts are 30 and 10. Find χ².
Show the worked solution
- For the first class: (32−30)²/30=4/30.
- For the second: (8−10)²/10=4/10; add the terms.
Answer χ²=4/30+4/10≈0.53.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Observed counts are 44 and 16; expected counts are 45 and 15. Calculate χ².
Give me a hint
Set out the alleles or steps before deciding the outcome.
Compare my reasoning
- Calculate each (O−E)²/E term.
- Both deviations have magnitude one.
χ²=(44−45)²/45+(16−15)²/15=1/45+1/15≈0.089; each squared deviation is divided by its category expected count.
Look for these in your work
- Genetic terms used correctly
- Steps linked
A calculated χ² is smaller than the supplied critical value. State the conclusion correctly.
Give me a hint
Use the given evidence and state your assumptions.
Compare my reasoning
- The observed deviation is not statistically significant at the supplied level.
- Do not claim the hypothesis has been proved.
There is insufficient evidence to reject the null hypothesis at the stated significance level. The result is consistent with chance variation under the proposed ratio, but does not prove the genetic model.
Look for these in your work
- Prediction or interpretation supported
- Limits stated
A student uses the total observed count in every denominator. Explain the correction.
Give me a hint
Apply the inheritance or gene-control model to the new context.
Compare my reasoning
- Each term is category-specific.
- Divide by that category’s expected count.
For each category calculate (observed−expected)² divided by that category’s expected count, then sum the terms.
Look for these in your work
- Model applied accurately
- Unsupported certainty avoided
Common mistakes
- Using observed rather than expected values in the denominator
- Rejecting a hypothesis when calculated χ² is below the critical value
What can you explain now?
What does a large chi-squared value indicate relative to the expected counts?
Compare with the explanation
Observed deviations are comparatively large
The comparison with the critical value determines whether to reject the null hypothesis.
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Make it stick
How do you calculate and interpret χ² for offspring counts?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.