Using the Chi-Squared Test (OCR A-Level Biology A): Revision Notes

Using the Chi-Squared Test

What is the chi-squared test?

The chi-squared test (χ²) is a statistical method used to analyse categorical data — data that can be sorted into distinct groups or categories. This test determines whether differences between what you observe in an experiment and what you expect to happen are likely due to random chance or represent a genuine biological effect.

In genetics, the chi-squared test helps confirm whether experimental crosses produce offspring ratios that match predicted Mendelian ratios. It is particularly useful for testing whether two genes are linked on the same chromosome or assort independently during meiosis.

The null hypothesis

Before conducting a chi-squared test, you must formulate a null hypothesis. This is a statement about the biological system you are investigating that allows you to predict the expected numbers in each category.

For example, in a genetics experiment, your null hypothesis might state: "The two genes are located on different chromosomes and will show independent assortment, producing offspring in a 9:3:3:1 ratio."

The chi-squared test then determines whether your experimental results differ significantly from what this hypothesis predicts. If the difference is small, you accept the null hypothesis. If the difference is large, you reject it and conclude that another factor (such as gene linkage) is affecting the results.

Observed and expected data

Observed data (O) are the actual results you collect during your experiment. These are the real counts or measurements from your practical work.

Expected data (E) are the predicted values based on your null hypothesis. In genetic crosses, these come from theoretical ratios such as:

• 3:1 for a monohybrid cross
• 9:3:3:1 for a dihybrid cross with unlinked genes
• 1:1:1:1 for a dihybrid test cross with unlinked genes

To calculate expected values, multiply the total number of observations by the proportion predicted for each category. For example, if you have $324$ offspring and expect a 1:1:1:1 ratio across four categories, each expected value is $324 \div 4 = 81$.

The chi-squared formula

The chi-squared value is calculated using this formula:

$\chi^2 = \sum \frac{(O - E)^2}{E}$

Where:

• $\Sigma$ means "sum of" — you calculate the value for each category and add them together
• $O$ is the observed value for a category
• $E$ is the expected value for that category

Calculating degrees of freedom

Degrees of freedom (df) represent the number of categories that can vary independently in your data. This value determines which row of the critical values table you use.

Calculate degrees of freedom using:

$\text{df} = n - 1$

Where $n$ is the number of categories in your experiment.

For example, if you have four phenotypic categories in a genetic cross (such as four different combinations of traits), your degrees of freedom equals $4 - 1 = 3$.

Critical values and probability

After calculating your χ² value, compare it to a critical value from a statistical table. Biologists conventionally use the 5% significance level, which corresponds to a probability ($p$) of $0.05$.

This probability represents the chance that the differences between observed and expected values occurred purely by random chance. At $p = 0.05$, there is a 5% (or 1 in 20) probability that random variation alone could produce the observed results.

Interpreting chi-squared results

Compare your calculated χ² value to the critical value:

If calculated χ² < critical value:

• The difference between observed and expected results is not statistically significant
• The differences are likely due to random chance
• You accept the null hypothesis
• Example: If your calculated χ² is $3.23$ and the critical value is $7.82$, the probability of getting this result by chance is greater than 5%, so you accept the null hypothesis

If calculated χ² > critical value:

• The difference is statistically significant
• The differences are unlikely to be due to chance alone
• You reject the null hypothesis
• A biological factor (such as gene linkage) is affecting the results
• Example: If your calculated χ² is $181.83$ and the critical value is $7.82$, the probability of getting this result by chance is less than 5%, so you reject the null hypothesis

Application to genetics: Worked example

Chi-squared and genetic linkage

Chi-squared tests help determine whether genes are linked (on the same chromosome) or show independent assortment (on different chromosomes).

When two genes are on different chromosomes, they assort independently during meiosis. In metaphase I, homologous chromosome pairs can align in two equally probable orientations. This produces four types of gametes in equal proportions ($0.25$ or $\frac{1}{4}$ each). Random fertilisation then produces the characteristic Mendelian ratios.

Example with independent assortment

Students crossed pure-breeding wild-type flies (long wings, red eyes) with flies having vestigial wings and sepia eyes. All $F_1$ offspring showed the wild-type phenotype. The $F_1$ flies were then crossed together, producing these $F_2$ offspring:

Performing a chi-squared test on these data:

The calculated χ² is $5.39$ with 3 degrees of freedom. The critical value at $p = 0.05$ is $7.82$. Since $5.39 < 7.82$, the null hypothesis is accepted. The genes for wing length and eye colour are on different chromosomes and show independent assortment.

Example with gene linkage

When the same students tested wing length alongside a different eye colour gene (cinnabar), they found very different results. In a test cross, they obtained:

The expected ratio for unlinked genes would be 1:1:1:1 (approximately $71$ flies in each category). However, the observed results show two large classes ($136$ and $119$) and two very small classes ($15$ and $14$).

The two larger classes ($136$ and $119$) represent parental combinations — these are the original combinations of alleles present in the heterozygous parent. The two smaller classes ($15$ and $14$) represent recombinant phenotypes produced by crossing over during prophase I of meiosis.

Remember!