Statistics for biology: study guide

Key ideas

Biologists test whether observed counts fit a prediction, and they report how precisely a mean is known. The chi-square test and the standard error cover most of what is asked.

• The expected count for a category is the total times the predicted fraction, for example 75 % of 100 for a 3 : 1 ratio.
• Chi-square is χ² = Σ(O − E)² ÷ E, summed over categories. Degrees of freedom are the number of categories minus 1.
• If χ² is below the critical value (3.84 for 1 degree of freedom at p = 0.05), the difference from the prediction can be due to chance and the null hypothesis is not rejected.
• The standard error of the mean is SE = SD ÷ √n. It falls as the sample grows, while the standard deviation does not.

Key equations

• χ² = Σ (O − E)² ÷ E
• df = categories − 1
• SE = SD ÷ √n
• expected = total × ratio fraction

Worked example

A cross predicts 3 : 1, and 100 offspring give 70 dominant and 30 recessive. Expected values are 75 and 25. χ² = (70 − 75)² ÷ 75 + (30 − 25)² ÷ 25 = 0.33 + 1.00 = 1.33, with 1 degree of freedom. This is below 3.84, so the data fit the 3 : 1 ratio. A sample of 36 with SD = 6 has SE = 6 ÷ 6 = 1.

Common mistakes

• Using the observed counts as the expected counts, or percentages in place of counts.
• Taking the number of individuals minus 1 as the degrees of freedom. It is the number of categories minus 1.
• Saying that a non-significant result proves the prediction. It only fails to reject it.

Practice

Test yourself with practice questions on this topic: Statistics for biology