A.1.9 - Student's t-test in biological investigations

Choose a test for the biological question

The question being investigated and the way observations are collected determine which statistical test is appropriate.

Suppose a biologist asks whether seedlings grown at two temperatures differ in their mean shoot-length increase over 14 days. Length increase is a quantitative measurement with meaningful intervals, and there are two treatment groups. An independent-samples t-test can address this comparison if its assumptions are reasonable. It assesses the population means represented by the samples, rather than requiring every warmed seedling to grow more than every control seedling.

For a temperature experiment, six seedlings sharing one heated cabinet are not six independent applications of the temperature treatment. A cabinet-specific light or humidity difference could explain their response. Repeating a ruler reading on one seedling likewise improves measurement information without creating more independent seedlings. Identify the unit to which the treatment is independently applied before counting the sample size.

Choose among these different biological questions:

The word paired has different roles in those examples. A correlation deliberately links two different variables measured on the same patch. A before-and-after measurement of the same organisms creates a paired mean-comparison design, which needs an appropriate paired analysis. It must not be put into the independent t-test taught here. Nor does having equal group sizes make independent observations into matched pairs.

Set the assumptions and hypotheses

A test needs a defensible model of the observations before its numerical result can be trusted.

Here we use the pooled-variance Student independent-samples t-test. This particular version assumes independent observations within and between groups, approximately normal distributions of the response within each population, and a common population variance. A normal distribution is a symmetric bell-shaped distribution; the condition concerns the response within each group, not a requirement that the two groups have the same mean. With small samples, severe skew or extreme outliers can make the simple test unreliable.

The worked investigation below is an original illustrative exercise, not a reported experiment. Twelve independently controlled growth units each contain one seedling of the same cultivar. Seedlings are randomly allocated to six units at 18°C and six at 22°C. Each unit has its own temperature control; positions are randomised, and light, soil, watering and the 14-day interval are kept comparable. One shoot-length increase is recorded per unit. Random allocation helps prevent initial plant differences being systematically assigned to one temperature, while independent units avoid confusing one cabinet with one treatment.

For this exercise, assume that prior pilot work using the same cultivar and protocol supports approximately normal growth responses and similar population variances at these temperatures. Together with the independent design, those stated conditions justify the pooled version. The small illustrative samples will have similar spread, but six observations per group cannot establish normality or prove equal population variances. In a real investigation, inspect the raw data and distribution, check unusual values against the records and consider the design and prior evidence. Do not delete a genuine extreme value just to improve significance.

If a common variance is not reasonable, a suitable unequal-variance method such as Welch's t-test uses its own equation and degrees of freedom. If the design is paired, it needs a paired method. Those are different analyses; do not attach the pooled degrees-of-freedom rule to an unrelated formula. Changing a formula cannot repair a lack of independent replication.

Let $\mu_{18}$ and $\mu_{22}$ denote population mean shoot-length increases over 14 days under the two conditions represented by the experiment. State the hypotheses before examining the results:

• Null hypothesis, $H_0$: $\mu_{18}=\mu_{22}$; the population mean increases do not differ.
• Alternative hypothesis, $H_1$: $\mu_{18}\ne\mu_{22}$; the population mean increases differ.

This is a two-sided test: either a sufficiently large increase or decrease counts against the null. Choose the significance level in advance, here $\alpha=0.05$, or 5%. A justified directional question set in advance could use a one-sided alternative and its matching rule. Choosing the direction after seeing the data would make that rule misleading. Temperature can accelerate some biological processes but can also damage them, so a difference in either direction is meaningful here.

Calculate means and sample variation

Keep the individual measurements so that an average does not hide spread or an unusual observation.

These are all the original illustrative measurements for the independently replicated design. The columns contain separate seedling groups; rows are only a compact way to display them, not matched pairs. Increase was measured consistently to the nearest millimetre.

The sample sizes are $n_{18}=6$ and $n_{22}=6$. Use the arithmetic mean separately for each group:

The observed mean difference is $26-21=5\,\mathrm{mm}$, but this number alone does not establish a population difference. We also need variation within each group.

Every deviation and square is shown below. Subtract each group's own mean; subtracting one overall mean would mix the between-group difference into the within-group spread.

For each group, $s^2=22/(6-1)=4.4\,\mathrm{mm}^2$, so $s=\sqrt{4.4}=2.0976\ldots\,\mathrm{mm}$, approximately 2.1 mm. Variance has squared length units; taking the square root gives SD in millimetres. Retain the unrounded variance for later working instead of squaring a prematurely rounded SD.

Both ranges are 6 mm: $24-18=6$ and $29-23=6$. Range depends only on the extremes; SD uses every observation's distance from the mean. Neither summary proves a distribution is normal. These deliberately simple teaching groups have identical sample variances; real independent samples need not have identical observed variances even when their populations share a common variance.