use the t-test to compare the means of two different samples (the formula for the t-test will be provided, as shown in the Mathematical requirements)

Statistical Analysis of Biological Variation (Optional Supplement)

Learning Objectives (with Assessment Objectives)

• AO1 – Knowledge: understand the biological reasons for measuring variation and the statistical concepts behind the t‑test.
• AO2 – Data handling: select, organise, analyse and evaluate data using the independent‑samples t‑test (or Welch’s t‑test) and report results correctly.
• AO3 – Experimentation: plan and carry out a simple experiment, check assumptions, and evaluate the reliability and limitations of the statistical conclusions.

Why Study Variation in Biology?

• Variation is the raw material for natural selection (Topics 17 & 18).
• Differences in genotype, phenotype, disease susceptibility or treatment response are quantified using statistical tools.
• Statistical tests tell us whether an observed difference is likely to be a real biological effect or just random chance.

Link to the Cambridge 9700 Syllabus

Statistical Foundations (AO1)

p‑value: the probability of obtaining a test statistic at least as extreme as the observed value, assuming the null hypothesis is true. A small p‑value (≤ α, usually 0.05) leads to rejection of H₀.

Type I & Type II errors:

• Type I – false positive (rejecting a true H₀). Risk = α.
• Type II – false negative (failing to reject a false H₀). Risk = β; power = 1 – β.

Confidence interval (CI): a range of values that is likely to contain the true population mean difference with a given confidence (usually 95 %). Reporting a CI alongside a p‑value gives information about the magnitude and precision of the effect.

Effect size (Cohen’s d): quantifies the biological importance of a difference.

\[

d = \frac{\bar{x}1-\bar{x}2}{s_{\text{pooled}}}

\]

where \(s{\text{pooled}}=\sqrt{sp^{2}}\). Interpretation: small (≈0.2), medium (≈0.5), large (≥0.8).

Multiple‑testing correction: when several hypotheses are tested on the same data set, the chance of a Type I error rises. Simple methods such as the Bonferroni correction (divide α by the number of tests) can be mentioned.

Non‑parametric alternative: if normality or homogeneity of variance cannot be satisfied, the Mann‑Whitney U test (also called Wilcoxon rank‑sum) may be used.

Data visualisation: box‑plots or dot‑plots give a quick visual check of normality, spread and outliers before running a t‑test.

When to Use the Independent‑Samples t‑Test (AO2)

• Two independent groups (e.g., two genotypes, two treatments, two populations).
• The response variable is continuous and approximately normally distributed in each group.
• Variances are equal (homoscedastic). If not, use Welch’s t‑test.
• Sample sizes are moderate (≥ 5 per group) – the test is robust when assumptions are met.

Statistical Checklist (Before Running the Test) – AO2

1. Independence: Ensure observations in one group do not influence the other.
2. Normality: Examine histograms, Q‑Q plots, or perform Shapiro‑Wilk/Kolmogorov‑Smirnov tests.
3. Equality of variances: Conduct Levene’s test or an F‑test.
4. Choose the test:

   • Equal variances → Student’s (pooled) t‑test.
   • Unequal variances → Welch’s t‑test.
   • Severe non‑normality → Mann‑Whitney U.

Statistical Toolbox Overview (AO1)

• Student’s (pooled) independent‑samples t‑test – equal variances.
• Welch’s t‑test – unequal variances.
• Chi‑square test – association between categorical variables.
• One‑way ANOVA – compare >2 group means.
• Simple linear regression – relationship between two continuous variables.
• Mann‑Whitney U test – non‑parametric alternative for two independent samples.

Formulae (AO1)

Student’s (pooled) independent‑samples t‑test

\[

t = \frac{\bar{x}1 - \bar{x}2}{\sqrt{sp^{2}\!\left(\frac{1}{n1}+\frac{1}{n_2}\right)}}

\]

\[

sp^{2}= \frac{(n1-1)s1^{2}+(n2-1)s2^{2}}{n1+n_2-2}

\]

Welch’s t‑test (unequal variances)

\[

t = \frac{\bar{x}1 - \bar{x}2}{\sqrt{\frac{s1^{2}}{n1}+\frac{s2^{2}}{n2}}}

\]

\[

df = \frac{\left(\frac{s1^{2}}{n1}+\frac{s2^{2}}{n2}\right)^{2}}

{\frac{(s1^{2}/n1)^{2}}{n1-1}+\frac{(s2^{2}/n2)^{2}}{n2-1}}

\]

Effect size (Cohen’s d)

\[

d = \frac{\bar{x}1-\bar{x}2}{\sqrt{s_p^{2}}}

\]

Step‑by‑Step Procedure (Student’s t‑test) – AO2 & AO3

1. State hypotheses:

   
   \(H{0}:\mu{1}=\mu_{2}\) (no difference)

   
   \(H{A}:\mu{1}\neq\mu_{2}\) (two‑tailed) or \(>\) / \(<\) (one‑tailed).

2. Collect data and enter into a spreadsheet.
3. Calculate descriptive statistics – means \(\bar{x}1,\bar{x}2\) and variances \(s1^{2},s2^{2}\).
4. Check assumptions using the checklist. If variances differ, repeat steps 5‑9 with Welch’s formula.
5. Compute pooled variance \(s_p^{2}\) (or skip this for Welch’s).
6. Calculate the t‑value** using the appropriate formula.
7. Determine degrees of freedom:

   
   Student’s: \(df = n1+n2-2\)

   
   Welch’s: use the approximation formula above.

8. Obtain the critical t (or p‑value) from a t‑distribution table or software for the chosen \(\alpha\) (usually 0.05) and the appropriate tail(s).
9. Decision:

   • If \(|t| > t{\text{crit}}\) (or \(p \le \alpha\)) → reject \(H{0}\).

   • Otherwise, fail to reject \(H_{0}\).

10. Report the result in the standard format, e.g.

    t(10) = -3.42, p = 0.008, d = 1.2.

11. Interpretation (AO3): translate the statistical conclusion into a biological statement, discuss effect size, confidence interval and any limitations.

Worked Biological Example (AO1–AO3)

Biological context (Topic 12 – Enzymes): A mutant allele is suspected to increase the activity of a key metabolic enzyme.

• \(n1=n2=6\)
• \(\bar{x}1 = 12.1\) U mg⁻¹, \(\bar{x}2 = 15.1\) U mg⁻¹
• \(s1^{2}=0.07\), \(s2^{2}=0.09\)
• Levene’s test: p = 0.62 → equal variances → use Student’s t‑test.
• Pooled variance: \(s_p^{2}= \frac{5(0.07)+5(0.09)}{10}=0.08\)
• \[

  t = \frac{12.1-15.1}{\sqrt{0.08\left(\frac{1}{6}+\frac{1}{6}\right)}}=

  \frac{-3.0}{\sqrt{0.08\times0.333}}=

  \frac{-3.0}{0.163}= -18.4

  \]

• Degrees of freedom: \(df = 6+6-2 = 10\)
• Critical t (two‑tailed, α = 0.05, df = 10) = 2.228.
• \(|t| = 18.4 > 2.228\) → reject \(H_{0}\).
• Effect size: \(d = \dfrac{-3.0}{\sqrt{0.08}} = -10.6\) (very large).
• 95 % CI for the mean difference (using software) ≈ –3.2  to –2.8 U mg⁻¹.

Result reporting: t(10) = -18.4, p < 0.001, d = -10.6, 95 % CI = [-3.2, -2.8]

Interpretation & Evaluation (AO3)

• Biological meaning: The mutant allele produces a markedly higher enzyme activity; the effect is both statistically significant and biologically large.
• Assumption check: Shapiro‑Wilk p > 0.2 for both groups (normality); Levene’s test p = 0.62 (equal variances). Sample size meets the minimum requirement for a t‑test.
• Limitations:

  • Only six replicates per group – limited power to detect smaller effects.
  • Experiment performed under a single set of growth conditions; results may not generalise to other environments.
  • Potential hidden confounders (e.g., slight differences in protein extraction efficiency).

• Further work: increase n, test additional mutant alleles, perform a dose‑response assay, or use ANOVA to compare more than two genotypes.

Suggested Classroom Activity (AO3)

Objective: Test whether two seed genotypes differ in mean germination time.

1. Formulate a hypothesis (e.g., “Genotype A germinates faster than Genotype B”).
2. Plant at least 8 seeds of each genotype under identical conditions.
3. Record the time (hours) each seed takes to germinate.
4. Enter data into a spreadsheet; produce a box‑plot to visualise spread and check normality.
5. Run the appropriate t‑test (Student’s or Welch’s) or Mann‑Whitney U if assumptions fail.
6. Report the result in the format t(df) = value, p = …, d = … and discuss the biological implication.

Key Take‑aways (AO1 & AO2)

• The independent‑samples t‑test (or Welch’s version) is the standard method for comparing two group means in biological research.
• Checking assumptions (independence, normality, equal variances) is essential before interpreting the test.
• Report the test statistic, degrees of freedom, p‑value, effect size and, where possible, a confidence interval.
• Always link the statistical conclusion back to the underlying biological question and evaluate the reliability of the result.

Summary Table – What Students Should Be Able to Do

Suggested Flowchart (to be drawn on board or slide)

Hypothesis → Data collection → Visual check (box‑plot) → Test assumptions (normality, equal variance) → Choose Student’s, Welch’s or Mann‑Whitney U → Calculate test statistic & df → Obtain p‑value (or CI) → Decision (reject/fail to reject H₀) → Biological interpretation & evaluation.