# Statistical analysis in ecology (t-test, Spearman, Pearson, Simpson index, mark-release-recapture)

> Biology · CIE A-Level 9700
> Source: https://www.owlsprep.com/study/cie-9700-u19-statistical-analysis-in-ecology/

This module walks you through 5 core ecology statistical methods required for CIE A Level Biology, including test selection, calculation steps, significance interpretation, and field data limitations.

**Prerequisites:** [Understand basic ecological sampling methods (quadrats, transects)](https://www.owlsprep.com/study/cie-9700-u19-ecological-sampling-methods/); [Recall null hypothesis and 95% significance level definitions](https://www.owlsprep.com/study/cie-9700-u12-null-hypothesis-basics/)

## Learning objectives

- Select appropriate statistical tests for different ecology field data sets
- Calculate and interpret results for t-test, Spearman's rank, Pearson correlation and Simpson's diversity index
- Apply mark-release-recapture methodology to estimate wild animal population size
- Evaluate limitations of each statistical method for real-world ecological data

## Test Selection for Ecological Datasets

CIE exam questions frequently require you to justify your choice of statistical test for a given scenario, rather than just performing calculations. The first step is to identify if you are comparing two groups, testing for correlation, measuring diversity, or estimating population size.

| Scenario | Appropriate Test |
| --- | --- |
| Compare mean plant height in shaded vs sunlit sites | Unpaired t-test |
| Test if soil pH correlates with lichen coverage | Spearman's Rank |
| Test if temperature correlates with metabolic rate | Pearson Correlation |
| Quantify diversity of insect species in a meadow | Simpson's Index |
| Estimate total rabbit population in a woodland | Mark-Release-Recapture |

**Worked example:** A student collects data on number of earthworms in 10 quadrats from a pasture and 10 quadrats from an arable field. Select the correct statistical test and justify your choice.

1. Step 1: Identify the two independent groups: pasture and arable field earthworm counts
2. Step 2: Confirm you are comparing the mean of two unrelated datasets
3. Step 3: Select unpaired t-test, as it is designed to test for significant difference between two independent sample means

**Check your understanding**

Test your understanding of test selection

1. Which test is most appropriate for measuring correlation between percentage vegetation cover and distance from a river?

   - A) Unpaired t-test
   - B) Spearman's Rank
   - C) Simpson's Index
   - D) Mark-Release-Recapture

   *Why:* Spearman's rank is ideal for non-linear transect data that is not normally distributed.

> **Exam tip:** Always state your justification for test selection explicitly in exams, even if not asked directly, to earn full method marks.

## Unpaired t-test for Habitat Comparison

**Unpaired t-test** — Calculates the difference between two sample means relative to the spread of the data

*Notation:* t = \frac{|\bar{x_1} - \bar{x_2}|}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

*Example:* Comparing mean species richness of two different woodland sites

**Worked example:** Sample 1 mean = 12, variance = 3, n=10; Sample 2 mean = 8, variance = 2, n=10. Calculate the t statistic.

1. Step 1: Calculate absolute difference between means: |12 - 8| = 4
2. Step 2: Calculate denominator: sqrt((3/10)+(2/10)) = sqrt(0.5) ≈ 0.707
3. Step 3: Divide difference by denominator: t = 4 / 0.707 ≈ 5.66
4. Step 4: Compare to critical t value for 18 degrees of freedom at p=0.05: critical t ≈ 2.10, so result is significant

*Calculator:* allowed

## Correlation Tests: Pearson vs Spearman's Rank

**Comparing methods**

These two correlation tests are often confused, but they have very different use cases in ecology

- **Pearson Correlation** — Requires normally distributed continuous data, measures linear correlation only
  - Pros: Higher sensitivity for linear relationships
  - Cons: Gives misleading results for non-linear or skewed data

- **Spearman's Rank** — Non-parametric, ranks data points before calculation, works for ordinal or non-linear data
  - Pros: Robust to outliers, suitable for most field ecology data
  - Cons: Less sensitive for perfectly linear normally distributed data

**Worked example:** Calculate Spearman's rank for 5 paired data points where sum of squared difference between ranks = 6

1. Step 1: Use formula r_s = 1 - \frac{6\sum d^2}{n(n^2 -1)}
2. $$r_s = 1 - \frac{6 * 6}{5*(25 -1)} = 1 - \frac{36}{120} = 0.7$$
3. Step 3: Interpret result: strong positive correlation between the two variables

*Calculator:* allowed

## Simpson's Diversity Index

**Simpson's Diversity Index** — n = number of individuals of each species, N = total number of individuals of all species

*Notation:* D = 1 - \frac{\sum n(n-1)}{N(N-1)}

*Example:* A D value of 0.8 indicates very high diversity, while 0.2 indicates very low diversity dominated by one species

**Worked example:** A sample has 3 species: 10 daisies, 5 dandelions, 2 clover. Calculate Simpson's D.

1. Step 1: Calculate total N = 10 +5 +2 =17
2. Step 2: Calculate sum of n(n-1): 10*9 +5*4 +2*1 = 90 +20 +2 =112
3. Step 3: Calculate denominator N(N-1) =17*16=272
4. Step 4: D = 1 - (112/272) ≈ 0.59, indicating moderate diversity

*Calculator:* allowed

## Mark-Release-Recapture Population Estimation

This method for estimating mobile animal populations relies on 3 core assumptions: no births, deaths or migration during the sampling period, marks do not fall off, and marked individuals mix fully back into the population before recapture.

**Worked example:** You mark 50 snails, release them, then recapture 40 snails 3 days later, 8 of which are marked. Estimate total population size.

1. Step 1: Use formula N = \frac{M * R}{m}, where M = number marked initially, R = total recaptured, m = number of marked recaptured
2. $$N = \frac{50 * 40}{8} = 250$$
3. Step 3: Interpret result: estimated total snail population is 250 individuals

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using Pearson correlation for non-linear transect data
  - Why it fails: Pearson only measures linear correlation, so it will return artificially low r values for clear non-linear relationships
  - Correct: Use Spearman's rank for all ecological transect data unless you confirm normality and linearity
- **Wrong:** Forgetting to compare calculated test statistic to critical value
  - Why it fails: You cannot confirm statistical significance without referencing the 95% confidence threshold for your degrees of freedom
  - Correct: Always cross-reference your calculated statistic against the provided critical value table before drawing conclusions
- **Wrong:** Calculating Simpson's index for samples with fewer than 10 total individuals
  - Why it fails: Small sample sizes skew D to artificially low values that do not reflect real habitat diversity
  - Correct: Collect a sufficiently large representative sample before calculating diversity metrics
- **Wrong:** Extending mark-release-recapture sampling over more than 7 days
  - Why it fails: Population turnover (births, deaths, migration) breaks the core assumptions of the method
  - Correct: Complete recapture within 48 hours to minimise changes to the population
- **Wrong:** Using a paired t-test for two separate independent habitat datasets
  - Why it fails: Paired t-test is designed for matched samples from the same site, not two unrelated sites
  - Correct: Use unpaired t-test when comparing biodiversity of two geographically separate locations

## Cheatsheet

| Method | Core Use Case | Key Formula | Significance Rule |
| --- | --- | --- | --- |
| Unpaired t-test | Compare two independent sample means | $t = \frac{\|\bar{x_1} - \bar{x_2}\|}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$ | Significant if calculated t > critical t at p=0.05 |
| Spearman's Rank | Test correlation for non-linear data | $r_s = 1 - \frac{6\sum d^2}{n(n^2-1)}$ | Significant if $r_s$ > critical value at p=0.05 |
| Pearson Correlation | Test linear correlation for normal data | $r = \frac{\sum (x-\bar{x})(y-\bar{y})}{s_x s_y}$ | Significant if r > critical value at p=0.05 |
| Simpson's Index | Measure species diversity | $D = 1 - \frac{\sum n(n-1)}{N(N-1)}$ | No significance test, D ranges 0 to 1 |
| Mark-Release-Recapture | Estimate mobile population size | $N = \frac{M * R}{m}$ | No significance test, estimate only |

## What's next

Mastering these statistical methods is critical for scoring full marks on Paper 5 practical analysis questions, which make up 20% of your total A Level Biology grade. You will now be able to confidently answer questions that ask you to justify test selection, perform calculations, and draw valid conclusions from field ecology data. Next, you can build on this knowledge by exploring how to design a full controlled ecological investigation, and learn to identify sources of error in field sampling that skew your statistical results. You will also connect these methods to broader biodiversity conservation assessments, where Simpson's index is routinely used to measure the impact of habitat restoration projects.

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