Statistical tests and provided formulae

Introduction to Statistics in Biology

In Biology, data can be "messy." If you measure the heights of two different groups of plants, the averages might look different, but how do you know if that difference is significant or just down to random chance? That is where statistical tests come in!

The best part about this chapter for your OCR AS Level is that you do not need to memorize these complex formulas. They will be provided for you in the exam. Your job is to know which test to use, how to plug in the numbers, and how to explain what the result means.

Note: For other mathematical skills like magnification or calculating rates, please see the chapters on "Magnification, scale and surface area to volume ratio" and "Rates, Q10 and biological formulae to recall".

1. Standard Deviation (\(s\))

Standard deviation tells us how spread out our data is around the mean (average).

• Low Standard Deviation: The data points are all very close to the mean. This suggests your results are consistent and reliable.
• High Standard Deviation: The data is spread out widely. This suggests the results are more variable.

The Provided Formula:
\(s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}\)

What the symbols mean:
\(\sum\) = "sum of" (add them all up)
\(x\) = an individual value
\(\bar{x}\) = the mean of the values
\(n\) = the total number of values in the sample

Quick Tip: If you see "error bars" on a graph in your exam, they are often showing the standard deviation. If the bars for two different means overlap, the difference between them is likely not significant.

2. Student's t-test

The t-test is used when you want to compare the means of two sets of data to see if there is a significant difference between them.

Paired vs. Unpaired

• Unpaired t-test: Used when the two groups are different. Example: Comparing the mass of shells on a sheltered shore vs. an exposed shore.
• Paired t-test: Used when you measure the same individuals twice. Example: Measuring the resting heart rate of 10 students before and after they drink caffeine.

Key Takeaway: If your calculated \(t\) value is greater than the critical value (found in a table), we say the difference is significant. There is less than a 5% probability that the difference is due to chance.

3. Spearman's Rank Correlation Coefficient (\(r_s\))

Use Spearman's Rank when you want to see if there is a correlation (a relationship) between two different variables.

The Scale:
The result will always be between \(+1\) and \(-1\).
\(+1\) = A perfect positive correlation (as one goes up, the other goes up).
\(0\) = No correlation at all.
\(-1\) = A perfect negative correlation (as one goes up, the other goes down).

The Provided Formula:
\(r_s = 1 - \frac{6\sum d^2}{n(n^2 - 1)}\)

Step-by-Step Logic:
1. Rank both sets of data from highest to lowest.
2. Find the difference (\(d\)) between the ranks.
3. Square those differences (\(d^2\)) and add them up (\(\sum d^2\)).
4. Plug the numbers into the formula!

4. Chi-squared Test (\(\chi^2\))

The Chi-squared test is used for categorical data. It compares the frequencies you observed in an experiment with the frequencies you expected to see.

Example: You expect a 1:1 ratio of male to female kittens in a litter, but you actually get 4 males and 1 female. Is this difference just luck, or is something else going on?

The Provided Formula:
\(\chi^2 = \sum \frac{(O - E)^2}{E}\)

What the symbols mean:
\(O\) = Observed value (what you actually counted)
\(E\) = Expected value (what you predicted based on a theory or ratio)

Summary Table: Which test should I use?

Are you looking for a relationship/link? \(\rightarrow\) Spearman's Rank Correlation.
Are you comparing two means? \(\rightarrow\) Student's t-test.
Are you comparing observed counts vs. expected counts? \(\rightarrow\) Chi-squared test.

5. Simpson's Index of Diversity (\(D\))

In Module 4 (Biodiversity), you learn about measuring how diverse a habitat is. You don't need to memorize this formula either!

The Provided Formula:
\(D = 1 - \left( \sum \left( \frac{n}{N} \right)^2 \right)\)

What the symbols mean:
\(n\) = total number of organisms of a particular species
\(N\) = total number of organisms of all species

Interpretation: The value of \(D\) is always between \(0\) and \(1\).
Close to 1: High biodiversity (the habitat is stable and healthy).
Close to 0: Low biodiversity (the habitat is dominated by few species and is easily damaged).

6. Understanding "Significance" and Critical Values

When you finish a calculation, you get a "test statistic" (like a \(t\) value or a \(\chi^2\) value). To know what it means, you must compare it to a Critical Value from a table provided in the exam.

The Null Hypothesis (\(H_0\))

In statistics, we always start by assuming nothing interesting is happening. This is the Null Hypothesis.
Example: "There is no significant difference between the means of group A and group B."

The 5% Threshold (\(p = 0.05\))

In Biology, we usually use the 0.05 (5%) probability level.
If your calculated value is higher than the critical value in the table at \(p = 0.05\):
1. We reject the Null Hypothesis.
2. The result is significant.
3. There is a less than 5% probability that the results are due to chance.

Degrees of Freedom (\(df\))

To find the correct critical value in a table, you need to know the "degrees of freedom." The exam will usually tell you how to calculate this for the specific test, but common ones include:
- For Chi-squared: \(number \: of \: categories - 1\)
- For Spearman's Rank: \(n\) (number of pairs)

Common Mistake to Avoid: Don't confuse "significant" with "important." In stats, significant simply means the result is unlikely to have happened by accident.

Quick Review:
- Standard Deviation = spread of data around the mean.
- t-test = comparing two means.
- Spearman’s Rank = looking for a correlation.
- Chi-squared = comparing observed vs. expected frequencies.
- Critical Value = the "target" number you must beat to prove your result is significant.