Handling Data (AQA A-Level Biology): Revision Notes

Statistical Tests

Statistical tests help biologists determine whether observed differences in data are due to genuine effects or simply random chance. In biology, any probability greater than 5% suggests results could be due to chance alone, while probabilities of 5% or below indicate the data differ significantly and a real cause must be influencing the outcome.

Understanding statistical significance

Statistical significance occurs when the probability that results are due to chance alone is 5% or less (p ≤ 0.05). This means we can be at least 95% confident that observed differences represent real effects rather than random variation.

The null hypothesis assumes there is no significant difference between observed and expected results. Statistical tests help determine whether to accept or reject this null hypothesis based on calculated probability values.

Chi-squared (χ²) test

The chi-squared test compares patterns in collected data with patterns expected by chance. This test determines how much observed frequencies deviate from expected frequencies.

When to use chi-squared tests

Use chi-squared tests when comparing observed frequencies with expected frequencies, particularly for checking genetic cross results. For example, when testing whether a die is fair by comparing actual throws with expected equal frequencies for each number.

Chi-squared formula and calculation

The formula for the chi-squared test is:

$\chi^2 = \sum \frac{(O-E)^2}{E}$

Where:

• O = observed values
• E = expected values
• Σ = sum of all categories

Interpreting chi-squared results

Calculate degrees of freedom as the number of categories minus 1. Here: 2 categories - 1 = 1 degree of freedom.

Compare the calculated χ² value (0.77) with the critical value from statistical tables. For 1 degree of freedom at p = 5%, the critical value is 3.84. Since 0.77 < 3.84, the probability is between 10% and 50%, so we accept the null hypothesis - no significant difference exists between observed and expected results.

Student t test

The Student t test judges whether differences between means of two data sets are statistically significant. This test requires normally distributed data with sufficient sample sizes (ideally more than 15 in each group).

When to use t tests

Use t tests when comparing means from two independent groups to determine if observed differences are statistically significant. Sample sizes do not need to be equal between groups.

Student t test formula and calculation

The formula for an unpaired t test is:

$t = \frac{\bar{x\_1} - \bar{x\_2}}{\sqrt{\frac{s\_1^2}{n\_1} + \frac{s\_2^2}{n\_2}}}$

Where:

• $\bar{x\_1}$ and $\bar{x\_2}$ = means of each group
• $s\_1^2$ and $s\_2^2$ = variances of each group
• $n\_1$ and $n\_2$ = sample sizes of each group

Correlation coefficient (Pearson's product moment correlation coefficient)

The correlation coefficient (r) measures the strength and direction of linear relationships between two variables. Values range from -1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no correlation.

When to use correlation analysis

Use correlation analysis when examining relationships between two continuous variables. Always plot data on scatter graphs first to visualise potential relationships before calculating correlation coefficients.

Correlation coefficient formula and calculation

The formula is:

$r = \frac{\sum (x-\bar{x}) \times (y-\bar{y})}{\sqrt{\sum (x-\bar{x})^2 \times \sum (y-\bar{y})^2}}$

Where:

• x = values of first variable, $\bar{x}$ = mean of first variable
• y = values of second variable, $\bar{y}$ = mean of second variable
• Σ = sum of all values

Using statistical tables effectively

Understanding how to calculate and use degrees of freedom is crucial for interpreting statistical results correctly.

Degrees of Freedom Calculations:

• Chi-squared: number of categories - 1
• t test (unpaired): (n₁ + n₂) - 2
• Correlation: n - 2

Compare calculated values with critical values from appropriate statistical tables. If calculated values exceed critical values at p = 0.05, results are statistically significant.