Difficulty: Introductory | Prerequisites: Basic algebra, familiarity with scatterplots
Linear regression tests whether two numerical variables are associated (e.g. clam size and burrowing depth). The p-value tells you how likely the observed pattern is under the null hypothesis, and R² tells you how much of the variation in one variable is explained by the other. Together they let you decide whether a relationship is statistically significant and how strong it is.
p-value
The probability of observing data as extreme as (or more extreme than) what was collected, assuming the null hypothesis is true. In simple terms, a small p-value means the pattern you see is unlikely to be caused by chance alone.
R² (coefficient of determination)
The proportion of variation in the response variable that is explained by the explanatory variable. Think of it as a measure of how tightly the data cluster around the best-fit line. An R² of 0.21 means 21% of the variation is explained.
Null hypothesis (H₀)
The default assumption that there is no relationship between the two variables. For a regression, this typically states that the slope of the best-fit line is zero, or equivalently, that the variables are not associated.
Alternative hypothesis (Hₐ)
The hypothesis that the two variables are associated. You accept this when the p-value falls below your significance threshold (usually 0.05).
Linear regression
A statistical test that fits a straight line (the best-fit line) through a scatterplot to quantify the relationship between a numerical explanatory variable and a numerical response variable.
Best-fit line (regression line)
The straight line that minimises the total squared distance between itself and all the data points. Its slope tells you the direction and rate of the relationship.
The p-value is a probability, always between 0 and 1.
A p-value of 0.29 means: "If there were truly no relationship, there is a 29% chance of seeing a best-fit line this steep or steeper." That is not unusual, so we fail to reject the null.
A p-value < 0.05 means the pattern is unlikely under the null hypothesis. We reject the null and conclude the variables are associated.
The standard threshold (alpha) is 0.05, but this is a convention, not a law of nature.
R² ranges from 0 to 1 (sometimes expressed as a percentage).
R² = 0.02 means only 2% of the variation in the response variable is explained by the explanatory variable. The relationship, even if it existed, would be very weak.
R² = 0.21 means 21% of the variation is explained. That is a modest but meaningful relationship.
A low R² with a significant p-value means the relationship is real but other factors also matter.
Null hypothesis: "Variable X is not associated with variable Y" (e.g. "Depth is not associated with length").
Alternative hypothesis: "Variable X is associated with variable Y" (e.g. "Depth is associated with length").
Keep hypotheses about association, not causation, unless the study design supports causal claims (e.g. a controlled experiment).
Combine p-value and R² to form a complete conclusion.
State the direction of the relationship (positive or negative) when the result is significant.
Note which group or condition the conclusion applies to. In the clam experiment, only the crab-scented group showed a significant relationship between shell length and burrowing depth.
A non-significant result does not prove there is no relationship. It means we lack sufficient evidence to detect one.
In a scatterplot, each dot (or symbol) represents one individual observation. In the clam study, each dot represents one individual clam's shell length and burrowing depth.
Students often think a large p-value means the null hypothesis is true. It does not. It means we do not have enough evidence to reject it.
Students confuse p-value and R². The p-value tells you whether a relationship exists; R² tells you how strong it is. You need both.
Students sometimes think R² = 0.21 means "21% of individuals show the pattern." It means 21% of the variation in the response variable is accounted for by variation in the explanatory variable.
Students phrase the p-value as "the probability that the null hypothesis is true." That is incorrect. It is the probability of seeing data this extreme if the null hypothesis were true. The distinction matters.
⚠️ You will be given p-values and R² values and asked to interpret them in words. Practice translating numbers into plain-English sentences.
⚠️ Expect questions that give you two groups with different p-values and R² values and ask you to compare conclusions across groups.
⚠️ You must be able to write null and alternative hypotheses for a regression scenario you have not seen before.
⚠️ Know that "linear regression" is the name of the statistical test used when both variables are numerical and you are looking for a linear association.
True or False: A p-value of 0.03 means there is a 3% chance the null hypothesis is true.
False. It means there is a 3% chance of seeing data this extreme if the null hypothesis were true.
Fill in the blank: R² = 0.35 means that ___% of the variation in the response variable is explained by the explanatory variable.
35%
True or False: If p = 0.50 and R² = 0.01, you should reject the null hypothesis.
False. The p-value is well above 0.05.
Fill in the blank: The statistical test used to assess a linear relationship between two numerical variables is called ___.
Linear regression
True or False: A significant p-value with a low R² means the relationship is both real and strong.
False. It is real (statistically significant) but weak (low R²).
Q: A researcher reports p = 0.12 and R² = 0.08 for the relationship between plant height and number of flowers. What do you conclude?
A: We fail to reject the null hypothesis. There is a 12% probability of seeing a pattern this strong by chance if there were no real relationship. Even if there were a relationship, only 8% of the variation in flower number would be explained by height.
Q: In an experiment, treatment group A shows p < 0.01 and R² = 0.45, while treatment group B shows p = 0.40 and R² = 0.03. Describe what this tells you.
A: In group A, the explanatory and response variables are significantly associated, and 45% of the variation is explained. In group B, there is no significant association, and almost none of the variation is explained. The relationship exists in group A but not in group B.
Q: Write a null and alternative hypothesis for a study testing whether body mass predicts running speed in lizards.
A: Null hypothesis: body mass is not associated with running speed. Alternative hypothesis: body mass is associated with running speed.
Q: A scatterplot shows individual birds. What does each dot represent?
A: Each dot represents one individual bird (specifically, that bird's values for both measured variables).
This material connects directly to experimental design: regression is one of the tools you use to analyse results from controlled experiments (like the clam study with treatment vs. control groups). It also links to evolution topics, since biologists use regression to test whether traits are associated with fitness measures.
Understanding p-values and hypothesis testing reappears in every data-analysis context in biology, from genetics to ecology.
Linear regression, p-value, R-squared, R², coefficient of determination, null hypothesis, alternative hypothesis, best-fit line, regression line, scatterplot, statistical significance, alpha level, hypothesis testing, explanatory variable, response variable, correlation vs. causation, BIOL 180, intro biology statistics