Difficulty: Introductory | Prerequisites: Basic algebra, familiarity with scatterplots
Biology is an empirical science, and the ability to design experiments, read graphs, and interpret statistical output sits at the centre of every topic you will cover in BIOL 180. This section focuses on the tools researchers use to decide whether a pattern in data is real or just noise: p-values, R² values, null and alternative hypotheses, and regression. If you can read a scatterplot and explain what its statistics mean in plain English, you can tackle any data-interpretation question on the exam. You should already be comfortable with the idea of variables (independent vs dependent) and what it means to plot one against the other.
A p-value tells you how likely your data would be if there were no real effect; a small p (below 0.05) means the pattern is probably real. R² tells you how much of the variation in your dependent variable is explained by the independent variable. Together, these two numbers let you judge 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, it answers the question: "If nothing interesting is going on, how weird is my result?"
R² (coefficient of determination)
The proportion of variation in the dependent variable that is explained by the independent variable, expressed as a value between 0 and 1. Think of it as a measure of how tightly your data points cluster around the best-fit line.
Null hypothesis (H₀)
A statement that there is no relationship or no difference between the variables being studied. It is the default assumption that the researcher tries to reject.
Alternative hypothesis (Hₐ)
A statement that there is a relationship or a difference between the variables. This is what the researcher suspects is true and hopes the data will support.
Linear regression
A statistical method that fits a straight line through data points on a scatterplot to describe the relationship between two continuous variables. Think of it as drawing the "best" line through a cloud of dots.
Statistical significance
A result is statistically significant when the p-value falls below a pre-set threshold (typically 0.05). This means the observed pattern is unlikely to have arisen by chance alone.
Treatment group
The group in an experiment that receives the experimental condition (e.g., seawater from a tank with crabs).
Control group
The group that receives the baseline or standard condition (e.g., plain seawater), used as the point of comparison.
Each dot (or symbol) on a scatterplot represents one individual organism or one observation.
The x-axis shows the independent variable (e.g., clam shell length), and the y-axis shows the dependent variable (e.g., burrowing depth).
Different symbols or colours distinguish treatment groups (e.g., plain seawater vs crab-water).
A best-fit line drawn through the data shows the overall trend.
p < 0.05: the result is statistically significant. You reject the null hypothesis. The observed relationship is unlikely to be due to chance.
p ≥ 0.05: the result is not statistically significant. You fail to reject the null hypothesis. You cannot conclude there is a real relationship.
A p-value does not tell you the size or importance of an effect, only whether it is likely to be real.
R² = 0.02 means 2% of the variation in Y is explained by X. That is a very weak relationship.
R² = 0.21 means 21% of the variation in Y is explained by X. That is a moderate relationship.
R² ranges from 0 (no explanatory power) to 1 (perfect prediction).
A high R² without a significant p-value can happen with small sample sizes, and a significant p-value with a low R² means the relationship is real but weak.
The null hypothesis always states "no effect" or "no relationship." For example: "There is no relationship between clam shell length and burrowing depth."
The alternative hypothesis states that there is a relationship or effect. For example: "There is a relationship between clam shell length and burrowing depth."
Hypotheses are written before collecting data, not after seeing results.
When both variables are continuous (numerical) and you want to test for a linear relationship, a linear regression is appropriate.
The regression produces both a p-value (is the slope significantly different from zero?) and an R² (how much variation does the line explain?).
Look at each treatment group separately. A significant relationship in one group but not the other tells you something about the experimental condition.
In the clam example: plain seawater shows no significant relationship between shell length and burrowing depth (p = 0.29, R² = 0.02). But in crab-water, larger clams burrow deeper (p < 0.05, R² = 0.21). The presence of predator cues changes clam behaviour, and the effect depends on body size.
Students often think a non-significant p-value means the null hypothesis is true. It does not. It means you lack sufficient evidence to reject it. The relationship might still exist but be too small or the sample too small to detect.
Students often confuse statistical significance with practical importance. A p-value below 0.05 with R² = 0.02 means the relationship is real but explains almost nothing. Significant does not mean strong.
Students sometimes believe R² tells you whether a result is significant. It does not. R² tells you the strength of the relationship; the p-value tells you whether it is statistically reliable.
Students often write hypotheses after seeing the data and tailor them to match the results. Hypotheses must be stated before analysis.
⚠️ You will be asked to interpret p-values and R² from a graph or dataset. Be ready to explain both in one or two sentences.
⚠️ Know how to write null and alternative hypotheses for a regression. The null always says "no relationship."
⚠️ Be prepared to identify the correct statistical test. If both variables are continuous, the answer is linear regression.
⚠️ Expect a question asking what each data point on a scatterplot represents. The answer is always one individual or one observation.
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 = 0.45 means that ___% of the variation in the dependent variable is explained by the independent variable.
45%
True or false: If p = 0.29, you should reject the null hypothesis.
False. You fail to reject the null hypothesis because p > 0.05.
Fill in the blank: When testing for a linear relationship between two continuous variables, the appropriate statistical test is ___.
Linear regression
True or false: A statistically significant result always means the effect is large and important.
False. Significance tells you the result is unlikely due to chance; R² tells you how strong it is.
Q: In a scatterplot of clam shell length vs burrowing depth, what does each dot represent?
A: Each dot represents one individual clam, showing that clam's shell length and its burrowing depth.
Q: A researcher reports p = 0.29 and R² = 0.02 for a regression. Is the relationship statistically significant? How strong is it?
A: The relationship is not statistically significant (p > 0.05), so we fail to reject the null hypothesis. R² = 0.02 means only 2% of the variation in Y is explained by X, which is a very weak relationship.
Q: A different group shows p < 0.05 and R² = 0.21. Interpret these values.
A: The relationship is statistically significant (p < 0.05), so we reject the null hypothesis and conclude there is a real relationship. R² = 0.21 means 21% of the variation in Y is explained by X, a moderate but meaningful relationship.
Q: Write the null and alternative hypotheses for a regression testing whether clam shell length predicts burrowing depth.
A: Null hypothesis: There is no relationship between clam shell length and burrowing depth. Alternative hypothesis: There is a relationship between clam shell length and burrowing depth.
Q: What statistical test would you use to determine whether there is a linear relationship between two continuous variables?
A: Linear regression.
Statistical reasoning connects directly to every experimental question in BIOL 180. When you study natural selection, genetic drift, or any evolutionary mechanism, the evidence comes from data analysed with the same tools covered here. Understanding p-values and R² will also help you evaluate published research papers and the primary literature assignments in later biology courses.
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