Bivariate data

1. 1Identify the explanatory and response variables.
2. 2Read the instruction first. If the question names row or column totals, or says which percentages to compare, use those. 2020 Paper 1 Question 22 asked for column totals when the staff group, in the rows, was the explanatory variable.
3. 3Otherwise percentage within each category of the explanatory variable, so each of those categories totals 100%.
4. 4Compare the same response category across the groups.
5. 5Answer with calculate, compare, conclude. Quote both percentages, say how big the difference is, and conclude in context.

1. 1Put the explanatory variable on the horizontal axis and the response variable on the vertical axis.
2. 2Give the features the question names. If it just says describe, give all three: direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak).
3. 3Judge strength by how tightly the points cluster around the pattern. Strength is not steepness.
4. 4Name both variables in the question’s own words. The marks are for the features, and the context makes your answer unambiguous.

A complete description reads: there is a strong, negative, linear association between screen brightness and battery life.

1. 1Get $r$ from your calculator’s statistics mode, with the explanatory values as $x$.
2. 2The sign gives the direction and the size gives the strength. Quote the value of $r$ beside your word.
3. 3Look at the scatterplot before you trust $r$. On curved data $r$ is not an appropriate measure, whatever its value.
4. 4From $r$ to $R^2$, square it and the sign disappears. From $R^2$ to $r$, take the square root, then attach the sign from the slope or the scatterplot.
5. 5Put $R^2$ into the sentence in Part 4, with the response variable as the thing whose variation is explained.

1. 1From raw data, use your calculator’s regression function, with the explanatory values as $x$.
2. 2From summary statistics, find $m = r\dfrac{s_y}{s_x}$, then $c = \bar{y} - m\bar{x}$.
3. 3If $s_x$ and $\bar{x}$ are not given, check whether the $x$ values are just 1, 2, 3 and so on. Then you calculate them yourself, as in 2022 Paper 2 Question 3.
4. 4Rewrite the final equation in the letters the question uses. In 2022 Paper 1 Question 16 the marking guide accepted the equation only in $n$ and $t$, so $y = 42.6x + 55.4$ lost the mark.
5. 5Interpret the slope and intercept with the sentences below, using the question’s names and units.

1. 1Substitute into the equation, keeping your calculator’s full precision.
2. 2Check whether the value you substituted sits inside the range of the explanatory variable in the data.
3. 3Inside is interpolation. Outside is extrapolation. Say which.
4. 4For extrapolation, explain the danger. The arithmetic is identical, but you have no evidence the relationship continues outside the observed range, and the further out you go the less support the prediction has. Say how far out you have gone.
5. 5If the question says evaluate the reasonableness of a claim, calculate the prediction, compare it with the claim, then conclude.

1. 1Residual $=$ actual value $-$ predicted value. A point above the line has a positive residual.
2. 2Random scatter above and below zero, with no pattern, means a linear model is appropriate.
3. 3A curve, a steady rise or fall, every point on one side of zero, or any other pattern means a linear model is not appropriate, however high $r$ is.
4. 4Given the residuals and asked for an actual value, work backwards: actual value $=$ predicted value $+$ residual. 2021 Paper 2 Question 5 needed this reverse reading.

1. 1An association, however strong, does not show that one variable causes the other.
2. 2Name a specific non-causal explanation in this context: another variable that drives both (a confounding variable), or coincidence.
3. 3For a confounding variable, say how it drives both. In 2024 Paper 1 Question 20 it was time of day, with temperature and vehicle numbers both low early and late and both high in the middle of the day.
4. 4Also check whether the cause could run the other way from the one claimed.
5. 5“Correlation does not imply causation” is true, but on its own it does not engage with the context, which is what the marks reward.