Bivariate Data Analysis

Bivariate data analysis is about two variables measured on the same people or things, and whether they are related. Most questions ask you to describe a scatterplot, use a line of best fit to make a prediction, or comment on how reliable a conclusion is.

• Univariate and bivariate data, and when each is needed.
• Independent and dependent variables.
• Association and causation: the difference between variables that are associated and a causal relationship.
• Scatterplots: construct them, and describe the association by its form (linear or non-linear), direction (positive or negative) and strength (strong, moderate or weak).
• Lines of best fit: draw by eye or with digital tools, use them to make predictions, and recognise the limitations of interpolation and extrapolation.

Calculating Pearson's correlation coefficient and the least-squares regression line by formula is Standard 2 content. In Standard 1 you describe strength in words and may be given an equation found with technology.

• Univariate data has one variable, such as the heights of a class. You describe it with a mean, median, histogram or box plot (Year 11 Data analysis).
• Bivariate data has two variables for each individual, such as each student's height and arm span. You use it to investigate whether one variable is related to the other.

• The independent variable (explanatory variable) is the one used to explain or predict. It goes on the horizontal axis ($x$).
• The dependent variable (response variable) responds to it. It goes on the vertical axis ($y$).

Ask: "Which one could depend on the other?" The temperature might affect ice-cream sales; ice-cream sales can't affect the temperature. So temperature is independent.

A scatterplot plots each pair of values as one point $(x,y)$. Label both axes with the variable and its units, use an even scale, and plot every point carefully.

Describing the association

A full description uses all three, for example: "There is a strong, negative, linear association between the age of a car and its resale value." Then put it into words in context: "Older cars tend to be worth less."

If the points show no pattern at all, say there is no association. An outlier is a point well away from the rest; mention it if there is one, as it can affect the line of best fit.

A line of best fit is a straight line that summarises a linear association.

Drawing one by eye

• Follow the direction of the points.
• Have roughly the same number of points above and below the line, spread along its length.
• It doesn't have to pass through any data point, or through the origin.

Graphing software and spreadsheets can also draw the line ("trendline") and give its equation.

Finding the equation from two points

Choose two points on the line (not data points, unless they happen to be on it), far apart.

Worked example. A line of best fit relating the number of hours a student works in a part-time job each week ($x$) to their average mark ($y$) passes through (4, 82) and (16, 58).

Use one point to find $c$:

So $y=-2x+90$.

Interpreting. The gradient $-2$ means that, on average, each extra hour of work per week is associated with an average mark about 2 lower. The intercept 90 is the predicted mark for a student who doesn't work, which only makes sense if 0 hours is within or near the data.

Substitute into the equation, or read from the graph: go up from the $x$-value to the line, then across to the $y$-axis.

Example. Using $y=-2x+90$, a student who works 10 hours a week has a predicted mark of $-2\times 10+90=70$.

To find $x$ for a given $y$, substitute $y$ and solve: a predicted mark of 76 gives $76=-2x+90$, so $x=7$ hours.

Interpolation and extrapolation

• Interpolation: predicting within the range of the data. Usually reasonable, if the association is strong.
• Extrapolation: predicting outside the range of the data. Less reliable, because the pattern may not continue.

If the data in the example above came from students working 2 to 20 hours, predicting for 10 hours is interpolation. Predicting for 50 hours is extrapolation, and the line gives $-2\times 50+90=-10$: an impossible mark. That shows the line can't be extended that far.

The strength of the association also matters: predictions from a weak association are unreliable even within the data.

Two variables can be strongly associated without one causing the other. Possible reasons:

• A third variable (confounding variable) affects both. Towns with more police officers have more crime; both increase with the size of the town.
• Coincidence, especially with small datasets.
• The cause might run the other way.

To claim causation, you would need more evidence, such as a controlled experiment. In an exam, when a question says "Does this show that X causes Y?", the answer is usually: no, it shows an association; then give a likely third variable.

Some associations do involve causation (for example, the hours a heater runs and the electricity it uses), but you know that from the situation, not from the scatterplot.

• Swapping the axes: the independent variable goes on the horizontal axis.
• Describing only one feature: give form, direction and strength.
• Using data points that aren't on the line to find its equation.
• Finding run over rise for the gradient instead of rise over run.
• Saying a strong association proves one variable causes the other.
• Making predictions far outside the data without a warning about extrapolation.

• Describe questions: use the three words (for example "moderate, positive, linear") and one sentence in context.
• Predict questions: show the substitution, give units, and round sensibly (a number of people should be a whole number).
• Comment on the reliability questions: mention interpolation or extrapolation and the strength of the association.
• If asked to draw a line of best fit, use a ruler and draw it across the whole range of the data.

• Data analysis (Year 11): univariate data, types of variables and displays.
• Linear relationships (Year 11): gradient and intercept, and $y=mx+c$.
• Algebraic relationships (Year 12): a line of best fit is a linear model with the same strengths and limitations.