Bivariate Data Analysis

Bivariate data analysis looks at two numerical variables together to see whether, and how, they are related. It was worth 8 marks in NESA's sample HSC paper and about 7 marks a year in recent HSC papers. Many marks come from calculator work (finding $r$ and the regression line) and from explaining what the numbers mean.

• Bivariate datasets: when you need one variable or two; independent and dependent variables; correlation versus causation.
• Scatterplots and lines of best fit, by eye and with technology.
• Describe an association by its form (linear or non-linear), strength (strong, moderate, weak) and direction (positive or negative).
• The gradient and intercept of a line of best fit, and the equation of the line.
• Pearson's correlation coefficient $r$ and the least-squares regression line, using a calculator or spreadsheet.
• Predictions with a line, and the limits of interpolation and extrapolation.

Univariate data has one variable per person or item (such as heights). Bivariate data has two variables measured on each one (such as height and arm span), and is needed when you want to know whether one variable is related to the other.

• The independent variable is the one that might explain or predict the other. It goes on the horizontal ($x$) axis.
• The dependent variable is the one being predicted. It goes on the vertical ($y$) axis.

To decide which is which, ask "which depends on which?" Sleep might affect reaction time, so sleep is independent.

Correlation is not causation

Two variables can be strongly associated without one causing the other. A third (lurking) variable may drive both, or the association may be coincidence. For example, the number of air conditioners sold and the number of bushfires in a month are positively associated, but both are caused by hot weather. You can only call a relationship causal when there is a good reason (often from a controlled experiment) to believe changing one variable actually changes the other.

A scatterplot shows each pair of values as a point. Describe the association with three words:

• Form: do the points follow a straight line (linear) or a curve (non-linear)?
• Direction: positive if $y$ tends to increase as $x$ increases; negative if $y$ tends to decrease.
• Strength: how closely the points follow the pattern: strong, moderate or weak.

A line of best fit by eye goes through the middle of the points, with about as many points above as below, following the trend. Its equation is $y=mx+c$, where the gradient $m$ is found from two points on the line (not data points unless they lie on it) and $c$ is the $y$-intercept.

Pearson's correlation coefficient $r$ measures the strength and direction of a linear association. It is always between $-1$ and $1$.

The sign gives the direction. Note that $r$ only measures linear association: data on a clear curve can have a small $r$. Always look at the scatterplot too.

The least-squares regression line is the line of best fit calculated exactly from the data. Your calculator gives it in the form $y=a+bx$ (some calculators use $y=ax+b$; read the letters carefully).

Worked example: the price of used cars

A dealer recorded the age and price of eight cars of the same model.

(a) Which variable is independent? Age, because price depends on age (not the other way around).

(b) Find $r$ and the equation of the least-squares regression line.

Calculator steps: choose statistics mode, then the linear regression type ($y=a+bx$). Enter ages in the $x$ column and prices in the $y$ column. Then read $a$, $b$ and $r$ from the regression results.

(c) Describe the association. $r=-0.994$, so there is a strong, negative, linear association between the age and price of these cars.

(d) Interpret the gradient. The gradient is −2.81, so on average the price falls by about $2810 for each extra year of age. Remember the prices are in thousands of dollars.

(e) Predict the price of a 6.5-year-old car.

The predicted price is about $17 940. This is interpolation (6.5 is inside the data range of 2 to 9 years), so it is fairly reliable.

(f) Why would it be unwise to predict the price of a 15-year-old car? The line gives $36.20-2.81\times 15=-5.95$ a negative price, which is impossible. Fifteen years is far outside the data (extrapolation), and the trend won't stay linear: old cars lose value more slowly and never become worth less than nothing.

The intercept 36.20 suggests a new car (age 0) costs about $36 200, but age 0 is also outside the data, so treat this with care.

• Interpolation: predicting within the range of the data. Usually reasonable when the association is strong.
• Extrapolation: predicting outside the range. Risky, because the pattern may not continue.
• Predictions are less reliable when the association is weak, when there are few data points, or when there are outliers.
• If a question gives $y$ and asks for $x$, substitute $y$ into the line and solve for $x$; say whether this is interpolation or extrapolation.

A spreadsheet can draw the scatterplot and add the least-squares line (a trendline) with its equation and $r^2$; take the square root of $r^2$ and give it the sign of the gradient to get $r$.

• Putting the dependent variable on the $x$-axis.
• Saying a strong correlation proves that one variable causes the other.
• Calling $r=-0.9$ "weak" because it is negative. The sign is direction; the size is strength.
• Mixing up $a$ and $b$ from the calculator, so the gradient and intercept are swapped.
• Forgetting the units of the variables, such as prices in thousands of dollars.
• Extrapolating far beyond the data without comment.

• Describe an association with all three: form, strength and direction (for example "strong, negative, linear").
• Interpret the gradient as "for each extra 1 [unit of $x$], [$y$] increases/decreases by about [gradient] [units]".
• Write $r$ and the line from your calculator to the accuracy asked, then use the rounded line consistently.
• For "how reliable is this prediction", mention interpolation or extrapolation and the strength of the association.
• Correlation and causation questions want a possible lurking variable or a reason in context.

• Data analysis (Year 11): univariate data, statistical investigations and calculator statistics mode.
• Linear relationships (Year 11): gradient, intercept and $y=mx+c$.
• Algebraic relationships: when data looks curved, an exponential or quadratic model may fit better.