Two-Variable Data and Models

Two-Variable Data and Models

A scatterplot shows the relationship between two numerical variables. A line of best fit, $y = mx + b$, summarizes that relationship and predicts a $y$-value for a given $x$-value. The slope $m$ is the predicted change in $y$ for each one-unit increase in $x$; the $y$-intercept $b$ is the predicted value when $x = 0$.

A residual is the difference between an actual value and the predicted value. Residual $=$ actual $-$ predicted, not the reverse. A residual close to zero means the line of best fit predicted that point well; a large residual means the actual value fell far from what the model expected.

Correlation describes both the direction and the strength of a linear relationship. Points trending upward show positive correlation; points trending downward show negative correlation. Tightly clustered points along the trend indicate a strong correlation; loosely scattered points indicate a weak one. A strong correlation between two variables never proves that one causes the other; a plausible third variable can drive both at once.

A linear model is most appropriate when the scatterplot follows a roughly straight-line pattern. Clear curvature is a warning that a line may miss the relationship even when the variables are strongly associated.

Reading the line in context

Many questions give a line in words and ask what a piece of it means. Always answer with units and with the real quantities.

If the slope is negative, $y$ is predicted to decrease as $x$ increases. The slope also scales: if $m = 4$ per hour, then a 3-hour increase predicts a change of $4 \times 3 = 12$. Watch the units. If $x$ is measured in minutes and the question talks about hours, convert first.

Predicted values and residuals

Worked Example. A line of best fit is $y = 4x + 3$. An actual data point is $(5, 28)$. Find its residual.

Find the predicted value first, then subtract it from the actual value.

Predicted value: $4(5) + 3 = 23$

Residual $=$ actual $-$ predicted $= 28 - 23 = 5$

The residual is positive, so the actual point lies above the line. A negative residual means the actual point lies below the line. A habit that prevents sign mistakes: always say "actual minus predicted" out loud before you subtract.

Correlation, causation, and a third variable

When two variables show a strong correlation, three explanations are possible: one variable causes the other, the relationship runs in the opposite direction, or a third variable influences both. Without a randomized experiment, correlation alone cannot distinguish between these three explanations.

The classic example is ice cream sales and drowning incidents, which rise and fall together. Hot weather plausibly increases both, and neither causes the other. When the ACT asks for the most likely explanation of a correlation between two everyday quantities, look for the answer that names a plausible lurking third variable.

Describing a scatterplot

Describe direction first (upward is positive, downward is negative), then strength (tight clustering is strong, loose scatter is weak). Points scattered with no visible trend have correlation near zero. Do not confuse the steepness of the line with the strength of the correlation: a steep line with loose scatter is a weak correlation, and a shallow line with tight clustering is a strong one.

Evaluating a model

A model is not automatically good just because it produces a number. Check whether its predictions match the pattern in the data, whether residuals are randomly scattered around zero, whether the model is being used inside the range where the data were actually collected, and whether its parameters make sense in context. A systematic pattern in the residuals, instead of random scatter, means the model has missed real structure in the data.

Two ideas deserve special attention:

• Residual patterns. If residuals bounce randomly above and below zero, the line has captured the pattern. If they form a curve (for example positive, then negative, then positive again), the true relationship is curved and a different type of model would fit better.
• Interpolation versus extrapolation. Predicting inside the range of the observed $x$-values is interpolation and is usually reasonable. Predicting outside that range is extrapolation, where no data confirms that the relationship continues. A model of height built from ages 5 to 15 that predicts a 40-year-old is 9 feet tall has been stretched far beyond its data.

Strategy checklist

• To predict, substitute the $x$-value into the line.
• For a residual, compute the prediction and then subtract: actual $-$ predicted.
• For slope and intercept questions, say what changes per one unit of $x$ and what the value is at $x = 0$, with units.
• For correlation, give direction and strength separately.
• For cause-and-effect claims from observational data, look for a third variable.
• Check whether a prediction lies inside the data range before trusting it.