Bivariate Data & Correlation — Print
Fit a line, read its slope in context, and judge how well it fits — then resist the urge to claim a cause.
What the practice questions cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Slope — How much y changes for each 1 that x increases.
- Range — The set of outputs that actually occur.
- Outlier — A dot far away from the rest of the pattern.
- Extrapolation — Predicting outside the range. Risky.
- Residual — The gap between an actual value and the line's prediction.
- Regression line — The line of best fit computed by a formula rather than by eye.
- Residual plot — A plot of residuals against x, used to judge whether a line was the right model.
- Prove — The statement you must arrive at. Always the last line.
Two students fit a line to the same scatter plot. Their lines are slightly different. Who is right?
Both might be. But you can measure which line fits better, and the measurement is simple.
Drop a little line from each dot to your fitted line. That gap is how much the line missed that dot by — its residual.
Above the line is a positive residual, below is negative, and for a least-squares line they cancel out to zero.
Now plot the residuals on their own. If the line was right, they should look like a formless jumble.
If instead they make an arch — negative at both ends, positive in the middle — the data curves and a line was the wrong shape. This is the one place in mathematics where finding no pattern is the good outcome.
Reading r
r close to 1: strong positive linear relationship. Close to -1: strong negative. Close to 0: little or no linear relationship.
r = 0.9 is strong; r = 0.3 is weak. The sign matches the slope's sign, always.
r measures only how linear the pattern is. A perfect U-shaped curve can have r near zero.
Sign then size
Read the sign for direction, then the size for strength. r = -0.95 is a very strong relationship, not a weak one.
Residuals
Residual = actual - predicted. A positive residual means the real value was above the line.
Add up the residuals for a least-squares line and you get zero — the line splits the misses evenly.
A small residual means the line predicted that case well.
Actual minus predicted
Always in that order. If the real score was 82 and the line predicted 78, the residual is +4.
The residual plot
Plot each residual against its x value. If a line was the right model, the residuals look like random scatter around zero.
A clear curve or arch in the residual plot means the true relationship is not linear, even if r looked respectable.
This is the honest check, and it catches bad models that r alone will let through.
Pattern is bad news
In a residual plot, structure means the model is wrong. You want a formless cloud — the one place in mathematics where no pattern is the good outcome.
Correlation is still not causation
A strong r says the two variables moved together. It says nothing about why.
Three explanations always remain: x causes y, y causes x, or something else causes both.
Only a randomised experiment can rule out the third. Observational data never can.
Worked examples
Interpreting r
A study finds r = -0.87 between hours of television and test score. Describe the relationship.
- Negative sign: more television goes with lower scores. The direction.
- |-0.87| is close to 1, so it is strong. The strength.
- So: a strong negative linear association. Both pieces reported.
- But television may not be the cause — a third factor could drive both. Correlation is not causation.
A strong negative linear association; no cause established
A residual
A regression line predicts y = 3x + 10. For x = 5 the actual value was 28. Find the residual.
- Predicted: 3(5) + 10 = 25 Substitute into the line.
- Residual = actual - predicted = 28 - 25 Always in that order.
- = +3 Positive, so the actual value sat above the line.
+3
Judging a model
A residual plot shows residuals that are negative at both ends and positive in the middle. What does that tell you?
- There is a clear pattern — an arch. Residuals should look random.
- The data curves, but the line cannot. The line under-predicts in the middle and over-predicts at the ends.
- So a linear model is the wrong choice here. A quadratic or exponential model would fit better.
The relationship is not linear — try a different model
Words to know
- Regression line
- The line of best fit computed by a formula rather than by eye.
- Correlation coefficient
- r, between -1 and 1. Its sign gives the direction, its size the strength.
- Residual
- Actual value minus predicted value.
- Residual plot
- A plot of residuals against x, used to judge whether a line was the right model.
Mistakes to watch for
Calling r = -0.9 weak because it is negative. Read the sign for direction and the size for strength. -0.9 is nearly as strong as a relationship gets.
Computing predicted minus actual for a residual. It is always actual minus predicted, in that order, so that a point above the line gives a positive residual. The other order reverses every sign.
Treating a high r as proof of cause. Three explanations always survive: x causes y, y causes x, or something else causes both. Only a randomised experiment can rule out the third.
Ignoring a curved residual plot because r looked respectable. r measures how linear the pattern is; the residual plot shows whether a line was the right model at all. The residual plot is the honest one.
The correlation coefficient r was developed by Karl Pearson in the 1890s, building on Galton's work.
Its sign and size are all it reports: r = -0.95 is a very strong relationship, not a weak one, because strength lives in the size and direction in the sign.
r also only sees straight lines. A perfect U-shaped relationship — every point exactly on a parabola — can give r of almost exactly zero.
Problem 1
r = 0.92 describes a relationship that is
- strong and positive
- weak and positive
- strong and negative
- non-existent
Worked answer
strong and positive
Positive sign
Close to 1, so strong
Problem 2
r = -0.88 describes a relationship that is
- strong and negative
- weak and negative
- strong and positive
- non-existent
Worked answer
strong and negative
Negative sign, so it falls
|-0.88| is close to 1, so strong
Problem 3
r = 0.05 means
- almost no linear relationship
- a strong relationship
- a perfect relationship
- a negative relationship
Worked answer
almost no linear relationship
0.05 is very close to 0
So almost no linear relationship
Problem 4
A line predicts 50 and the actual value is 56. Find the residual.
Worked answer
6
56 - 50 = 6
Problem 5
A line predicts 120 and the actual value is 113. Find the residual.
Worked answer
-7
113 - 120 = -7
Problem 6
A regression line is y = 4x + 6. For x = 7 the actual value was 30. Find the residual.
Worked answer
-4
Predicted: 4(7) + 6 = 34
30 - 34 = -4
Problem 7
A regression line is y = -2.5x + 80. Predict y when x = 12.
Worked answer
50
-2.5(12) = -30
-30 + 80 = 50
Problem 8
For y = 1.8x + 12, with x in hours and y in dollars, interpret the slope.
Worked answer
1.80 more per hour
Dollars per hour
Each extra hour is associated with $1.80 more
Problem 9
A residual plot shows a clear curved arch. This means
- a line is the wrong model
- the line fits perfectly
- r must be zero
- the data has no outliers
Worked answer
a line is the wrong model
Structure in the residuals means the model missed something
A curve needs a curved model
Problem 10
A regression line has a negative slope. What can you say about the sign of r?
Worked answer
negative
Negative slope means negative r
Problem 11
A strong r between two observed variables proves
- nothing about cause
- that the first causes the second
- that the second causes the first
- that a third variable is involved
Worked answer
nothing about cause
Correlation measures how they move together
Cause needs a controlled experiment
Problem 12
For a least-squares line, what do all the residuals add up to?
Worked answer
0
Positives and negatives cancel
The sum is 0
Problem 13
A regression line for score against hours studied is y = 6.5x + 41. A student studies 4 hours and scores 74. Find the residual.
Worked answer
7
Predicted: 6.5(4) + 41 = 67
74 - 67 = 7
Problem 14
Study A has r = 0.6; study B has r = -0.8. Which has the stronger linear relationship?
Worked answer
B
|0.6| = 0.6
|-0.8| = 0.8
B is stronger
Problem 15
For y = -0.4x + 32, where x is age in years and y is hours of sleep, interpret the slope in context.
Worked answer
sleep drops 0.4 hours per year of age
Negative, so sleep falls
Each extra year is associated with 0.4 fewer hours
Problem 16
A regression on data for x between 5 and 20 gives y = 3x + 10. Is predicting at x = 200 reliable? Answer yes or no.
Worked answer
no
200 is far outside 5 to 20
That is extrapolation
So not reliable