10.3 Bivariate Data: Scatter Plots, Correlation, Two-Way Tables & Linear Fits

10.3 Bivariate Data: Scatter Plots, Correlation, Two-Way Tables & Linear Fits

Skills 6 and 7 of Competency 4 handle paired data — every observation carries two values. Skill 6 covers both numerical pairs (scatter plots) and categorical pairs (frequency tables); skill 7 adds fitting a line and interpreting its intercepts.

Describing a scatter plot

Every scatter plot description names three things.

Direction.

• Positive association: as x increases, y tends to increase (study hours and test scores)
• Negative association: as x increases, y tends to decrease (car age and resale value)
• No association: no consistent pattern

Form. Linear if the points follow a straight-line pattern; nonlinear if they curve. Fitting a line to clearly curved data is a modeling error, and items test whether you notice.

Strength. How tightly the points cluster around the pattern. Tight clustering is a strong association; a diffuse cloud is weak.

Also note outliers — individual points far from the pattern — and clusters, which may indicate distinct subgroups.

Correlation

The correlation coefficient r summarizes the direction and strength of a linear association on a scale from −1 to 1.

+---------------------------------------------------------------------------+
|   r = -1     perfect negative        points exactly on a falling line     |
|   r ~ -0.8   strong negative                                              |
|   r ~ -0.4   weak negative                                                |
|   r = 0      no linear association                                        |
|   r ~ +0.4   weak positive                                                |
|   r ~ +0.8   strong positive                                              |
|   r = +1     perfect positive        points exactly on a rising line      |
+---------------------------------------------------------------------------+

The sign gives direction; the absolute value gives strength. So r = −0.85 describes a stronger association than r = +0.40, even though it is negative. Ranking correlations by strength while ignoring the sign is a standard item, and treating −0.85 as "weaker than 0.40 because it is smaller" is the standard error.

An r near 0 means no linear relationship, which is not the same as no relationship at all: data lying on a perfect parabola can have r ≈ 0.

Correlation does not imply causation

[!WARNING] A strong correlation between two variables does not establish that one causes the other.

Three alternative explanations must be ruled out before claiming causation:

1. Lurking (confounding) variable. Ice cream sales and drowning deaths correlate strongly, but neither causes the other — hot weather drives both.
2. Reverse causation. The direction may be backwards from what was assumed.
3. Coincidence. With enough variables examined, some will correlate by chance.

Only a well-designed randomized experiment supports a causal conclusion; observational data supports association. Items give a correlational finding and ask which conclusion is justified, and the correct answer names an association rather than a cause.

Fitting a line and interpreting it

A line of best fit (trend line) summarizes a linear pattern. Drawn informally, it should pass through the middle of the cloud with roughly equal numbers of points above and below; it need not touch any actual data point.

Once you have two points on the line, find its equation exactly as in section 5.3.

A trend line passes through (2, 34) and (8, 76). Slope = (76 − 34)/(8 − 2) = 42/6 = 7. Using point-slope: y − 34 = 7(x − 2) → y = 7x + 20.

Interpreting slope and intercepts in context is what skill 7 asks for explicitly.

A model for a plant's height in centimeters after x weeks is h = 2.4x + 5. Slope 2.4: the plant grows about 2.4 cm per week. y-intercept 5: the plant was 5 cm tall when measurement began (x = 0). x-intercept: solving 2.4x + 5 = 0 gives x ≈ −2.08, meaning the model projects zero height about 2 weeks before observation started. This is extrapolation outside the data range and has no physical meaning — the plant did not have negative height.

That last point is a genuine assessment target. Always state slope with units ("2.4 centimeters per week"), and check whether an intercept falls inside the range where the model was fit. Interpolation — predicting within the observed range — is reliable; extrapolation beyond it is not.

Two-way frequency tables

For paired categorical data, a two-way table cross-classifies observations.

+---------------------------------------------------------------------------+
|                    Plays sport   No sport   TOTAL                         |
|   Grade 7               84          66       150                          |
|   Grade 8              105          45       150                          |
|   TOTAL                189         111       300                          |
+---------------------------------------------------------------------------+

• Joint frequency: a cell count, such as 84 seventh graders who play a sport.
• Marginal frequency: a row or column total, such as 189 students who play a sport.
• Relative frequency: a count divided by a total. Divided by the grand total it is a joint relative frequency; divided by a row or column total it is a conditional relative frequency.

To judge association, compare conditional relative frequencies within rows:

Grade 7: 84/150 = 56% play a sport. Grade 8: 105/150 = 70% play a sport.

Because the two percentages differ substantially, grade level and sport participation appear associated in this data. If the row percentages had been nearly equal, there would be little evidence of association.

Comparing raw counts instead of percentages is only valid when the row totals are equal, as they happen to be here. When group sizes differ — say 150 seventh graders and 400 eighth graders — raw counts mislead badly and percentages are mandatory. That is the trap items are built around.