Bivariate Data

8.I.1Scatter plots01/04

Construct and interpret scatter plots.

Step01of 04

A scatter plot displays paired numerical data. Each dot is one data point: x-coordinate from one variable, y-coordinate from the other. The cloud's pattern reveals whether the variables are related.

8.I.1 — Construct and interpret scatter plots.

Step 01 of 04

A scatter plot displays paired numerical data. Each dot is one data point: x-coordinate from one variable, y-coordinate from the other. The cloud's pattern reveals whether the variables are related.

Step 02 of 04

Three things to look for.

Direction	→positive (up to right), negative (down to right), or no direction
Strength	→tight cluster (strong) vs. loose scatter (weak)
Form	→straight (linear) vs. curved (nonlinear)

Step 03 of 04

Outliers and clusters.

Outlier	→a point far from the main cloud — investigate, don't ignore
Cluster	→group of points concentrated in one region
Gap	→region with no data — could indicate missing population

Step 04 of 04

Visualize. Hours studied vs. test score — typically a positive, roughly linear association.

Positive, fairly strong, linear association.

Key insight

Scatter plots show paired numerical data. Look for direction (positive / negative / none), strength (tight / loose), and form (linear / nonlinear). Outliers stand apart from the main cloud.

8.I.2 — Line of best fit.

Step 01 of 04

For a roughly linear scatter, draw a single straight line that "balances" through the cloud — this is the line of best fit (also called regression line or trend line).

By eye, aim for a line that:

Goes through the middle of the cloud	→roughly equal points above and below
Follows the cloud's general direction	→matches the trend
Is NOT just connecting two extreme points	→should fit the overall pattern

Step 02 of 04

Once you have a line, write its equation using slope-intercept form. Pick two points ON THE LINE (they don't need to be data points!) and compute slope, then find intercept.

For our hours-vs.-score data, a reasonable line might be $y = 4 x + 52$ : slope 4 (each hour studied predicts 4 more points), intercept 52 (predicted score with 0 hours studied).

Step 03 of 04

Use the line to predict. Plug in any x value to get the predicted y.

\overset{y}{^} = 4 (7.5) + 52 = 82

Predicted score for 7.5 hours of study: about 82. The hat over y indicates a PREDICTED value (not actual data).

Step 04 of 04

Limits of prediction.

Within data range	→usually reliable
Extrapolating far outside	→risky — the linear pattern may not continue
Strong correlation	→predictions are tighter
Weak correlation	→predictions are loose

Don't predict scores for 50 hours of study using a line built from 0-10 hour data — the linear trend is unlikely to hold that far.

Key insight

Line of best fit balances through the data cloud. Slope = predicted change in y per unit of x. Predict by plugging in x. Don't extrapolate too far past the data range.

8.I.3 — Two-way frequency tables.

Step 01 of 04

A two-way frequency table cross-tabulates two CATEGORICAL variables. Each cell counts how many subjects fall into BOTH categories.

Survey: 100 students asked about owning a smartphone and playing a sport.

	Plays sport	No sport	Total
Smartphone	32	48	80
No phone	8	12	20
Total	40	60	100

Step 02 of 04

Three flavors of "fraction" come from this table.

Joint	→cell / grand total — "what % play sports AND own a phone?" → $32/100 = 32%$
Marginal	→row or column total / grand total — "what % play sports overall?" → $40/100 = 40%$
Conditional	→cell / row (or column) total — "what % of phone owners play sports?" → $32/80 = 40%$

Step 03 of 04

Conditional probability reveals association.

P (sport ∣ phone) = 32/80 = 40%

P (sport ∣ no phone) = 8/20 = 40%

Same percentage in both rows — owning a phone DOESN'T affect the rate of playing sports. The two variables are INDEPENDENT in this dataset.

Step 04 of 04

If conditional rates DIFFER between rows, the variables ARE associated. The size of the difference tells you how strong the association is.

"and"	→joint — divide by grand total
"of all students"	→marginal — divide by grand total
"given that" / "of"	→conditional — divide by the SUBSET total

Key insight

Two-way tables organize categorical data. Joint = cell / grand. Marginal = margin / grand. Conditional = cell / margin. Match the wording of the question ("and" / "of all" / "given") to the right denominator.

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