Scatter Plots & Bivariate Data

How to do it:

1. Plot each pair $(x, y)$ as a point.

2. Name direction: as $x$ increases, does $y$ tend to rise (positive), fall (negative), or stay messy (no clear linear association)?

3. Comment on strength (tight cloud vs faint drift) and call out outliers.

Worked example: Weekly mileage versus finishing time often slopes down: more miles tend to pair with fewer minutes. A few injured-week points sit off the cloud.

Check: a plot shows co-movement, not causation. Reserve "caused" for experiments.

How to do it:

1. Ask whether the cloud follows a straight path, curves, or has no path at all. A line is only a good model for a straight path.

2. Look for clusters (two separate groups often mean two kinds of cases) and gaps.

3. Describe strength: strong if the points hug a line tightly, weak if they scatter widely around it.

Worked example: hours since charging ($x\text{)}$ versus phone battery percent ($y\text{)}$ drops in a straight, tight path: a strong negative linear association. Weeks ($x\text{)}$ versus seedling height ($y\text{)}$ rises fast and then levels off: a curved pattern, so no single line fits. Shoe size versus quiz score is a shapeless cloud: no association.

Check: "nonlinear" is a real description, not a failure. If the points curve, say so instead of forcing a line through them.

How to do it:

1. Sketch a straight line that cuts through the cloud, not just the first and last points.

2. Use it as a model for estimates.

3. Judge the fit by vertical distances above and below the line.

Worked example: Walk-a-thon miles versus dollars pledged slopes up. A line through the cloud estimates pledges for someone who walked $7$ miles but forgot to enter data.

Check: real points rarely land on the line. Check outliers (a $2$-hour shift with $\$140\text{)}$ before you let them yank the line.

How to do it:

1. Pick two points on your drawn line (not necessarily data points), far apart for accuracy.

2. Slope $m=\dfrac{y_2-y_1}{x_2-x_1}\text{.}$

3. Substitute one point into $y=mx+b$ to find $b\text{,}$ or read $b$ where the line crosses the $y$-axis.

Worked example: a fit line for miles walked ($x\text{)}$ and dollars pledged ($y\text{)}$ passes through $(2,50)$ and $(8,140)\text{.}$ Slope $m=\dfrac{140-50}{8-2}=15\text{.}$ Then $50=15(2)+b$ gives $b=20\text{,}$ so $y=15x+20\text{.}$

Check: a third point on the line should satisfy the equation: $(5,95)$ gives $15(5)+20=95\text{.}$ Connecting the first and last data points is not a fit line; it ignores everyone in between.

How to do it:

1. The slope is the predicted change in $y$ for each one-unit increase in $x\text{,}$ with both units named.

2. The $y$-intercept is the predicted $y$ when $x=0\text{;}$ say whether $x=0$ makes sense in the story.

3. Write each interpretation as a full sentence a non-mathematician could read.

Worked example: $y=15x+20$ for the walk-a-thon: each extra mile walked is associated with about $\$15$ more in pledges, and a walker with zero miles would still bring in about $\$20$ (a flat sponsor gift). For $y=-2.5x+95$ with $x$ hours of screen time and $y$ quiz score, each extra hour is associated with a drop of about $2.5$ points.

Check: the slope is "associated with," not "causes." An intercept can be meaningless when $x=0$ is far outside the data, such as a height-versus-age line predicting that a newborn is $30$ inches tall.

How to do it:

1. Substitute the $x$ value into the fit line to get a predicted $y\text{.}$

2. Residual $=$ actual $-$ predicted. Positive means the real point is above the line.

3. Predicting inside the range of the data (interpolation) is reasonable; predicting far outside it (extrapolation) is risky.

Worked example: with $y=15x+20\text{,}$ a walker who logged $7$ miles is predicted to raise $15(7)+20=125$ dollars. She actually raised $108\text{,}$ so the residual is $108-125=-17\text{:}$ she sits $17$ dollars below the line. Predicting for $50$ miles ($770$ dollars) extrapolates far past the longest walk in the data, so treat it as a guess.

Check: if most residuals on one side of the plot are positive and most on the other side are negative, the line is tilted wrong or the pattern is curved.

How to do it:

1. When both variables are categories (grade versus Spirit Week theme), count joint outcomes in a table.

2. Add row and column totals (marginals).

3. Use relative frequencies when groups have unequal size.

Worked example: Cafeteria vegetarian choice by grade is a two-way table, not an $x$-$y$ plot. Shirt color by grade is the same tool.

Check: tables for categories, scatter plots for paired numbers. Do not force counts onto a coordinate plane.