Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 1 correlation and regression? Then this Algebra 1 lesson on Reading Two-Way Tables and Scatter Plots explains each idea step by step, with worked examples.
Quick answer: A two-way table shows how two categories relate, using joint, marginal and conditional relative frequencies. For two numerical variables, a scatter plot and a line of best fit describe the trend. The slope gives the predicted change per unit, residuals measure errors, and the correlation r shows strength and direction, but correlation alone never proves causation.
Standards: S-ID.B.5, S-ID.B.6, S-ID.C.7, S-ID.C.8, S-ID.C.9 · Level: Grade 9 Algebra 1
How do two-way tables show relationships between categories?
A two-way table counts people in two categories at once. In a survey of 120 students, 38 ninth graders and 30 tenth graders buy lunch in the cafeteria, while 22 ninth graders and 30 tenth graders pack lunch.
| Cafeteria | Packed | Total | |
|---|---|---|---|
| 9th grade | 38 | 22 | 60 |
| 10th grade | 30 | 30 | 60 |
| Total | 68 | 52 | 120 |
- Joint: a cell divided by the grand total. 38 ÷ 120 ≈ 31.7% are ninth graders who buy lunch.
- Marginal: a row or column total divided by the grand total. 68 ÷ 120 ≈ 56.7% buy lunch.
- Conditional: a cell divided by its row or column total. Of the ninth graders, 38 ÷ 60 ≈ 63.3% buy lunch.
How do you fit a line to a scatter plot?
A scatter plot pairs two numerical variables. If the points follow a straight trend, a line of best fit describes it. The line is written ŷ = mx + b, where ŷ is the predicted value. For the study-hours data below, a calculator gives ŷ = 5.43x + 48.4.
- Enter the data pairs in a calculator or spreadsheet.
- Run linear regression to get the slope m, the intercept b and the correlation r.
- Interpret m as the predicted change in y for each 1-unit increase in x.
- Interpret b as the predicted y when x = 0, if that makes sense in context.
The slope 5.43 means each extra hour of study is associated with about 5.4 more points. At 7 hours the line predicts 5.43(7) + 48.4 = 86.41, but the actual score is 88, so the residual is 88 − 86.41 = 1.59.
A residual is observed minus predicted. If the residuals look random around zero, a line is a good model. A curved pattern means a line is not appropriate.
What does the correlation coefficient tell you?
The correlation coefficient r is a number from −1 to 1. Its sign gives the direction, and its distance from 0 gives the strength of the straight-line relationship.
Does correlation mean causation?
No. Two variables can move together because a third one drives both. Ice cream sales and sunburns both rise in hot weather, yet neither causes the other. Only a well-designed experiment can show cause and effect.
Using a line far outside the data. Predicting beyond the data is extrapolation, and the trend may not continue.
Say “associated with” instead of “causes” unless the data come from an experiment.
- Joint, marginal and conditional relative frequencies come from a two-way table.
- The line of best fit has the form ŷ = mx + b; the slope is the predicted change in y per 1 unit of x.
- Residual = observed value − predicted value.
- The correlation r lies between −1 and 1; values near ±1 show a strong linear relationship.
- Correlation does not prove causation, and extrapolating is risky.
Reading Two-Way Tables and Scatter Plots: frequently asked questions
What is a conditional relative frequency?
It is a cell count divided by a row total or a column total instead of the grand total. It answers questions such as “Of the ninth graders, what fraction buy lunch?”
What does the slope of a regression line mean?
The slope is the predicted change in y for each increase of 1 in x. In the study data, a slope of 5.43 means about 5.4 extra points for each extra hour studied.
What is a residual?
A residual is the actual y-value minus the value the line predicts. A positive residual means the point lies above the line, and a negative residual means it lies below.
What do values of r near 1, 0 and −1 mean?
A value near 1 means a strong positive linear relationship, near −1 a strong negative one, and near 0 little or no linear relationship.
Why does correlation not imply causation?
A lurking variable, such as hot weather, can affect both variables. The variables then change together even though neither one causes the other.
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