Algebra 1 Correlation Regression Lesson – Free PDF

Contents

This algebra 1 correlation regression lesson explains two-way tables, scatter plots and correlation in one printable page, with definitions, a method and a worked example.

A scatter plot shows how two numerical variables relate, and a line of best fit turns that pattern into predictions. A two-way table organizes categorical data into joint, marginal and conditional relative frequencies. The correlation coefficient \(r\) measures linear strength from \(-1\) to \(1\), but a strong correlation never proves causation.

Algebra 1Common Core: S-ID.B.5 · S-ID.B.6 · S-ID.C.7 · S-ID.C.8 · S-ID.C.9

Algebra 1 Correlation Regression Lesson: key facts

Joint, marginal and conditional relative frequencies come from a two-way table and are found by dividing by a grand total or by a row or column total.
A residual equals the actual value minus the predicted value; a random residual plot means a line fits well.
In \(y=mx+b\), the slope is the predicted change in \(y\) per 1-unit change in \(x\), and the y-intercept is the predicted \(y\) when \(x=0\).
The correlation coefficient \(r\) ranges from \(-1\) to \(1\) and measures only the strength and direction of a linear relationship.
Correlation does not prove causation, and predictions far outside the data are unreliable.

What do the relative frequencies in a two-way table mean?

A two-way table counts individuals in every combination of two categorical variables. Dividing a count by a total gives a relative frequency.

Three kinds of relative frequency

Joint: a cell count divided by the grand total.

Marginal: a row or column total divided by the grand total.

Conditional: a cell count divided by the total of its own row or column.

A two-way table of 150 students. The highlighted cell (24 students) is used for the joint relative frequency.

The joint relative frequency of a job and a GPA of 3.0 or higher is \(\frac{24}{150}=0.16\), or 16%. The marginal relative frequency of having a job is \(\frac{60}{150}=0.40\). The conditional relative frequency of a 3.0 or higher given a job is \(\frac{24}{60}=0.40\), but given no job it is \(\frac{63}{90}=0.70\). Because 40% and 70% differ so much, the data suggest an association between having a job and GPA.

How do you fit a line to a scatter plot?

Plot the points. Describe the direction, the strength and the form (linear or curved).
Choose a line that follows the points, or use technology to compute one.
Write it as \(y=mx+b\) and use it to make predictions.
Interpret the slope \(m\) as the predicted change in \(y\) for each 1-unit increase in \(x\), and the y-intercept \(b\) as the predicted \(y\) when \(x=0\).

The line is \(y=4.0x+57.8\). The dashed segment is the residual at 3 hours: actual 74 minus predicted 69.8.

Worked example. For the line \(y=4.0x+57.8\), each extra hour of studying adds about 4.0 points to the predicted score, and a student who studies 0 hours is predicted to score 57.8. At 3 hours the prediction is \(4.0(3)+57.8=69.8\).

What is a residual, and what does a residual plot show?

A residual is the actual value minus the predicted value. The student who studied 3 hours scored 74, so the residual is \(74-69.8=4.2\): the point lies 4.2 above the line. A residual plot graphs each residual against \(x\). Random scatter around zero means a line is a good fit. A curved pattern means another function, such as a quadratic or an exponential, may fit better.

Left: no pattern, so a line is a good model. Right: a clear arch means a curve (quadratic) would fit better.

What does the correlation coefficient r tell you?

The correlation coefficient \(r\) is a number from \(-1\) to \(1\) that measures how closely points follow a straight line. A positive \(r\) means a rising pattern and a negative \(r\) means a falling pattern. The closer \(r\) is to \(-1\) or \(1\), the stronger the linear relationship. For the study data, \(r\approx 0.96\). Remember that \(r\) only measures straight-line patterns.

Negative, near zero, positive: the sign of \(r\) matches the direction, and its size shows how tightly points hug a line.

Does correlation mean causation?

No. A strong \(r\) shows that two variables move together, not that one causes the other. A lurking variable may drive both: hot weather raises ice cream sales and also sunburns. Only a well-designed experiment with random assignment can support a causal claim. Also avoid extrapolation, which means predicting far outside the data.

Ice cream sales and sunburns are correlated because hot weather drives both.

Two-Way Tables, Scatter Plots and Correlation: frequently asked questions

What is the difference between joint, marginal and conditional relative frequency?

A joint relative frequency divides one cell by the grand total. A marginal relative frequency divides a row or column total by the grand total. A conditional relative frequency divides a cell by the total of its own row or column.

How do you interpret the slope of a line of best fit?

The slope is the predicted change in \(y\) for every 1-unit increase in \(x\). For the line \(y=4.0x+57.8\), each extra hour of studying is associated with about 4.0 more points on the quiz.

What is a residual?

A residual is the actual \(y\)-value minus the predicted \(y\)-value. A positive residual means the point is above the line, and a negative residual means it is below the line.

What does a correlation coefficient of \(-0.9\) mean?

It means a strong negative linear relationship: as \(x\) increases, \(y\) tends to decrease, and the points lie close to a falling line. It does not mean that \(x\) causes \(y\) to decrease.

Why can you not conclude causation from correlation?

A lurking variable can influence both variables, and the direction of cause can run either way. Only a randomized experiment can support a causal conclusion.