Lesson 1: Linear Regression

New Concept

Linear regression helps us model real-world data that shows a linear trend but isn't perfectly aligned. We'll find the equation for a regression line, or "line of best fit," to make predictions about the data.

What’s next

You're ready to put this into practice! Next, you'll work through interactive examples to draw regression lines and use them to solve real-world problems.

Property

A line that fits the data in a scatterplot is called a regression line. To find an equation for a line of best fit, we choose two points on the line whose coordinates we can estimate fairly accurately and use the point-slope formula. These two points need not be any of the original data points.

Examples

A regression line for minimum wage data passes through $(5, 1.25)$ and $(25, 3.35)$. The slope is $m = \frac{3.35 - 1.25}{25 - 5} = 0.105$. The equation is $y = 1.25 + 0.105(x-5)$, which simplifies to $y = 0.105x + 0.725$.

For data on study time versus test scores, a line of best fit passes through $(1, 75)$ and $(4, 90)$. The slope is $m = \frac{90-75}{4-1} = 5$. The equation is $y = 75 + 5(x-1)$, or $y = 5x + 70$.

Property

Using a regression line to estimate values between known data points is called interpolation.

Examples

Cocoa sales data is collected for temperatures between $2^\circ$C and $18^\circ$C. Using the regression line $C = -2.5T + 52$ to predict sales at $9^\circ$C is interpolation. We get $C = -2.5(9) + 52 = 29.5$ cups.

A baby whale's length is recorded at birth (0 months) and 7 months. Using a linear model to estimate its length at 4 months is interpolation, because 4 is between 0 and 7.

Property

Making predictions beyond the range of known data is called extrapolation.

Examples

Using the cocoa sales model, $C = -2.5T + 52$, to predict sales at $24^\circ$C is extrapolation. The model gives $C = -8$, an unreasonable result, showing the limits of extrapolation.

An engine's temperature is tracked for 7 minutes. Using a linear model to predict its temperature after 2 hours (120 minutes) is extrapolation and may be inaccurate, as the engine's temperature will not rise indefinitely.

Property

A scatterplot is a graph of data points that can show a linear trend. Data points that lie far from the regression line are called outliers.

Examples

• Study Time vs. Test Scores: If the trend is that more study time leads to higher scores, a student who studies for 10 hours but gets a very low score would be an outlier.

• Height vs. Shoe Size: In a group where taller people generally have larger shoe sizes, a person who is 5 feet tall and wears a size 13 shoe would be an outlier.