Line of Best Fit: Linear Regression and Correlation Explained

A line of best fit is the straight line that follows a scatter of points as closely as possible. The least-squares line makes the total of the squared vertical gaps (residuals) as small as it can be. The correlation coefficient $r$, between $-1$ and $1$, says how tightly the points hug a straight line.

The formulas

The line always passes through the mean point $(\bar{x}, \bar{y})$.

Worked example 1

Working through the sums gives $m = 2.02$ and $c = -0.02$:

Each step of 1 in $x$ adds about 2 to $y$, and $r$ is very close to 1.

Worked example 2: a negative trend

For $x = 10, 20, 30, 40, 50$ and $y = 95, 80, 70, 52, 40$, the line is $y = -1.38x + 108.8$. The negative slope means $y$ falls as $x$ rises.

Reading r and r²

$r^2$ is the fraction of the variation in $y$ that the line explains: $r^2 = 0.95$ means 95%.

Predicting with the line

Substitute an $x$-value: with $y = 2.02x - 0.02$, at $x = 7$ we predict $y \approx 14.12$. Predicting far outside your data range (extrapolation) is risky.

Common mistakes

• Swapping $x$ and $y$: the line for predicting $y$ from $x$ is different.
• Trusting a line when $r$ is near 0.
• Extrapolating far beyond the data.

Practice questions

1. If $r = -0.9$, what kind of relationship is it?
2. If $r = 0.6$, what is $r^2$?
3. Using $y = 2.02x - 0.02$, predict $y$ at $x = 10$.

Answers: 1) strong negative 2) 0.36 3) 20.18

Paste your own data into the linear regression calculator to get the line, $r$, and a scatter plot with residuals. For single-variable summaries, see mean, median and mode.

Frequently asked questions

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