Interpreting the Line of Best Fit
A line of best fit summarizes an entire scatterplot with one equation, . Instead of dozens of scattered points, you get a single line you can use to describe the trend and predict new values. The catch: on a test, you are rarely asked to find the line — you are asked what its parts mean.
Both parts have a plain-English job. The slope is the predicted change in for each 1-unit increase in . The -intercept is the predicted -value when — the starting value. Interpret them with the units of the actual problem and you are done.
What the slope means in context
Slope is a rate. In a model like , where is hours worked and is earnings in dollars, the slope means earnings are predicted to rise about 6 dollars for each additional hour worked. Always state slope as amount of per one unit of , using the problem's units.
Watch the sign. A negative slope means is predicted to drop as grows — for example, a slope of in a model of battery percent versus hours of use means the battery loses about percentage points per hour.
In the scatterplot below, the data points scatter around a rising line of best fit. The line's slope is the rate this trend climbs — how much the predicted goes up for each step right in .
What the -intercept means in context
The -intercept is the model's prediction at . In , the intercept means about 20 dollars are predicted before any hours are worked — maybe a base payment or tips already collected.
Sometimes does not make real-world sense (a person with zero height, a house with zero square feet). The intercept is still part of the equation, but its interpretation may be meaningless outside the math. Say so when asked.
Predictions: interpolation vs. extrapolation
To predict, substitute an -value into the equation and simplify. Predicting at an inside the range of the collected data is called interpolation, and it is usually reliable — the line was built from data in that region.
Predicting at an outside the range of the data is called extrapolation, and it is risky. If you measured a plant's growth for 10 days, the trend may not continue to day 25 — growth slows, batteries do not drain below zero, and lines eventually stop matching reality. Treat extrapolated predictions with suspicion.
Worked examples
Example 1: interpret the slope and intercept
A coffee shop models a barista's earnings with , where is hours worked and is earnings in dollars. Interpret the slope and the -intercept.
Answer: Slope: about 6 dollars more per hour. Intercept: about 20 dollars at zero hours.
Example 2: make a prediction
Use to predict the earnings after hours.
Answer: About 62 dollars
Example 3: spot the extrapolation
A scientist measures a plant's height for to days and fits . Is using the line to predict the height at days reliable?
Answer: No — this is extrapolation, so the prediction is unreliable.
Try one yourself
Common questions
Why do predictions use the word about?
Because the line of best fit is a summary, not a law. Real data points sit above and below the line, so the line's output is a predicted value, not a guaranteed one. Saying about 62 dollars signals that honestly.
What is the difference between interpolation and extrapolation?
Interpolation predicts at an -value inside the range of the collected data and is usually reliable. Extrapolation predicts outside that range and can fail badly, because the pattern may not continue past what was measured.
What if the -intercept doesn't make sense in the situation?
That happens often. A model relating house size to price might have a negative intercept — no house has zero square feet, so the intercept has no real-world meaning there. It still anchors the line mathematically; just say the interpretation is not meaningful in context.
Is the line of best fit the same as a trend line?
Essentially, yes. A trend line is any reasonable line drawn through the middle of the cloud; the line of best fit is the specific one a calculator computes to fit the data best. On this lesson's problems you interpret them the same way.
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