SHSAT: Scatter plots and lines of fit
Use a line of fit to make predictions, compare observed and predicted values, interpret slope, and judge which claims a model supports.
How do you use a line of fit to make predictions, and how do you compare them with what was actually observed?
Short answer
Substitute x into the line's equation to get the predicted y, then subtract: observed minus predicted tells you how far the real point is above or below the line.
Know first: Evaluating y = mx + b, Reading a table of values
Step 1 of 5
A line of fit predicts; the data may differ
A scatter plot shows pairs of measurements, such as temperature and sales. If the points rise from left to right, the association is positive; if they fall, it is negative.
A line of fit summarizes the trend with an equation y = mx + b. The slope m is the predicted change in y for each 1-unit increase in x. Plugging in an x gives a prediction, and real observations usually sit a little above or below it.
| Term | Meaning |
|---|---|
| predicted value | the y the line gives for a given x |
| observed value | the y actually measured |
| residual | observed − predicted; positive means the point is above the line |
| interpolation | predicting inside the range of x-values in the data |
| extrapolation | predicting outside that range, which is less reliable |
A strong association doesn't prove that one quantity causes the other.
Step 2 of 5
Predict, then compare
Worked example: Above or below the line
A line of fit for test scores is S = 6h + 52, where h is hours of study. A student studied 5 hours and scored 79. What is the observed score minus the predicted score?
- A−3
- B3
- C79
- D82
- 1
Predict
S = 6(5) + 52 = 30 + 52 = 82.
- 2
Compare in the right order
Observed − predicted = 79 − 82 = −3.
- 3
Interpret
The student scored 3 points below the line's prediction.
- 4
See why the other choices are there
3 subtracts in the reverse order, predicted − observed. 79 is the observed score and 82 is the prediction; neither is the difference.
Answer
−3
Find the wrong step
A line of fit y = −3x + 45 was built from data with x-values from 2 to 10.
- 1
At x = 6, the model predicts y = −18 + 45 = 27.
- 2
The slope says y is predicted to drop 3 for each 1-unit increase in x.
- 3
At x = 20 the model predicts −15, and since the line fits well from 2 to 10, this is a reliable interpolation.
What went wrong
x = 20 is outside the data's range, so this is extrapolation. The trend may not continue that far.
The fix
The prediction at x = 20 is extrapolation and should be treated with caution. Predictions for x between 2 and 10 are interpolation.
Step 3 of 5
Read what the model does and doesn't say
Check yourself · Question 1
A line of fit for a plant's height is P = 5d + 12, where d is the day. On day 6, the observed height was 45 centimeters. How many centimeters above the prediction was the observed height?
Answer: B
Predicted: 5(6) + 12 = 42. Observed − predicted = 45 − 42 = 3 centimeters above.
Check yourself · Question 2
A model y = −2x + 50 was built from data with x-values from 5 to 15. Which statement does the model support?
Answer: B
The slope is −2, so each 1-unit increase in x lowers the prediction by 2. An increase of 3 lowers it by 6.
Trap answer: Because the line fits the data, increasing x causes y to fall.
Reading cause into a trend
- Why it looks right
- When points line up neatly, it feels like one quantity must be driving the other.
- Why it's wrong
- The data show only that the two quantities move together. Another factor could explain both.
- The right answer
- The slope statement: when x increases by 3, the predicted y decreases by 6.
Common mistake
Computing predicted minus observed.
- Why it's tempting
- The prediction is the first number you calculate, so it comes first in the subtraction.
- Do this instead
- Use observed − predicted. A positive result means the actual point is above the line; a negative result means it's below.
Step 4 of 5
What to remember
Remember
- Substitute x into the line of fit to get a prediction, then find observed − predicted.
- The slope is the predicted change in y for each 1-unit increase in x.
- Predictions outside the data's range are extrapolation, and an association alone doesn't prove cause.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what did the line predict at 25°C, and how did you decide whether the observed sales were above or below it?