Interpreting the slope and intercept
A straight line has been fitted through a cloud of data points. What do its slope and its intercept actually say about the things that were measured?
Where you are 5 in Least squares regression
When you reach for it
Ten adults on an online typing course logged their hours of practice and then took a speed test (the numbers are made up). On a scatter plot the points drift upwards from left to right, and a straight line has been fitted through them, a Line of best fit:
where x is the hours of practice and y is the predicted speed in words per minute. What do the 4 and the 22 say about typing?
Reading m and b in y = mx + b is familiar from Slope-intercept form, and Reading a linear function from its graph gives them meaning in context for an exact line. A fitted line differs in one way that changes every sentence: the points do not sit on it. So the slope becomes a predicted change, true on average rather than for each individual, and the intercept sometimes means nothing at all. Drawing the line belongs to Line of best fit, describing the cloud to Describing a scatter plot.
The procedure
The line gives predictions. In y = 4x + 22, the y is the speed the line predicts, not a speed anyone measured. The learner with 4 hours of practice typed 39 words per minute, while the line gives 4(4) + 22 = 38. Statistics books mark a predicted value with a hat, \hat{y}, read “y-hat”, to keep it apart from a measured one. In this lesson the y of a fitted line always means the prediction.
The slope is the predicted change for one more unit of x. Slope is rise over run, so on a run of 1 the rise is the slope itself. From 5 hours to 6 hours the line rises from 42 to 46, which is 4 words per minute. In words: for each extra hour of practice, the predicted typing speed is about 4 words per minute higher.
The sentence names both variables, the size and direction of the change, and its units: words per minute per hour, the units of y per unit of x. Over a longer run the predicted change grows in step: 3 extra hours predict 3 \times 4 = 12 words per minute more.
Why “predicted” and “on average”. Look at learners one hour apart in the data. From 5 to 6 hours the speed went from 43 to 47, up 4; from 6 to 7 hours, from 47 to 49, up only 2; from 7 to 8 hours, from 49 to 56, up 7. No single gain is the slope. The slope is what the line, running through the middle of the cloud, says about the typical gain. So the sentence says “predicted” or “on average”, and “about”, never “will”.
The intercept is the prediction at x = 0. Putting x = 0 into y = 4x + 22 gives 22: a learner with no practice has a predicted speed of about 22 words per minute. Whether that means anything depends on two questions. Is x = 0 possible for these individuals? Is it near the data? Beyond the data nobody knows whether the trend carries on (this is extrapolation). Here both answers are yes: an adult can sign up and not practice, and 0 hours is only 1 hour below the data. So 22 is a fair, rough baseline. When either answer is no, the intercept only sets the height of the line and has no meaning of its own (Example 3).
A slope is not a cause. The slope describes how two variables go together in these data. It does not show that changing x would change y: in observed data something else can drive both (keener learners may both practice more and type faster), as Correlation is not causation: lurking and confounding variables and Observational studies and experiments explain.
Worked examples
Practice and typing speed
Use the typing line y = 4x + 22, fitted to ten learners with between 1 and 10 hours of practice.
(a) Interpret the slope.
(b) Interpret the intercept and say whether it makes sense.
(c) Rohini practiced 3 hours more than Pavel. What does the line say about their speeds?
Solution(a) The slope is 4, in words per minute per hour. For each extra hour of practice, the predicted typing speed is about 4 words per minute higher.
(b) At x = 0 the line gives y = 4(0) + 22 = 22. A learner who does no practice on the course has a predicted speed of about 22 words per minute. That makes sense: 0 hours is possible and lies just below the data, which start at 1 hour.
(c) Over a run of 3 hours the line rises 3 \times 4 = 12, so Rohini’s predicted speed is 12 words per minute higher than Pavel’s. It is a prediction, not a promise: the learners at 4 and 7 hours actually typed 39 and 49 words per minute, 10 apart, while the line predicts 38 and 50, which are 12 apart.
Rain and library visitors
Ottoline, who runs a town library, recorded the rainfall and the number of visitors on 40 days (made-up data). Rainfall ran from 0 mm, on several dry days, to 30 mm. The fitted line is y = -14x + 520, where x is the day’s rainfall in millimeters and y is the predicted number of visitors.
(a) Interpret the slope.
(b) Interpret the intercept.
(c) How do the predictions for a dry day and for a day with 10 mm of rain compare?
Solution(a) The slope is -14 visitors per millimeter, and negative means the line runs downhill. The sentence uses its size with a word such as “lower” or “fewer”: for each extra millimeter of rain, the predicted number of visitors is about 14 lower.
(b) At x = 0 the line gives y = 520: on a dry day it predicts about 520 visitors. This means something, because dry days are possible and several are in the data.
(c) The run is 10 mm, so the predicted change is 10 \times (-14) = -140. The line predicts 140 fewer visitors on the wet day: 520 - 140 = 380, against 520.
Speed and fuel, where the intercept means nothing
Marisol drove one car round a test track at steady speeds from 60 to 120 km/h, recording the fuel used in liters per 100 km (made-up data). The fitted line is y = 0.05x + 2.4, with x the speed in km/h and y the predicted fuel use.
(a) Interpret the slope, choosing a step that gives a readable number.
(b) Does the intercept mean anything?
Solution(a) Taken literally: for each extra kilometer per hour, the predicted fuel use is 0.05 liters per 100 km higher. One km/h is too small a step to picture, so scale it up. Over 10 km/h the line rises 10 \times 0.05 = 0.5: for each extra 10 km/h, the predicted fuel use is about half a liter per 100 km higher.
(b) No; both questions fail. A car at 0 km/h covers no distance, so “liters per 100 km” has no value for it: an idling car burns fuel and goes nowhere. And 0 km/h is 60 km/h below the slowest test, so the whole dashed stretch has no data under it. The 2.4 only sets the height of the line so that it runs through the data between 60 and 120 km/h.
Crate weight and delivery cost
A delivery firm’s records for crates weighing from 5 to 60 kg (made-up data) give the fitted line y = 1.8x + 12, where x is the weight in kilograms and y is the predicted delivery cost in dollars.
(a) Give the units of the slope and interpret it.
(b) Sorcha’s crate weighs 25 kg more than Pavel’s. Compare their predicted costs.
(c) Interpret the intercept.
Solution(a) The units are dollars per kilogram. For each extra kilogram, the predicted delivery cost is about 1.80 dollars higher.
(b) The run is 25 kg, so the predicted cost rises by 25 \times 1.8 = 45: Sorcha’s crate is predicted to cost about 45 dollars more. Neither weight is needed, because the predicted difference depends only on the difference in weight.
(c) At x = 0 the line gives y = 12. A crate cannot weigh 0 kg, but 0 is only 5 kg below the lightest crate in the data, so the line is not stretched far. A cautious reading: about 12 dollars of the predicted cost does not grow with weight. That part could be something like a fixed handling charge, but the data cannot show what it is; only the firm’s price list could.
Floor area and price, from two predictions
In one town, the houses sold over a year (made-up data) had floor areas from 80 to 250 square meters. The fitted line predicts a price of 305{,}000 dollars for 100 square meters and 383{,}000 dollars for 130 square meters. Find the line, with price in thousands of dollars, and interpret its slope and intercept.
SolutionThe predictions are points on the line, (100, 305) and (130, 383), so the slope is rise over run between them:
Putting (100, 305) into y = 2.6x + b gives 305 = 2.6(100) + b, so b = 305 - 260 = 45, and the line is y = 2.6x + 45. Check: 2.6(130) + 45 = 338 + 45 = 383.
The slope is 2.6 thousand dollars per square meter, which is 2{,}600 dollars. For each extra square meter of floor area, the predicted price is about 2{,}600 dollars higher.
The intercept would say that a house with no floor area has a predicted price of 45{,}000 dollars. No house has 0 square meters of floor, and 0 is 80 square meters below the smallest house in the data, so the intercept has no meaning here.
Which sentences does the line support?
Ottoline’s library line from Example 2 is y = -14x + 520, fitted to days with 0 to 30 mm of rain. Say whether the line supports each sentence, and why.
(a) “Every millimeter of rain keeps 14 people away from the library.”
(b) “On days with 10 mm more rain, the line predicts about 140 fewer visitors.”
(c) “On a day with 45 mm of rain, the library should expect -110 visitors.”
(d) “The library had 520 visitors on each dry day.”
Solution(a) Not supported, twice over. “Every” claims the same change each time, but the slope is only the predicted, average change. “Keeps people away” claims that rain causes the drop, but the data were only observed, and rainy days may differ in other ways, such as the season. A supported version: “For each extra millimeter of rain, the predicted number of visitors is about 14 lower.”
(b) Supported: 10 \times (-14) = -140, and the sentence says “predicts”.
(c) Not supported. The arithmetic is right, 520 - 14(45) = 520 - 630 = -110, but 45 mm is beyond the data, which stop at 30 mm, and a negative number of visitors is impossible. That is extrapolation, taken up in Predicting with a regression line, and extrapolation.
(d) Not supported. 520 is the predicted number for a dry day; the real dry days scatter above and below it.
Boundaries
- A fitted slope describes a trend, not an individual. It is the predicted, or average, change in y for one more unit of x; one individual may change by more, by less, or the other way.
- Only inside the data. Beyond the data’s range of x the trend may bend or stop (Where a model holds: domain, interpolation and extrapolation).
- The intercept means something only when x = 0 is possible and near the data. Otherwise it only sets the line’s height.
- The line has to suit the cloud. If the points follow a curve, or show no trend, a straight line’s slope summarizes something that is not there.
- Association, not cause. A slope from observed data never shows that changing x changes y.
- Units travel with the slope. Changing the units of x or y changes the number (Exercise 16) but not the relationship.
Where this goes next
Predicting with a regression line, and extrapolation puts values of x into the line and asks how far a prediction can be trusted. A residual is the gap between a real point and the line, the scatter this lesson kept pointing at. Later, the The least squares criterion and its line says which line counts as best, Slope from r and the spreads, and regression to the mean builds the slope from the Correlation and the spreads, and Test for a slope asks whether a sample’s slope could be 0 in the whole population.
Exercises
Reading the slope
Exercises 1 to 5 use the lines from the worked examples.
- Typing, y = 4x + 22. Pavel practiced 2 hours and Sorcha 9 hours. How much higher is Sorcha’s predicted speed?
- Crates, y = 1.8x + 12. How much more is the predicted cost of a crate 10 kg heavier than another?
- Houses, y = 2.6x + 45, with price in thousands of dollars. How much higher, in dollars, is the predicted price of a house 15 square meters larger than another?
- Library, y = -14x + 520. Compare the predicted visitors on a day with 5 mm of rain and a day with 12 mm.
- Fuel, y = 0.05x + 2.4. Compare the predicted fuel use at 95 km/h and at 115 km/h.
- On a second typing course, with learners practicing from 0 to 12 hours, the fitted line is y = 3.5x + 18. Write a sentence interpreting the slope, with units.
Reading the intercept
- For the course in Exercise 6, interpret the intercept and say whether it makes sense.
- A second delivery firm handles crates from 30 to 200 kg, and its fitted line is y = 0.9x + 40, cost in dollars. Does the 40 mean anything? Explain.
- A second library’s line is y = -9x + 310, from days with 0 to 25 mm of rain, many of them dry. What does the line predict for a dry day, and for a day with 20 mm?
- For flats in a city, with floor areas from 40 to 120 square meters, the fitted line is y = 4.1x - 30, price in thousands of dollars. What would the intercept say, and why can it be set aside? What does the line predict for the smallest flat in the data?
Lines from two predictions, and falling lines
- A library’s line predicts 448 visitors on a day with 3 mm of rain and 328 visitors on a day with 18 mm. Its data include dry days. Find the line and interpret its slope and intercept.
- A line for houses of 70 to 200 square meters predicts 250{,}000 dollars at 90 square meters and 340{,}000 dollars at 120 square meters. Find the line, with price in thousands of dollars, and interpret its slope and intercept.
- A firm’s line for crates of 10 to 80 kg predicts 30.50 dollars for 15 kg and 57.50 dollars for 45 kg. Find the line and interpret its slope.
- For the learners in the typing example, a second line predicts errors per 100 words typed: y = -0.6x + 9. Interpret the slope. How do the predicted errors compare for two learners whose practice differs by 5 hours? ### Reasoning
- Sorcha says: “The slope of the typing line is 4, so if I practice one more hour, my speed will go up by 4 words per minute.” Explain what the line does and does not support in her sentence.
- Rohini records the typing practice in minutes instead of hours. What does the slope become, and why has the relationship between practice and speed not changed?
- Ninian says the negative slope of Ottoline’s library line proves that rain keeps readers away. Give a different explanation that would produce the same slope.
- Suppose a fitted line had slope 0. What would it predict, and what would that say about the two variables?
Answers
- The predicted speed is 28 words per minute higher. The run is 9 - 2 = 7 hours, and 7 \times 4 = 28.
- About 18 dollars more in predicted cost, since 10 \times 1.8 = 18.
- About 39{,}000 dollars higher in predicted price. The run is 15 square meters, and 15 \times 2.6 = 39 thousand dollars.
- About 98 fewer visitors predicted on the 12 mm day. The run is 7 mm, and 7 \times (-14) = -98.
- About 1 liter per 100 km more predicted at 115 km/h. The run is 20 km/h, and 20 \times 0.05 = 1.
- For each extra hour of practice, the predicted typing speed is about 3.5 words per minute higher.
- At x = 0 the line gives y = 18: a learner with no practice has a predicted speed of about 18 words per minute. It makes sense, since 0 hours is possible and is in the data.
- No. A crate of 0 kg is impossible, and 0 is 30 kg below the lightest crate, far outside the data. The 40 only sets the line’s height.
- A predicted 310 visitors on a dry day, and on a 20 mm day 310 - 9(20) = 310 - 180 = 130. Both are inside the data.
- It would say a flat with no floor area has a predicted price of -30{,}000 dollars, which is impossible; 0 is far below the smallest flat, so the intercept only sets the line’s height. For 40 square meters, y = 4.1(40) - 30 = 164 - 30 = 134, a predicted price of 134{,}000 dollars.
- y = -8x + 472. The slope is \displaystyle \frac{328 - 448}{18 - 3} = \frac{-120}{15} = -8: for each extra millimeter of rain, the predicted number of visitors is about 8 lower. The intercept is 448 + 8(3) = 472: about 472 visitors predicted on a dry day, which means something because dry days are in the data.
- y = 3x - 20. The slope is \displaystyle \frac{340 - 250}{120 - 90} = \frac{90}{30} = 3: for each extra square meter, the predicted price is about 3{,}000 dollars higher. The intercept, 250 - 3(90) = -20, is a predicted price of -20{,}000 dollars for a house with no floor, which means nothing.
- y = 0.9x + 17. The slope is \displaystyle \frac{57.5 - 30.5}{45 - 15} = \frac{27}{30} = 0.9: for each extra kilogram, the predicted cost is about 0.90 dollars higher. The intercept is 30.5 - 0.9(15) = 30.5 - 13.5 = 17.
- For each extra hour of practice, the predicted number of errors is about 0.6 lower per 100 words. Over 5 hours, 5 \times (-0.6) = -3: the learner with more practice is predicted to make 3 fewer errors per 100 words.
- The line supports “learners with one more hour of practice type about 4 words per minute faster, on average”. It does not support “my speed will”. One person’s gain can differ: between learners one hour apart in the data, the gains ran from 1 to 7 words per minute. And the data compare different learners, so they do not show that the extra hour causes the gain.
- The slope becomes \displaystyle \frac{4}{60} = \frac{1}{15} \approx 0.067 words per minute for each minute of practice, because the same rise of 4 is now spread over a run of 60. The points, the line and every prediction are unchanged; only the unit of x is different.
- A third variable could produce it. If rainy days mostly fell in a season when regular visitors were away, or when the library kept shorter hours, wet days would have fewer visitors even if rain kept nobody away. Observed data cannot tell these apart.
- It would predict the same value, the intercept, for every x. Knowing x would not change the prediction of y: the line shows no linear trend between the two variables.