Line of best fit
Draw a line of best fit on a scatter plot, use it to make predictions by interpolation, and evaluate whether a given line is a good fit for the data.
Worked examples
Using a line of best fit to estimate a value
Straightforward
Problem
A scatter plot shows the relationship between the number of hours of exercise per week () and resting pulse rate (, in beats per minute). The line of best fit has equation .
(a) Use the line of best fit to estimate the resting pulse rate of a person who exercises for 10 hours per week.
(b) The data were collected from people who exercise between 2 and 20 hours per week. Is the estimate in part (a) an interpolation or an extrapolation?
(a) Use the line of best fit to estimate the resting pulse rate of a person who exercises for 10 hours per week.
(b) The data were collected from people who exercise between 2 and 20 hours per week. Is the estimate in part (a) an interpolation or an extrapolation?
1
Identify the value of to substitute.
The person exercises for 10 hours per week, so .
2
The estimated resting pulse rate is 65 beats per minute.
Substitute into the equation of the line of best fit.
The estimated resting pulse rate is 65 beats per minute.
3
Determine whether this is an interpolation or an extrapolation.
The data range is to hours per week.
Since lies within this range, the estimate is an interpolation — it uses the line within the observed data range rather than extending it beyond.
Since lies within this range, the estimate is an interpolation — it uses the line within the observed data range rather than extending it beyond.
Answer
(a) 65 beats per minute. (b) Interpolation — lies within the data range of 2 to 20 hours per week.
Evaluating whether a line of best fit is appropriate
Moderate
Problem
A student draws a line of best fit on a scatter plot with 12 data points. The line passes through 4 of the 12 points exactly. Of the remaining 8 points, all 8 fall below the line.
(a) Explain why this is not a good line of best fit.
(b) Describe the two key features of a well-drawn line of best fit.
(a) Explain why this is not a good line of best fit.
(b) Describe the two key features of a well-drawn line of best fit.
1
Count the points above and below the line.
4 points lie exactly on the line and 8 points fall below it. No points fall above the line.
Distribution: 0 above, 4 on the line, 8 below.
Distribution: 0 above, 4 on the line, 8 below.
2
Explain why this distribution is a problem.
A good line of best fit should have roughly equal numbers of points above and below it. Here, all 8 remaining points fall below the line — none are above. This means the line is too high; it consistently overestimates the data values. A better line would be shifted downward until approximately 6 points fall above and 6 below.
3
State the two key features of a well-drawn line of best fit.
Feature 1: Approximately equal numbers of data points fall above the line and below it.
Feature 2: The data points are spread along the full length of the line — not all clustered in the middle while the ends of the line are far from any data.
Feature 2: The data points are spread along the full length of the line — not all clustered in the middle while the ends of the line are far from any data.
Answer
(a) The line is too high — 8 of the 12 points fall below it and none fall above. A well-drawn line would be shifted downward so that roughly equal numbers of points fall above and below. (b) A good line of best fit: (i) has approximately half the points above and half below; and (ii) has the points spread evenly along its entire length.
Distinguishing association from causation
Challenging
Problem
A study of 100 adults finds a strong positive association between the number of credit cards owned and annual income. A financial journalist writes: "Getting more credit cards will increase your income."
(a) Identify the error in the journalist's reasoning.
(b) Suggest a more plausible explanation for the observed association.
(a) Identify the error in the journalist's reasoning.
(b) Suggest a more plausible explanation for the observed association.
1
Identify the type of error the journalist has made.
The journalist has concluded that one variable causes the other based on an association alone. This is the error of confusing correlation with causation — a statistical relationship between two variables does not tell us which one (if either) causes the other.
2
Explain why an association alone does not prove causation.
When two variables are associated, there are several possible explanations:
1. Variable A causes variable B.
2. Variable B causes variable A (reverse causation).
3. A third variable C causes both A and B (confounding variable).
A scatter plot and a line of best fit can show that two variables move together, but they cannot identify which explanation is correct.
1. Variable A causes variable B.
2. Variable B causes variable A (reverse causation).
3. A third variable C causes both A and B (confounding variable).
A scatter plot and a line of best fit can show that two variables move together, but they cannot identify which explanation is correct.
3
Suggest a more plausible explanation for the association.
A more plausible explanation: people with higher incomes are more likely to be approved for additional credit cards and may choose to hold more of them. The income enables the cards — not the other way around.
Alternatively, a third variable such as level of education or financial experience may independently drive both higher income and higher credit card ownership.
Alternatively, a third variable such as level of education or financial experience may independently drive both higher income and higher credit card ownership.
Answer
(a) The journalist has confused correlation with causation. An association between two variables does not prove that one causes the other. (b) Higher income likely enables people to qualify for and hold more credit cards — the direction of influence is probably the reverse of what the journalist claims. Alternatively, a third variable such as education level may drive both higher income and higher credit card ownership.
Practise
Q1·Straightforward
A scatter plot shows the relationship between the number of hours a student studies per week () and their score on a maths test (). The line of best fit drawn on the scatter plot has equation . Use the line of best fit to estimate the test score of a student who studies for 5 hours per week.
Explanation
Substituting into :
The line of best fit estimates a test score of 58 for a student who studies 5 hours per week.
The line of best fit estimates a test score of 58 for a student who studies 5 hours per week.
Q2·Straightforward
A scatter plot comparing daily maximum temperature (, in °C) and the number of ice creams sold by a beachside kiosk () has a line of best fit with equation . Use the line to estimate the number of ice creams sold on a day when the maximum temperature is 30°C.
Explanation
Substituting :
The line of best fit predicts 91 ice creams will be sold on a day with a maximum temperature of 30°C.
The line of best fit predicts 91 ice creams will be sold on a day with a maximum temperature of 30°C.
Q3·Straightforward
A scatter plot comparing a person's age (, in years) and their resting heart rate (, in beats per minute) has a line of best fit with equation . Use the line to estimate the resting heart rate of a 40-year-old.
Explanation
Substituting :
The line of best fit estimates a resting heart rate of 68 beats per minute for a 40-year-old.
The line of best fit estimates a resting heart rate of 68 beats per minute for a 40-year-old.
Q4·Straightforward
A factory records the number of items produced per hour () and the defect rate (, as a percentage). The line of best fit has equation . Use the line to estimate the defect rate when 25 items are produced per hour.
Explanation
Substituting :
The line of best fit estimates a defect rate of when 25 items are produced per hour.
The line of best fit estimates a defect rate of when 25 items are produced per hour.
Q5·Moderate
Which of the following best describes what a line of best fit is designed to do?
Explanation
A line of best fit is drawn through the middle of the data so that roughly equal numbers of data points fall above and below the line, with points spread evenly along its length. It does not have to pass through any specific data point, connect the first and last points, or pass through the origin.
Q6·Moderate
A scatter plot contains 10 data points. Student A draws a line where 8 points fall above and 2 fall below. Student B draws a line where 5 points fall above and 5 fall below, with points spread roughly evenly along the full length of the line. Which student has drawn the better line of best fit?
Explanation
Student B's line is better. A well-drawn line of best fit has roughly equal numbers of data points above and below it. Student B achieves this with 5 above and 5 below. Student A's line has 8 above and only 2 below, which means the line is too low for most of the data. A better line would shift upward to balance the distribution.
Q7·Moderate
A line of best fit is drawn on a scatter plot with 12 data points. The line fits the 6 central points closely, but the 3 leftmost points and the 3 rightmost points all sit well above the line. What does this suggest?
Explanation
A good line of best fit should have the data points spread roughly evenly along its entire length. When points at both ends are all above the line, this suggests the line is slightly too low at the extremes — it may need to be tilted or shifted to better represent the full data set. Having the line fit well only in the middle but poorly at the ends indicates the line is not a good overall fit.
Q8·Moderate
Which of the following best states the main characteristics of a well-drawn line of best fit?
Explanation
A well-drawn line of best fit has two key characteristics: (1) approximately equal numbers of data points fall above and below the line; and (2) the points are spread evenly along the entire length of the line from end to end. The line does not need to pass through any specific data point, the point with the highest -value, or the origin, and can have either a positive or negative gradient depending on the data.
Q9·Challenging
A study finds a strong positive association between the number of backyard swimming pools per suburb and the rate of skin cancer diagnoses in that suburb. Which of the following statements is most correct?
Explanation
The association does not prove causation. Both the number of swimming pools and the rate of skin cancer are higher in warmer, sunnier suburbs. The explanation is that these suburbs receive more sunshine hours, which encourages pool ownership and also increases residents' UV exposure. Climate is the confounding variable that drives both. The pools themselves are not causing the skin cancer.
Q10·Challenging
A researcher finds a strong negative association between the number of hours of television watched per day () and a student's score on an end-of-year exam (). A teacher concludes: "Watching television causes students to get lower exam scores." Which of the following statements is most correct?
Explanation
The association does not prove causation. A student who watches many hours of television may also be spending less time studying — reduced study time, not the television itself, may be the real cause of lower scores. Factors such as motivation, parental involvement, or home environment could independently affect both how much TV a student watches and how well they perform. A statistical association shows only that two variables tend to move together; it does not identify which one causes the other.
Q11·Challenging
A data analysis shows a strong positive association between the number of hospitals in a city and the total number of deaths recorded in that city per year. A journalist concludes: "More hospitals cause more deaths — we should close hospitals to save lives." Which of the following best explains the flaw in this reasoning?
Explanation
The journalist has confused correlation with causation. Larger cities have more people, which means they require more hospitals and also record more deaths simply because there are more residents. Population size is the confounding variable that drives both. Closing hospitals would not reduce deaths — it would likely increase them. The association exists not because hospitals cause deaths, but because both variables are higher in more populous cities.
Q12·Challenging
A study of primary school children finds a positive association between shoe size and reading ability score. A classmate says: "Bigger feet cause better reading." Which of the following is the most accurate statement?
Explanation
The classmate has confused association with causation. As children grow older, their feet get larger (increased shoe size) and their reading ability improves (through experience and instruction). Age is the confounding variable — it causes both. The shoes are not improving reading; both are driven by the child's development over time. This is a classic example of why association does not imply causation.