Scatter plots and lines of best fit
Building and interpreting scatter plots for bivariate data, and fitting and using a line of best fit
About three lessons of 45 to 60 minutes
Do students who study more actually score higher, or is that just a guess?
Plot each student's study time against their quiz score, one dot per student, and a pattern either appears or it does not. If the dots drift upward together, more study tends to mean a higher score, a positive association. A scatter plot turns a hunch into something you can actually see and measure.
Once a linear pattern is visible, a line of best fit summarises it with a single equation, letting you predict a score for a new amount of study time, and judge honestly how well that line really fits the scattered dots.
- Hours studied vs quiz scorea positive association: more study tends to mean a higher score
- Daily temperature vs ice cream saleshotter days tend to mean more sales, another positive association
- A car's age vs its resale valuea negative association: older cars tend to be worth less
- Shoe size vs favourite colourno real association, the scatter plot would show no pattern at all
What students will be able to do
Students will construct and interpret scatter plots for bivariate data, describe patterns such as clustering, outliers, positive, negative, or no association, and fit and use a line of best fit for data with a linear pattern.
- I can plot bivariate data as a scatter plot.
- I can describe a scatter plot's pattern as a positive, negative, or no association.
- I can identify clustering and outliers in a scatter plot.
- I can describe what a line of best fit represents, and use it to make a prediction.
- I can judge whether a line of best fit is a good match for the data, or whether the data does not really follow a line.
Standards this unit teaches
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 10 two-way tables & scatterplots worksheetthe closest scatter-plot practice available, on reading correlation from a scatter plot
- Grade 8 mean, median, mode and range worksheetgeneral statistics groundwork, describing a single data set before comparing two
- Coordinate plane worksheetsplotting an (x,y) point is the same skill used to plot every dot on a scatter plot
Words to teach and display
- Bivariate data
- data that pairs up two different measurements for each item, such as a student's study time AND their score
- Scatter plot
- a graph of bivariate data as individual points, one per pair of values, with no line connecting them
- Association
- whether two variables tend to move together (positive), move oppositely (negative), or show no clear pattern
- Clustering
- when points on a scatter plot group closely together in one area
- Outlier
- a point that sits far away from the overall pattern of the other points
- Line of best fit
- a straight line drawn through a scatter plot that best summarises a linear pattern in the data
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Building and reading a scatter plot
ConcreteA scatter plot pairs two measurements for the same item (bivariate data) and plots each pair as a single dot, with no connecting line. The overall shape formed by all the dots together, not any single dot, is what reveals the pattern.
If the dots generally trend upward from left to right, that is a positive association: as one quantity increases, the other tends to increase too. If the dots trend downward, that is a negative association. If the dots show no clear upward or downward drift at all, there is no association.
Look at the plant-height scatter plot above. Describe the overall association, and identify the outlier.
- Read the general trend of the dots from left to right (ignoring any single unusual point): the height rises from about 2 cm in week 1 to about 14 cm in week 8.
- That rising pattern is a positive association: as the week number increases, height tends to increase too.
- Compare each dot's position to the surrounding pattern: week 7's height, 3 cm, is far below where the trend predicts (the surrounding weeks are around 9 to 14 cm), so it is an outlier.
Answer: The scatter plot shows a positive association overall. Week 7 (height 3 cm) is an outlier.
- What would a scatter plot look like if there were no association between the two quantities?
- Why is an outlier identified by comparing it to the SURROUNDING pattern, not just by looking at one point alone?
2. Fitting and using a line of best fit
PictorialWhen a scatter plot's dots suggest a roughly linear pattern, a line of best fit is a single straight line drawn to sit as close as possible to all the points at once, not through any two points you happen to pick.
For 5 students' hours studied and quiz scores, (1,3), (2,4), (3,4), (4,6), (5,7), the line of best fit works out to y = x + 1.8 (a slope of 1, an initial value of 1.8), found using every point together, not just the first and last. That equation can then predict a score for a NEW amount of study time.
Five students' hours studied and quiz scores are (1,3), (2,4), (3,4), (4,6), (5,7). The line of best fit for this data is y = x + 1.8. Use it to predict the score for a student who studies 6 hours.
- The line of best fit is y = x + 1.8, found from all 5 points together.
- Substitute x = 6 (hours studied): y = 6 + 1.8 = 7.8.
- A predicted score of 7.8 is a reasonable estimate, since it continues the same trend just past the data's range (1 to 5 hours).
Answer: The predicted score for 6 hours studied is 7.8.
- Why is a line of best fit drawn using EVERY point, rather than just connecting the first and last?
- Would you trust this line's prediction for 20 hours studied as much as for 6 hours? Why or why not?
Common misconceptions and how to address them
MisconceptionEvery scatter plot must show some kind of association.
Why it happens: Students expect a pattern simply because they were asked to look for one, even when the data genuinely has no relationship.
How to address it: Some real data pairs, like shoe size and favourite colour, truly have NO association, and the honest scatter plot shows a random-looking spread with no trend. 'No association' is a valid, correct conclusion, not a sign something went wrong.
MisconceptionA line of best fit should be drawn through the first and last data points.
Why it happens: Connecting the two extreme points feels like the most natural way to draw 'a line through the data', but it ignores every point in between.
How to address it: A genuine line of best fit balances the distance to ALL the points at once (found mathematically, as in section 2), often passing near but not exactly through most points, including the first and last. Compare a first-to-last line against the true best-fit line on the same data to see they usually differ.
MisconceptionA positive association means the two quantities are increasing at the exact same rate.
Why it happens: Students conflate 'both go up together' with 'both go up by the same amount each time', which is a much stronger and often false claim.
How to address it: Positive association only means BOTH quantities tend to increase together, not that they increase by matching amounts. The steepness of the line of best fit (its slope) is what actually measures the rate, and it can be any positive number.
MisconceptionAn outlier should always be deleted from the data before finding a line of best fit.
Why it happens: Students learn that outliers 'mess up' patterns and conclude the fix is always to remove them.
How to address it: An outlier changes the position of the best-fit line, but that alone does not make it wrong or safe to delete. The better first step is to investigate WHY it occurred (like the plant's week 7 in section 1, perhaps damaged or measured incorrectly) before deciding whether it belongs in the analysis.
MisconceptionA line of best fit can be trusted to predict accurately no matter how far outside the original data you go.
Why it happens: Once an equation exists, students treat it as universally true rather than a summary of the specific data range it was built from.
How to address it: Predictions close to the data's original range (like 6 hours, just past 1 to 5) are reasonable; predictions far outside that range (like 50 hours) are extrapolation and become increasingly unreliable, since nothing guarantees the same pattern continues that far.
Guided practice (with answers)
1. A scatter plot shows a car's age (years) on the x-axis and its resale value on the y-axis, drifting downward from left to right. What kind of association is this?
Answer: A negative association, because as the car's age increases, its resale value tends to decrease.
2. A scatter plot of temperature vs ice cream sales shows one very hot day with unusually LOW sales, far from the rest of the upward-trending dots. What is that point called?
Answer: An outlier, because it sits far from the overall pattern formed by the rest of the data.
3. A line of best fit for a data set is y = 2x + 5. Predict y when x = 4.
Answer: 13, because y = 2(4) + 5 = 8 + 5 = 13.
4. Would you draw a line of best fit through data that shows no clear pattern at all (points scattered randomly)?
Answer: No, a line of best fit is only meaningful when the data shows a roughly linear association; scattered data with no pattern should not be forced onto a line.
5. A line of best fit is built from data ranging from x=2 to x=10. Is predicting y at x=9 or at x=40 more reliable, and why?
Answer: x=9 is more reliable, because it falls within the original data's range, while x=40 is far outside it (extrapolation), where the same pattern is not guaranteed to continue.
Independent practice worksheets
Practise building, describing and using scatter plots and lines of best fit, with computed, never-wrong answer keys.
Differentiation
- Start with a small data set (5 or 6 points) with a very clear, strong association, before introducing noisier or weaker patterns.
- Provide a simple 'up, down, or no pattern' sentence frame for describing association before requiring the words positive/negative in isolation.
- Circle the outlier's coordinates and the surrounding cluster's typical range side by side, so the comparison that identifies it is explicit.
- For predictions, always underline the given line-of-best-fit equation and the x-value being substituted before calculating.
- Compare two scatter plots with the same general trend but different amounts of scatter around the line, discussing which line of best fit is a 'better fit'.
- Investigate how a single added outlier point shifts a line of best fit's slope and intercept.
- Collect a small real data set (classmates' arm span vs height, for example) and build a genuine scatter plot and line of best fit from it.
- Discuss the difference between association and causation using a real example (e.g. ice cream sales and sunburns both rise in summer, but one does not cause the other; hot weather causes both).
Assessment: exit ticket
A three-question exit ticket sampling association, outliers, and using a line of best fit.
1. A scatter plot of hours of TV watched vs test score drifts downward from left to right. What association is this?
Answer: A negative association, because as hours of TV watched increases, test scores tend to decrease.
2. In a scatter plot where most points cluster tightly between x=10 and x=20, one point sits at x=75. What is that point?
Answer: An outlier, because it sits far outside the cluster formed by the rest of the data.
3. A line of best fit is y = 3x - 2. Predict y when x = 5.
Answer: 13, because y = 3(5) - 2 = 15 - 2 = 13.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 building and reading scatter plots, clustering and outliers (section 1), Lesson 2 fitting and using a line of best fit (section 2), Lesson 3 mixed practice plus the exit ticket.
- This unit assumes comfort plotting (x,y) coordinates. Revisit basic coordinate-plane plotting first if that foundation is shaky.
- Language to repeat: association describes the OVERALL trend, not any single point; an outlier is judged against the surrounding pattern; a line of best fit uses every point, never just two.
- The line of best fit equation quoted in section 2 (y = x + 1.8) is the actual least-squares line for the five given points, computed the same way the site's own ScatterPlot diagram engine derives it (never a hand-placed line), so the worked prediction and the drawn line always agree.
- Curriculum note: 8.SP.A.1 (Common Core) covers building and interpreting scatter plots; 8.SP.A.2 covers fitting and judging a line of best fit. They are grouped as one unit here since a line of best fit only makes sense once a scatter plot's pattern has already been read.
- Present and print both work: use the Print button for a clean handout, or build the plant-height scatter plot live on the board, asking students to spot the outlier themselves before it is named.