📊 Statistics Steppe · Statistics
Scatter Plots & Trends
Plot two measurements against each other, describe the association you see, and fit a straight line you can predict from.
In short
- Each point on a scatter plot is one item carrying two measurements, and the plot asks whether the two move together.
- Positive, negative or no association describes the overall drift left to right, not any single point.
- The slope of a line of best fit is rise over run, and in context it is a rate: the change in y per one unit of x.
- Predictions are sound inside the range of the data and speculative outside it, and association is never proof of cause.
Two measurements at once
A scatter plot puts two measurements of the *same thing* on one picture: hours of sunshine across, reeds ripened up. Every point is one day, carrying both of its numbers.
That is the whole idea, and it is what makes a scatter plot different from a bar chart. A bar chart compares separate categories; a scatter plot asks whether two quantities move together.
Read it left to right and ask one question: as the across-value grows, what does the up-value tend to do?
- Both grow together: a positive association.
- One grows while the other falls: a negative association.
- No overall drift either way: no association — at least, no straight-line one.
"Tend to" matters. Real data scatters, and a few points going the other way do not destroy a trend.
The line of best fit
When there is a straight-line trend, a line of best fit is drawn through the cloud of points — close to as many as possible, with roughly as many above it as below.
It is an ordinary straight line, so it has an ordinary equation, y = mx + b, and its slope is found the ordinary way from any two points on it:
m = (change in y) / (change in x)
If the line passes through (2, 14) and (7, 39), then m = (39 - 14) / (7 - 2) = 25 / 5 = 5.
Use two points on the line, not two data points — the line is the summary, and individual points sit above and below it. Rise over run, never run over rise: turning the fraction upside down measures how far you move sideways per step up, which is not what slope means.
What the slope and intercept mean
On a scatter plot the numbers in y = mx + b are not abstract. They carry the units of the data.
The slope is a rate: how much the up-value changes for every one-unit increase in the across-value. A slope of 5 in "hours of sunshine against reeds ripened" says each extra hour of sunshine goes with about 5 more reeds. A negative slope says the prediction falls as the across-value grows.
The intercept b is the prediction when x = 0. Sometimes that is meaningful — 0 hours of sunshine is a real day — and sometimes it is nonsense, such as the predicted weight of a person 0 cm tall. When x = 0 lies far outside the data, the intercept is a number the line needs, not a fact about the world.
Say "goes with", not "causes". Two quantities can move together because one drives the other, because both are driven by a third, or by coincidence. A scatter plot cannot tell those apart.
Predicting, and outliers
To predict, substitute into the equation. With y = 5x + 4 and x = 8: y = 5 x 8 + 4 = 44.
Predicting inside the range of the data is reasonable — the trend was measured there. Predicting far outside it is a guess dressed up as arithmetic; the trend has no evidence out there, and real relationships bend.
An outlier on a scatter plot is the point that breaks the pattern, not the point with the biggest number. A reading can be the largest on the plot and sit perfectly on the line, and a middling reading can be far off it. Lay a ruler along the points that do line up and see which one is nowhere near.
A single outlier can drag a line of best fit noticeably, which is worth knowing before you trust a prediction made from a small data set.
Worked examples
Example 1
A line of best fit passes through (3, 17) and (11, 49). Find its slope and say what it means, given x is km walked and y is minutes taken.
- Slope = (change in y) / (change in x).
- Change in y = 49 - 17 = 32 minutes. Change in x = 11 - 3 = 8 km.
- m = 32 / 8 = 4.
- In context: each extra kilometre walked goes with about 4 more minutes taken.
- Check the units: minutes divided by km gives minutes per km, which is exactly what a walking pace is.
Example 2
The trend line for a scatter plot is y = -3x + 58, where x is bog depth in cm and y is crossing time in minutes. Predict the crossing time at a depth of 9 cm, and comment.
- Substitute x = 9: y = -3 x 9 + 58.
- -3 x 9 = -27, so y = -27 + 58 = 31 minutes.
- The slope is negative, so deeper water goes with shorter crossings in this data — surprising, and worth asking about.
- A likely explanation is that a third quantity is at work: perhaps deeper crossings are the ones made by boat rather than on foot.
- The prediction is only trustworthy if 9 cm lies inside the range of depths that were actually measured.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 5Ferry hands plotted the depth of the bog against the time each crossing took. The plot shows bog depth (across) against crossing time (up). What kind of association does it show?
- a positive association
- a negative association
- no clear association
Answer: A. a positive association
- The points run from (2, 24) on the left to (12, 38) on the right.
- As the across-value increases, the up-value increases too.
- So the plot shows a positive association.
Problem 2
Difficulty 3 of 5A line of best fit is drawn through a scatter plot of load against sacks delivered. It passes through (0, 73) and (9, 85). What is its slope? Give a fraction in lowest terms (a whole number if it simplifies to one).
Answer: 1 1/3
- m = (y2 - y1) / (x2 - x1)
- m = (85 - 73) / (9 - 0) = 12/9
- m = 4/3
- In context: each extra kg goes with about 1.333 more sacks.
Problem 3
Difficulty 4 of 5Ferry hands plotted the depth of the bog against the time each crossing took. The line of best fit is y = 9x + 11, where x is the bog depth (cm) and y is the crossing time (minutes). Predict the crossing time when the bog depth is 4 cm.
Answer: 47 minutes
- The model is y = 9x + 11.
- Substitute x = 4: y = 9 x 4 + 11
- y = 36 + 11 = 47
- So about 47 minutes — a prediction from a trend, not a measurement.
Common mistakes
- Reading the direction backwards and calling a falling trend positive.
- Turning the slope fraction upside down, computing run over rise.
- Using two scattered data points instead of two points on the line of best fit.
- Naming the largest reading as the outlier when the outlier is the point furthest from the pattern.
- Treating an association as proof that one quantity causes the other.
What you should be able to do
- Describe an association as positive, negative or absent from a scatter plot.
- Find the slope of a line of best fit through two points on it.
- Use a trend line to predict a value, and say when the prediction is trustworthy.
- Interpret the slope and the intercept of a trend line in context.
Where this fits in the curriculum
Common Core
- 8.SP.A.1
Grade 8 — Construct and interpret scatter plots for bivariate measurement data, and describe patterns such as clustering, outliers and linear association.
- 8.SP.A.2
Grade 8 — Know that straight lines are used to model relationships between two quantitative variables, and informally fit a straight line to data.
- 8.SP.A.3
Grade 8 — Use the equation of a linear model to solve problems, interpreting the slope and the intercept.
Ontario
- MTH1W.D1
Grade 9 de-streamed — Describe the collection and use of data, and represent and analyse data involving one and two variables.
Ontario numbers its Data expectations per grade document and the specific numbering could not be verified line by line here, so this tag names the strand's OVERALL expectation rather than guessing at a specific one.
SAT
- Problem Solving and Data Analysis
Scatterplots, lines of best fit, and interpreting slope and intercept in context.