MISSION · Data and probability
Scatter Plots and Association
What a cloud of points can and cannot tell you
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A scatter plot puts two measurements of the same thing on two axes, so the shape of the cloud tells you how the measurements travel together.
Reading the cloud
Each dot is one individual measured twice — one student’s revision hours and their exam score, one car’s age and its price. The pattern the dots make is called the association.
- Positive — the cloud rises to the right. More of one goes with more of the other.
- Negative — the cloud falls. More of one goes with less of the other.
- None — the cloud is a shapeless blob. Knowing one measurement tells you nothing about the other.
Then there’s strength, which is separate from direction. A tight narrow band is a strong association; a wide fuzzy one is weak. Both can point the same way. Strength is about how confidently you could predict one value from the other, and a weak positive trend is a real trend that just isn’t much use for predicting.
The trend line
A line of best fit runs through the middle of the cloud, as close to as many points as possible. It’s a summary of the relationship, and it lets you estimate values you didn’t measure.
Two cautions come with it. First, a line drawn through a shapeless blob is meaningless — fit a line only when there’s actually a trend. Second, the line is trustworthy across the range of data you collected and nowhere else. A line fitted to children’s heights between ages 5 and 12 will cheerfully predict that a forty-year-old is nine feet tall. Stretching a trend past its data is called extrapolation, and it’s where most bad predictions come from.
Outliers
A point far from the rest is an outlier, and it’s worth a second look before it’s worth an explanation. Sometimes it’s a mistake — a decimal in the wrong place, a value entered in the wrong unit. Sometimes it’s the most interesting point on the chart.
Don’t delete outliers because they spoil the pattern. Investigate them, then decide.
Association is not cause
This is the one that matters outside the classroom. Ice cream sales and drowning deaths rise together every summer. Ice cream does not cause drowning; hot weather causes both.
A third factor driving two others is called a confounder, and it is the usual explanation for a surprising correlation. Two variables moving together licenses exactly one conclusion — that they move together. Causation needs an experiment, or at least an argument, not a chart.
Common mix-ups
- Reading causation from correlation. The headline error.
- Fitting a line to a blob, which manufactures a trend that isn’t there.
- Extrapolating beyond the data, where the line has no evidence behind it.
- Confusing direction with strength. A steep line is not necessarily a strong relationship; a tight cloud is.
Check yourself
A scatter plot of a car’s age against its price falls steadily to the right, tightly grouped. Describe the association, and say what it would be reckless to conclude.
A strong negative association: older cars cost less. It would be reckless to conclude that age alone sets the price — mileage, condition, and model all travel with age.
LEVEL 3 · REFLECT
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