Slope is the change in y per unit change in x — a rate, with units.
Slope The change in the vertical quantity per unit change in the horizontal one.
Step 1: Something to Notice
Watch first. The explanation comes later.
Ice cream sales and drowning deaths track each other closely across the year, with a correlation strong enough to publish.
Does banning ice cream save lives?
Step 2: Find Out
Isolate the slope and read what it means in the units of the two axes.
Step 1 — Predict
Ice cream sales and drowning deaths rise together. What follows?
Choose what you think will happen. You cannot see the experiment until you do — guessing first is what makes it worth watching.
Step 3: So Here Is Why
Now the explanation, after you have seen it happen.
The intercept is the value at zero
Sometimes meaningful, sometimes an extrapolation into unmeasured territory.
A trend line is a claim
It asserts that the relationship continues between and beyond the points you measured.
Correlation has four explanations
A causes B, B causes A, something causes both, or coincidence.
And the axes decide what a shape means
A truncated or logarithmic axis changes the picture without falsifying any point.
A line through points is a claim
Drawing a straight line asserts that the relationship is linear. If the points curve, the line misrepresents the data, and connecting points dot-to-dot asserts something different again.
Which variable goes where
The independent variable — what you chose — goes on the horizontal axis, and the dependent one on the vertical. Following the convention lets any reader interpret the graph without asking.
Step 4: A Common Mistake
Lots of people think
“If two quantities rise together, one causes the other.”
Step 5: Why the Axes Can Lie Without Anything Being False
The same idea somewhere new.
A vertical axis that starts at 98 instead of 0 turns a two percent rise into a dramatic climb; a logarithmic axis turns exponential growth into a straight line. Neither misrepresents a single data point. Reading the axes before reading the shape is the habit that protects you, because the shape is the part your eye trusts and the axes are the part that decides what the shape means.
Step 6: Think It Through
Practice makes it stick.
Ice Cream and Drowning
Problem 1 of 2
They rise together every summer. What follows?
The Steep Climb
Problem 2 of 2
A chart looks alarming. Its vertical axis starts at 98. What should you check?
Reading a Graph
1 of 5
What are the units of a slope?
2 of 5
What does an intercept represent?
3 of 5
How many explanations does a correlation leave open?
4 of 5
What separates correlation from causation?
5 of 5
A trend line asserts what?
Step 7: Quick Check
Show what you know.
Question 1 of 1
Two quantities rise together. Does one cause the other?
Step 8: Explain It in Writing
Claim, evidence, then reasoning.
The question
Explain what a slope means physically, and why correlation does not establish causation.
Fill in all three boxes. The reasoning box is the one that matters most.
What You Found Out
- Slope is a rate, with units of the y unit over the x unit.
- The intercept is the y value at x = 0, and may be an extrapolation.
- Correlation leaves four explanations open, and more data settles none of them.
- Axis choices change what a shape means without falsifying any point.