The best-fit line, straight from the GDC: The regression line of y on x is the best-fit line y = ax + b.
On Paper 2 you get a (gradient) and b (intercept) from the calculator's linear regression — no hand calculation.
[Diagram: math-scatter-regression] - Available in full study mode
IB-style question — find the line
Hours studied x and score y are (1,40), (2,50), (3,55), (4,70), (5,80).
Find the regression line of y on x.
Step by step
- Enter the pairs and run linear regression.
- Write the line.
Final answer
y = 10x + 29 (to 3 s.f. if needed).
a = change per unit, b = value at x = 0: In y = ax + b, the gradient a is the change in y for each 1-unit increase in x, and the intercept b is the predicted y when x = 0.
Always read them in context.
IB-style question — interpret a and b
A regression line for plant height y cm against weeks x is y = 1.8x + 4.
Interpret the gradient and the intercept.
Step by step
- Gradient = change in y per 1 unit of x.
- Intercept = y when x = 0.
Final answer
The plant grows about 1.8 cm per week, and was about 4 cm tall at the start (week 0).
Use the units: State the gradient with units ('per week', 'per °C') and the intercept as the starting value.
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Every regression line passes through (x̄, ȳ): The regression line always passes through the mean point (x̄, ȳ).
So if you know the line and one mean, you can find the other — and the point of the two means is guaranteed to be on the line.
IB-style question — use the mean point
A regression line is y = 10x + 29 and the mean of x is x̄ = 3.
Find the mean of y.
Step by step
- The mean point (x̄, ȳ) lies on the line.
- Substitute x̄ = 3.
Final answer
ȳ = 59.
A quick check: Substituting (x̄, ȳ) into the regression line should always work exactly — a handy way to check your line.
y on x to predict y; the two lines cross at the mean point: To predict y from x, use the line of y on x; to predict x from y, use x on y.
Both lines pass through (x̄, ȳ), so solving them simultaneously gives the two means.
IB-style question — find the means
The regression line of y on x is y = 2x + 1, and the line of x on y is x = 0.4y + 0.2.
Find the mean of x and the mean of y.
Step by step
- Both lines meet at the mean point — solve simultaneously.
- Solve for x̄, then ȳ.
Final answer
x̄ = 3, ȳ = 7.
Match the line to the direction: Use the line whose subject is the variable you want to predict — y on x for y, x on y for x.
Using the wrong one loses accuracy and marks.
IB-style question — two regression lines
For a data set, the regression line of y on x is y = 0.8x + 2, and the line of x on y is x = 0.9y − 1.
(a) Which line should be used to estimate y from a given x?
(b) Find the mean point (x̄, ȳ).
Step by step
- (a) To predict y FROM x, use the y-on-x line. (Using x-on-y the wrong way round loses marks.)
- (b) Both lines pass through the mean point, so solve them simultaneously.
- Back-substitute for ȳ.
Final answer
(a) the y-on-x line. (b) mean point ≈ (2.86, 4.29).
[Diagram: math-scatter-regression] - Available in full study mode