SME
Grade 9 Math · Algebra 1
Scatter plots, correlation, residuals and lines of best fit
Use this for “The line of best fit is ŷ = 5x + 47. Predict the points for 4.5 hours, and find a residual.”
Chart
Line of fit: ŷ = 5x + 47
Prediction for x of 4.5: ŷ = 5 × 4.5 + 47 = 69.5
Residual = actual − predicted: 2 positive (above the line), 4 negative
Sum of squared residuals = 12
Point 3 (3, 61): predicted 62, residual 61 − 62 = −1
Residual plot: no pattern around 0 means a line fits; a curve means it does not
This is an example. Type your own values to start a new problem.
ŷ
=
m
x
+
b
ŷ(
x
)
=
ŷ
k
Point number
example
e
Residual of point k
worked out (not typed)
Formulas
ŷ
=
m
×
x
+
b
69.5 = 5 × 4.5 + 47
e
=
yₖ
−
(
m
×
xₖ
+
b
)
for
point
k
−1 = yₖ − (5 × xₖ + 47) for point 3
Assumptions
•
Residual = actual − predicted; a point above the line has a positive residual.
•
A good line leaves residuals scattered about 0 with no pattern.
•
Predict only inside the data’s x values.
•
The line ŷ = 5x + 47 is drawn by eye; a calculator’s least-squares line is close to it.
Step-by-step
WE KNOW
x = 4.5, m = 5, b = 47, k = 3
FIND
predicted points (ŷ), residual of point k (e)
Step 1 · Find predicted points (ŷ)
Use ŷ = m × x + b
Put
x
into
the
line:
slope
times
x
,
plus
the
intercept.
ŷ
=
5
×
4.5
+
47
ŷ
=
22.5
+
47
ŷ
=
69.5
Step 2 · Find residual of point k (e)
Use e = yₖ − (m × xₖ + b) for point k
Point
3
is
(3,
61):
its
actual
points
minus
the
line’s
prediction
at
x
=
3.
e
=
61
−
(
m
×
3
+
b
)
e
=
61
−
(5
×
3
+
47)
e
=
61
−
(15
+
47)
e
=
−1
Check
Put the numbers back in. Both sides should match.
69.5 = 5 × 4.5 + 47 ✓
−1 = 61 − (5 × 3 + 47) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.