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Describe direction, strength and shape of association; distinguish correlation strength from gradient and recognise curved patterns and influential observations.
Before you startScatter diagrams and paired numerical data.
01 / Describe direction, strength and shape
Positive: larger x tends to accompany larger y. Negative: larger x tends to accompany smaller y.
“Tends to” allows individual exceptions.
Here, correlation describes linear association. First look for the direction, then how closely the points follow a straight trend. Also describe curvature or unusual observations rather than squeezing every diagram into a single label.
These are original constructed datasets for comparing patterns, not measurements from a real study.
The upward dataset has a strong positive linear association. Changing the horizontal units changes gradients, not the strength or sign of association.
02 / State the direction in context
An upward trend would be positive association
Longer journeys tend to use more fuel in that sample.
It need not rise between every pair of points
Scatter around a trend is expected.
For distance from a lamp and light reading, a downward pattern is negative association
Both measured quantities can remain positive.
All recorded temperatures and heater-use times are positive, but heater use tends to fall as temperature rises. Can their association be negative?
Direction compares changes, not whether observations lie below zero.
Yes. Larger temperatures tend to accompany smaller heater-use times, even though both variables have positive values.
Most points follow an upward trend, but one larger x has a smaller y than its neighbour. Does that rule out positive correlation?
A tendency is not a strict ordering of all observations.
No. Describe the overall pattern and mention unusual scatter if relevant.
03 / Judge closeness to a straight trend
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Strength concerns how closely the point cloud follows a linear pattern.
Gradient concerns the change in y per unit change in x.
Dataset A lies exactly on a shallow nonhorizontal line. Dataset B is widely scattered around a steep upward trend. Which shows stronger positive linear association?
Closeness matters, not steepness.
A has perfect positive linear association. B’s larger gradient does not make its correlation stronger.
Distances change from metres to centimetres, multiplying each x by 100 while y is unchanged. What happens to the gradient and correlation direction/strength?
The same horizontal physical change now has 100 times the numerical size.
The gradient per numerical x unit is divided by 100. Positive rescaling leaves the sign and strength of linear association unchanged.
04 / Look for a curved relationship
Points follow y = x² for x = −3,−2,−1,0,1,2,3. Is there a clear relationship despite no overall upward or downward straight trend?
Left and right halves move in opposite directions.
Yes. There is an exact curved relationship. Calling it no relationship would be wrong; the linear association is zero for these symmetric points.
The same curve is observed only for positive x. Must its association still be zero?
The sampled range changes the pattern.
No. Over positive x, y increases with x. Range and shape matter; the full symmetric-data conclusion cannot simply be reused.
05 / Notice unusual paired observations
Most points near x = 10 have y near 30. One has (10,4). What should you investigate?
Its response is far from the local pattern.
Check measurement, units, case matching and context. A genuine in-scope observation should not be removed merely to strengthen a trend.
One point has an x far beyond the rest. Can it strongly affect an apparent line or correlation?
It can have substantial influence on a fitted trend.
Yes. Examine the diagram and context, and distinguish an influential observation from a verified error. Do not infer its effect from distance alone.
06 / Use careful language
A sample shows strong positive correlation between two variables. Does strength alone show that one causes the other?
Consider common causes and study design.
No. Confounding, selection, reverse direction or coincidence may explain an association. Strong correlation is still not proof of causation.
Does a strong negative correlation guarantee the next observation lies exactly on a fitted line?
A trend can have scatter.
No. It supports a tendency within the observed context, not an exact individual guarantee.
07 / Write a complete description
A fictional sample of delivery trips shows a fairly tight upward linear cloud for distance and duration. Write a suitable description.
Name both variables, direction and strength, with sample scope.
In this sample, longer delivery distances tend to be associated with longer durations, with a strong positive linear association. This does not by itself establish a causal effect or guarantee each trip’s duration.
All points have y = 5 while x varies. Should this automatically be called perfect positive correlation because they lie on a line?
There is no variation in the response.
No. There is no upward or downward tendency; a numerical correlation coefficient is undefined when a variable has zero variation. Do not confuse any straight line with perfect positive or negative association.
08 / Separate the questions
Describe what happens as x increases, how closely the observations follow a straight trend and whether a curve or influential observation changes the interpretation. Units alter gradient, and a linear summary does not establish causality or rule out nonlinear relationships.
Section 1 of 8 · Describe direction, strength and shape