Scatter diagrams.
Plot paired observations, describe correlation and estimate using a carefully drawn line of best fit.
Recognise correlation · draw one ruled best-fit line · interpolate and recognise extrapolation risks
Plot paired observations as crosses
Sometimes we want to know how two measurements vary together, such as a student’s revision time and test score. Record both measurements for each student so the pairing is preserved.
Each paired observation becomes one cross at (x, y). Unlike a time-series plot, the points are not joined in observation order. The overall arrangement is what matters: do larger x-values tend to accompany larger y-values, smaller y-values, or no clear pattern?
A scatter diagram shows the relationship between two numerical variables measured on the same individuals or objects. Each pair (x, y) gives one plotted cross.
From the illustrative line: y ≈ 6.1. This is an estimate, not an observed point.
An observation is (3, 4). Explain how to plot it.
Find x = 3 on the horizontal axis and y = 4 on the vertical axis. Mark a small, clear cross at their intersection. Do not join successive observations.
Pause and explainWhy must the two measurements in a plotted pair come from the same person or object?
The graph examines their relationship. Pairing unrelated individuals would destroy the meaning of the observations.
Describe direction and strength
Correlation describes the direction and strength of a tendency in paired data. It is about the overall pattern, so one point that goes against the trend does not automatically remove the relationship.
An upward tendency is positive and a downward tendency negative. Closeness to a straight-line pattern describes strength. “No clear correlation” means the data do not support a clear tendency of the type being considered; it does not mean every y-value is identical.
| Pattern | Interpretation | Possible context |
|---|---|---|
| Positive | As x increases, y tends to increase. | Height and arm span |
| Negative | As x increases, y tends to decrease. | Age of a car and resale value |
| Zero / no clear correlation | No clear upward or downward trend. | Shoe size and a random lottery number |
A stronger straight-line relationship has points closer to a straight line. A weaker relationship has more scatter. A trend describes an overall tendency; it need not hold for every pair of points.
Pause and explainDoes strong negative correlation mean a weak relationship?
No. “Negative” names its direction. It can be very strong when the points lie close to a falling line.
Draw one ruled line of best fit
A line of best fit summarises a roughly linear trend without trying to pass through every point. It is a model for estimating typical outputs, not a claim that all observations lie exactly on it.
Place one ruled line across the data with a roughly balanced scatter on both sides. Read estimates by moving from an input to that line and then to the output axis. Stay aware of the observed range: extending the model far beyond it relies on an untested assumption.
Draw a single straight line by eye across the full spread of the data. Aim for a roughly even distribution of points on either side along its whole length. It need not pass through any plotted point or through the origin.
Use the positive-correlation line in the explorer to estimate y when x = 6.5.
Move vertically from 6.5 to the line, then horizontally to the y-axis. The illustrative line gives y ≈ 6.1.
Identify any outlier and check it, but do not automatically delete it. An outlier may be a recording error or a genuine unusual observation.
Pause and explainWhy is joining the first and last points not necessarily a good best-fit method?
Those two points may not represent the overall pattern. The line should account for the distribution of all the observations.
Your turn. Describe and estimate carefully.
Plot each pair as a clear cross, describe direction and strength separately, and avoid joining the data points.
When estimating, use the fitted line rather than the nearest observation unless asked otherwise. State when a prediction is outside the observed range and avoid treating correlation as proof of cause and effect.
As the age of a used car increases, its price tends to decrease. Name the correlation.
Reveal worked answer
Must a line of best fit pass through the origin or every plotted point?
Reveal worked answer
Data cover x values from 2 to 12. Is predicting y at x = 25 interpolation or extrapolation?
Reveal worked answer
A scatter diagram has points very close to an upward-sloping straight line. Describe the correlation.