POP

E9.5 Scatter Diagrams

Image · 914 × 270

01 — Explanation

How do I describe the correlation shown on a scatter diagram?

Describe it in two parts — direction, then strength:

Type	Pattern	Example
Positive	Both variables increase together	Study time vs. test score
Negative	One increases as the other decreases	Speed vs. journey time
Zero / none	No consistent pattern	Shoe size vs. test score

Points clustered tightly around a clear trend show strong correlation; a visible but loosely scattered trend is weak. A downward-sloping cluster is still negative correlation, not "no correlation" — only a genuinely random scatter with no visible trend qualifies as that.

Example: points rising from bottom-left to top-right in a tight pattern show a strong positive correlation.

02 — Explanation

What makes a good line of best fit?

A single ruled straight line, never a curve or a path connecting the points themselves.
Passes through the mean point $(\bar{x},\bar{y})$ .
Has a roughly even scatter of points above and below it, along its whole length.
Does not need to touch the origin or any individual plotted point.

Examiner Tip

A line of best fit should never be forced through $(0,0)$ unless the data genuinely supports it — let the trend of the points decide where the line goes.

03 — Explanation

How do I find and use the mean point?

\bar{x} = \frac{\sum x}{n}, \qquad \bar{y} = \frac{\sum y}{n}

Example: study time $x$ and test score $y$ for $10$ students, with $\sum x=50$ , $\sum y=650$ .

\bar{x}=\frac{50}{10}=5, \qquad \bar{y}=\frac{650}{10}=65

Plot $(5,65)$ , then draw a single straight line through it that follows the overall direction of the scattered points, with roughly equal numbers of points above and below along its length.

04 — Explanation

How do I use a line of best fit to predict a value?

Read directly off the line (not off any individual plotted point) at the given $x$ - or $y$ -value. Predictions made within the range of the original data (interpolation) are considered reasonably reliable; predictions made outside that range (extrapolation) are much less trustworthy, since the trend might not continue in the same way beyond the data actually collected.

Worked Example

Using the line of best fit through $(5,65)$ with gradient $6$ , estimate the test score for a student who studied for $8$ hours.

Equation of the line: $y - 65 = 6(x-5) \implies y = 6x+35$ .
Substitute $x=8$ : $y = 6(8)+35 = 83$ .
Since $8$ hours falls within the range of study times already recorded, this is a reasonably reliable interpolated estimate.

Examiner Tip: If a question asks you to predict a value outside the plotted range (e.g. $20$ hours of study, when the data only goes up to $10$ ), you must comment that the estimate is unreliable, since there's no evidence the same linear trend continues that far.

05 — Explanation

Why might two people draw slightly different lines of best fit for the same data?

Because a line of best fit is drawn "by eye" rather than calculated from a fixed formula, small variations between two reasonable lines are expected and both can be accepted — provided each one passes through, or very close to, the mean point, and keeps a roughly even scatter of points above and below it along its entire length. What examiners check for is whether the line's general position and slope reasonably represent the trend, not whether it matches one single "correct" line exactly.