Scatter diagrams

What a scatter diagram shows

A scatter diagram shows paired data: two measurements taken from each person or object, such as the height and the mass of each of 10 adults. Each person becomes one point, with one measurement across and the other up.

The point of a scatter diagram is to see whether the two measurements are related. Do taller people tend to be heavier? Do older laptops tend to have shorter battery lives? One pair of values tells you nothing, but twenty points together show a pattern, or show that there is none.

Usually the variable you think might influence the other goes across (height, age, temperature) and the one that might respond goes up (mass, battery life, sales). Either way round is acceptable unless the question fixes the axes.

Drawing a scatter diagram

To plot a scatter diagram:

1. Choose a scale on each axis that covers the data. The axes do not have to start at 0; a jagged break in the axis is not needed in the exam if the grid is printed for you.
2. Work out what one small square is worth on each axis before you plot anything. On a scale where 10 big units span 5 large squares with 5 small squares each, one small square is $10÷25=0.4$, not 1.
3. Plot each pair as one point, across then up, like coordinates.
4. Mark each point clearly, for example as a small cross ($\times$). The centre of the cross is the point, so a cross is more accurate than a large dot.

Do not join the points. A scatter diagram is a set of separate points; joining them suggests a pattern from one person to the next that does not exist. Check at the end that you have plotted as many points as there are pairs in the table.

Positive, negative and zero correlation

Correlation describes the relationship shown by a scatter diagram.

• Positive correlation: as one variable increases, the other tends to increase. The points slope upwards from left to right. Taller adults tend to be heavier.
• Negative correlation: as one variable increases, the other tends to decrease. The points slope downwards from left to right. Older laptops tend to have shorter battery lives.
• Zero correlation (no correlation): there is no pattern. Knowing one value tells you nothing about the other. The day of the month you were born has nothing to do with your height.

Correlation describes a tendency, not a rule. In a positive correlation, an individual pair can go against the trend: a shorter person can be heavier than a taller one. The pattern is in the points as a whole.

You can also describe how strong the correlation is: strong when the points lie close to a straight line, weak when they are loosely scattered around one. An exam answer of "positive" is enough where the question asks only for the type; "strong positive" is also correct.

When asked to interpret the correlation, say what it means in context: "the taller the adult, the heavier they tend to be", not just "positive".

Positive correlation: the points rise from left to right.

Negative correlation: the points fall from left to right.

Zero correlation: no pattern at all.

Correlation is not cause

A correlation shows that two things tend to change together. It does not show that one causes the other.

Ice cream sales and the number of people treated for sunburn show positive correlation, but ice cream does not cause sunburn. Both go up on hot, sunny days: a third factor, the weather, drives both. Sometimes a correlation is pure coincidence.

In an exam, if a statement claims that one variable causes the other, the safe answer is that the scatter diagram only shows a relationship, and that something else may explain it.

The line of best fit

When a scatter diagram shows positive or negative correlation, you can draw a line of best fit: a straight line that shows the trend of the points. The syllabus sets out exactly what a good one looks like. A line of best fit:

• should be a single ruled line drawn by inspection (by eye), not a curve, and not a zigzag joining the points;
• should extend across the full data set, from the leftmost points to the rightmost, not stop part of the way;
• does not need to pass through any of the points, but there should be a roughly even distribution of points either side of the line over its entire length.

That last rule is the one most often broken. Joining the first and last points, or two points that happen to line up, rarely balances the rest. Lay your ruler along the cloud of points and turn it until about as many points are above it as below it, at the left end, in the middle and at the right end.

A useful check: the line of best fit should pass through, or very near, the mean point $(\bar{x},\bar{y})$, the mean of the across values and the mean of the up values. If a question asks you to plot the mean point, your line must go through it.

Do not draw a line of best fit for zero correlation. There is no trend for it to show.

Using a line of best fit, and when not to trust it

To estimate one value from the other, go from the value you know to the line of best fit, then across (or down) to the other axis, as you would read any graph. Show the lines you use.

Because the line is drawn by eye, every reading is an estimate, and mark schemes accept a range of answers.

An estimate is reliable when:

• the correlation is strong, so the points lie close to the line;
• the value you start from is inside the range of the data. Estimating inside the data is called interpolation.

An estimate is unreliable when you go outside the range of the data, called extrapolation. Nobody knows whether the trend continues beyond the data. The laptop line, carried on to 12 years, predicts a negative battery life, which is impossible. Say this in context when asked: "16 years is outside the range of the data, so the trend may not continue".

Points that do not fit

Sometimes one point lies well away from the pattern of the others. It may be a measuring or recording error, or a genuinely unusual case: a student who was ill on the day of a test, or a house with a swimming pool in a list of prices.

When you are asked to identify such a point, give its coordinates and, if asked, a sensible reason in context. When you draw a line of best fit, ignore a point that clearly does not fit, because one unusual value should not pull the trend line away from all the others.