Scatter Diagrams

What a scatter diagram shows

A scatter diagram plots paired data: two measurements taken from the same item, one on the horizontal axis and one on the vertical. Each item becomes one point. If you measure the height and the shoe size of every student in a class, each student is a single point on the diagram, not two separate values.

The horizontal axis normally holds the variable that does the explaining, such as the number of hours of revision. The vertical axis holds the variable that responds, such as the test mark. Plot each item as a small cross rather than a blob - the syllabus says so, and a cross is easier to read and easier to line up with the axes.

A scatter diagram does a job no other diagram in this section does. A bar chart shows how many items fall in each category; a scatter diagram shows whether two measurements move together, which is a question about a relationship rather than a count.

• One point per item, plotted as a small cross
• Horizontal axis: the variable that does the explaining
• Vertical axis: the variable that responds
• Label both axes and give the scale, or the diagram cannot be read

Correlation: positive, negative or none

Correlation describes the pattern the points make. Positive correlation means that as one variable increases the other tends to increase too, so the points run up from left to right. Negative correlation means that as one variable increases the other tends to decrease, so the points run down from left to right. Zero correlation, also called no correlation, means there is no pattern at all: the points are scattered with no upward or downward trend.

The word 'tends' is doing real work in those definitions. Correlation is about the general pattern, not about every single point. In a strongly positive set of data one or two points can still sit below the rest, and the correlation is still positive.

Describe correlation with the words a mark scheme uses. 'The points go up' describes the picture but does not use the statistical term, and it is the term that earns the mark.

Strong and weak correlation

Once you have the type, describe the strength. If the points lie close to a straight line the correlation is strong; if they are widely scattered but still show a general trend, it is weak. A full description uses both words: 'strong positive correlation', 'weak negative correlation'.

Strength and type are separate decisions. You can have strong positive, weak positive, strong negative and weak negative data, and a question that asks you to describe the correlation fully is looking for both words.

The extreme case is worth knowing. If every point lies exactly on a straight line the correlation is perfect, and the line of best fit passes through every point. That almost never happens with real data, and Cambridge does not ask you to calculate a correlation coefficient at this level.

• Strong: the points lie close to a straight line
• Weak: a general trend, but the points are widely scattered around it
• Give both words: type first, then strength

The line of best fit

A line of best fit is a single straight line drawn by eye through the middle of the points. The syllabus states its three properties, and they are worth learning almost word for word. It must be a single ruled line, drawn by inspection; it must extend across the full data set, from the smallest x-value to the largest; and it does not need to coincide exactly with any of the points, but there should be a roughly even distribution of points either side of it over its entire length.

'Drawn by inspection' means you judge it by eye rather than by calculation, and 'ruled' means you use a ruler. A freehand curve, or a line that stops halfway across the diagram, breaks one of the stated rules and loses the mark.

A line that joins the lowest point to the highest point is not a line of best fit. It is tilted by wherever those two points happen to fall, and it usually leaves most of the other points on one side. The quick check is to count: roughly half the points above your line and half below, spread along its whole length.

• One straight line, ruled, drawn by eye
• It must reach across the full set of points, from the first x-value to the last
• It does not have to pass through any of the points
• The points should be roughly evenly spread above and below it over its whole length

Using the line to estimate a value

The line of best fit turns the diagram into a tool. To estimate a value, start from the given value on one axis, go straight across or straight up to your line, then straight across to the other axis. The answer is a reading from the line, not from the nearest plotted point.

If the value you are asked about lies between the plotted points you are interpolating, and the estimate is reliable, because your line was drawn through real data on both sides of it. If the value lies beyond the ends of the data you are extrapolating, and the estimate is much less reliable: nothing in the data tells you that the relationship carries on that way.

Show the construction lines on the diagram. The reading is shown by the working, and an answer with no lines drawn is hard to credit even when the number happens to be right.

• Go from the axis to the line, then from the line to the other axis
• Read from the line, not from the nearest plotted point
• Interpolation, inside the data, is reliable; extrapolation, outside it, is not
• Draw the construction lines so the reading can be seen

Correlation is not cause

Two variables can be strongly correlated without one causing the other. There are three reasons worth knowing, and a question that asks for 'a reason' is looking for one of them.

First, a third factor may be driving both. Sales of ice cream and the number of people swimming at a beach both rise in hot weather, but buying an ice cream does not cause anybody to swim. Second, the causation may run the other way round from the one you assumed. Third, the pattern may be coincidence, which is a real risk when the sample is small.

The syllabus also asks you to appreciate the restrictions on drawing conclusions from data. A scatter diagram of twenty items tells you about those twenty items. Before you claim anything about a whole population, ask how the sample was chosen and how large it was.

• A third, unmeasured factor may affect both variables
• The causation may run in the opposite direction
• The pattern may be coincidence, especially in a small sample
• Correlation describes a relationship; it does not explain it