0580 · 9.5Scatter diagrams

Paired data and plotting

0:00Students who spend longer revising might achieve higher test marks, but the results will not usually follow an exact rule. A scatter diagram helps us see whether there is an overall relationship. It also shows how much the individual results vary. By the end, you will be able to plot paired data, describe correlation, and use a straight line of best fit. The essential starting idea is paired data. Each pair contains two measurements for the same person or object.

0:25For a student, those measurements might be revision time and test mark. Keeping the measurements together matters, because mixing one student's revision time with another student's mark would create a false relationship. A scatter diagram represents each pair as one point on a coordinate grid. The horizontal position shows one measurement, and the vertical position shows the other. This point represents three hours of revision and a mark of fifty-three.

0:51It does not represent two separate students, because both measurements belong to the same student. Before plotting, read the axis labels and check the scales. Equal distances along an axis must represent equal numerical increases. The two axes can use different scales, and they do not always need to start at zero. Mark points clearly as small crosses so that their exact positions remain visible.

Describing correlation

1:14Our first worked example uses these eight illustrative pairs of revision times and test marks. The question asks us to draw a scatter diagram and describe the correlation. Put revision time on the horizontal axis and test mark on the vertical axis. Then plot each row as one point, without changing which measurements belong together. For student P, move to one hour horizontally and then thirty-eight marks vertically. Place student Q at two hours and forty-seven marks.

1:41The third pair gives a point with x equal to three and y equal to fifty-three. Continue through the remaining rows using exactly the same method. The points generally rise as we look from left to right. This is positive correlation, meaning that as one variable increases, the other tends to increase. Here, longer revision times tend to be associated with higher marks. The word tends matters, because the pattern does not have to hold for every individual pair.

2:09Notice that the student revising for four hours scored less than the student revising for three hours. That does not remove the positive correlation across the whole diagram. Correlation describes the overall pattern rather than an exact rule for every point. Individual differences are precisely why the points are scattered. Negative correlation means that as one variable increases, the other tends to decrease. In this illustrative vehicle example, older vehicles tend to have lower values.

2:35The points generally slope downwards from left to right. Negative describes the direction of the relationship, not whether the measurements themselves are negative numbers. Zero correlation means there is no overall tendency for one variable to increase or decrease as the other increases. These points show no consistent upward or downward trend. It does not mean that either variable has a value of zero. It also does not mean that the diagram must contain equal numbers of high and low points.

3:02The closeness of the points to a straight-line pattern helps us judge how strong a correlation is. Points clustered closely around that pattern suggest stronger correlation. More widely scattered points suggest weaker correlation. A negative correlation can be strong, because strength and direction describe different features. Correlation alone does not prove that one variable causes the other to change. Revision time and marks could also be connected with prior understanding or the quality of revision.

3:30A scatter diagram shows an association in the data, not a complete explanation for it. Keep that distinction when interpreting a real situation.

Lines of best fit

3:37When the points show an approximately straight-line trend, we can draw a straight line of best fit. This line summarises the overall relationship rather than following every observation. Think of it as running through the middle of the scattered points. It allows us to estimate a typical value for one variable from the other. Draw the line by eye, using a ruler to make one straight line. Extend it across the full data set, from the smallest to the largest horizontal value.

4:03Aim for a roughly even distribution of points on either side over its entire length. The line does not need to pass exactly through any plotted point. Check the balance along the whole line, not just the total number of points on each side. A line could have equal numbers above and below but still miss the trend in different sections. Its position and slope must both represent the point cloud sensibly.

4:25Slightly different lines can be reasonable because this is a judgement by inspection. Do not join consecutive crosses with a zigzag line. That would suggest a sequence between observations that the scatter diagram does not establish. Also, do not simply connect the first and last plotted points. Those two observations might lie away from the centre of the overall pattern. Our second worked example asks for the estimated mark after five hours of revision.

4:50Start at five on the horizontal axis and move vertically to the line of best fit. From that intersection, move horizontally to the mark axis. The reading is about sixty marks, so that is our estimate. The plotted student at five hours actually scored fifty-eight, but the question asks us to use the line of best fit. The line gives the overall estimate of sixty rather than repeating that particular observation.

5:14To estimate the revision time associated with seventy marks, begin on the vertical axis instead. Read across to the line and then down to obtain about seven hours. These answers are estimates, not guaranteed outcomes for an individual student. Estimating inside the range of the data is called interpolation. Both of our estimates are interpolation, because five hours and seventy marks lie inside the plotted data. Interpolation is fairly reliable, because the data show how the two quantities behave there.

5:44Predicting outside the range of the data is called extrapolation, and it is much less reliable. The trend might not continue beyond the plotted points. Our line rises five marks for each extra hour and reaches seventy-five marks at eight hours. Continuing it to fifteen hours would give one hundred and ten marks, which is impossible on a test out of one hundred. Our third worked example asks us to describe the correlation and estimate the value of a vehicle aged three point five years.

6:09The correlation is negative because value tends to decrease as age increases. Locate three point five on the age axis and move vertically to the drawn line. Reading across gives eleven on the value axis. The vertical axis measures thousands of pounds, so eleven represents eleven thousand pounds. Multiplying eleven by one thousand gives eleven thousand. Our estimated vehicle value is therefore eleven thousand pounds. Reading only the number eleven without checking the axis unit would give the wrong interpretation.

Exam technique and recap

6:40In an examination, you may be asked to draw or complete a scatter diagram, state the correlation, or draw and use a straight line of best fit. For a correlation answer, state positive, negative or zero as appropriate. If asked to interpret it, describe the tendency using the actual quantities. For example, as vehicle age increases, its value tends to decrease. These graphical methods must work without a calculator on Paper Two and can also appear on calculator Paper Four.

7:09Use clear crosses, a ruler, and careful readings from the given scales. Give estimates to a sensible precision supported by the grid or the question. Writing extra decimal places does not make a graphical estimate more accurate. The central idea is that each point keeps two related measurements together. The overall pattern reveals positive, negative or zero correlation, while the scatter shows individual variation. A single ruled line of best fit summarises an approximately straight-line relationship and supports estimates.

7:38Read the labels, follow the line carefully, and remember that a tendency is not a guarantee.