Scatter Graphs and Correlation
Statistics
Drawing one
- Once a diagram has been seen to slope one way or the other, its trend can be summarised by a line of best fit
- The syllabus is precise about what it must look like
- A single ruled straight line, drawn by inspection with a ruler
- It must extend across the full data set, not just the middle
- It need not pass through any of the points, but the points should be roughly evenly distributed on either side along its whole length
- Where one point clearly sits away from the pattern, an outlier, ignore it when placing the line
- Never draw a curve, and never join the points
Using one
- The line lets you estimate one quantity from the other: go up from the known value to the line, then across to the other axis
- Estimating within the range of the data is interpolation, and it is reasonably reliable
- Estimating beyond the data is extrapolation, and it is unreliable because the trend may not continue
- Where a question gives a value outside the plotted range, saying the estimate is unreliable is often the point being tested
Answer "what type of correlation" with one of exactly three phrases — positive, negative or zero correlation. The line of best fit is a single ruled straight line across the whole data set with the points balanced either side: not a curve, not a dot-to-dot, and not stopping in the middle. Ignore an obvious outlier when placing it, and show the lines you drew to take a reading. An estimate from beyond the data is extrapolation and is unreliable.
Reading an estimate from a line of best fit
A scatter diagram plots revision time in hours against test score. The line of best fit passes through (2, 38) and (8, 74). Estimate the score for a student who revises for 5 hours, and interpret the gradient.
Solution:
- The line rises from 38 to 74 as the hours go from 2 to 8
- Change in score = 74 − 38 = 36, over a change of 8 − 2 = 6 hours
- Gradient = 36 ÷ 6 = 6
- Five hours is 3 hours beyond the first point, so the score rises by 3 × 6 = 18
- Estimated score = 38 + 18 = 56
- The gradient means that each extra hour of revision is associated with about 6 more marks
- Five hours lies inside the plotted range, so this is interpolation and reasonably reliable
Naming and describing the correlation
A scatter diagram plots the age of a car in years against its value in pounds. The points run from high on the left down to low on the right. State the type of correlation and describe the relationship.
Solution:
- The points fall as you read from left to right, so a rise in one measurement goes with a drop in the other
- The type of correlation is negative
- To describe the relationship, name both quantities and say which way each moves
- As the age of the car increases, its value decreases
- Naming the quantities is what separates a describe answer from the one-word type answer