Comparing Data Sets
Comparing data sets means looking at their centres and their spreads side by side, then deciding what the difference means in context. This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑DAT‑C‑01.
You will compare means, medians, modes and ranges of two sets, read a smaller range as greater consistency, as for two batters with the same average, and spot sets with the same centre but very different spreads. You will find means from totals and from frequency tables, compare cities' temperatures and tests with dot plots, stem-and-leaf plots and histograms, use the mean and median to compare shapes, and choose between two bus companies or two bottling machines when the centre and spread point different ways. It is part of Year 9 Core Investigating Data.
Every question with a fully worked solution.
- Comparing Data Sets - Video - Comparing Data Sets Using Plots Watch
Theory
Comparing data sets means comparing three things: centre, spread and shape. This Year 9 maths page shows how to compare means or medians, use the range to judge which set is more consistent, compare shapes using each set's mean and median, and make a decision in context when the centre and the spread point different ways.
To compare two data sets, look at three things in order.
Centre says which set is higher overall. Compare the two means, or the two medians. Never compare one set's mean with the other set's median.
Spread says which set is more consistent. Compare the ranges. A smaller range means the scores are closer together and more predictable. The interquartile range (IQR) also measures spread, and one extreme score disturbs it far less than the range. The standard deviation, found on a calculator, is another measure of spread.
Shape says which way each set leans. Compare each set's own mean with its own median.
Two sets can have the same mean and be nothing alike. That is when the spread decides the question. Parallel displays, such as two dot plots drawn on the same scale, let you compare all three by eye.
| Compare | Using | It tells you |
|---|---|---|
| centre | the two means, or the two medians | which set is higher overall |
| spread | the two ranges (or IQRs) | which set is more consistent |
| shape | each set's mean against its own median | which way each set leans |
How to compare two data sets
- Find the same summary statistics for both sets: mean or median, and range.
- Compare the centres. Say which set is higher and by how much.
- Compare the spreads. Say which set is more consistent and quote both ranges.
- Compare the shapes if you can, using each set's mean and median.
- Conclude in context, with both parts: “Set \(B\) scores higher on average, but set \(A\) is more consistent.”
Find each mean, then subtract.
| \(\text{mean of } A\) | \(=\) | \(\dfrac{5+8+11+12}{4}\) |
| \(\) | \(=\) | \(\dfrac{36}{4} = 9\) |
| \(\text{mean of } B\) | \(=\) | \(\dfrac{7+9+13+15}{4}\) |
| \(\) | \(=\) | \(\dfrac{44}{4} = 11\) |
| \(\text{difference}\) | \(=\) | \(11 - 9 = \mathbf{2}\) |
\(B\)'s mean is \(2\) higher.
Compare the centres first, then the spreads.
| \(\text{mean } X\) | \(=\) | \(\dfrac{35}{5} = 7\) |
| \(\text{mean } Y\) | \(=\) | \(\dfrac{35}{5} = 7\) |
| \(\text{range } X\) | \(=\) | \(9 - 3 = 6\) |
| \(\text{range } Y\) | \(=\) | \(8 - 6 = 2\) |
The means are equal, so the spread decides. \(Y\) is far more consistent, and \(Y\)'s lowest score is \(6\), while \(X\) once scored \(3\). The coach should pick archer \(Y\).
Compare like with like: median with median, range with range.
| \(\text{median } C\) | \(=\) | \(26\) |
| \(\text{median } D\) | \(=\) | \(31\) |
| \(\text{range } C\) | \(=\) | \(33 - 21 = 12\) |
| \(\text{range } D\) | \(=\) | \(40 - 19 = 21\) |
\(D\)'s median is \(5\) higher, so \(D\) scores higher. But \(D\)'s range is \(9\) larger, so \(C\) is the more consistent set. A full answer gives both parts.
Compare each set's own mean with its own median.
| \(E\) | \(:\) | \(30 = 30\) |
| \(F\) | \(:\) | \(41 > 34\) |
\(E\)'s mean equals its median, so \(E\) is symmetrical. \(F\)'s mean is above its median, so a tail of high values has dragged it up: \(F\) is positively skewed.
The medians, \(30\) and \(34\), are the fairer comparison of centre here, because \(F\)'s mean is pulled by its tail.
Common pitfalls
Frequently asked questions
How do you compare two sets of data?
Compare their centres using the means or medians, their spreads using the ranges, and their shapes using each set's mean and median. Then write a conclusion that gives both the centre and the spread, in context.
What does a smaller range mean?
The scores are closer together, so the data are more consistent and more predictable. It does not mean the scores are lower.
Can two data sets have the same mean but be different?
Yes. \(7, 8, 8, 9, 9, 9, 10, 10, 11\) and \(3, 5, 7, 8, 9, 10, 11, 13, 15\) both have a mean of \(9\), but the second set has three times the range.
Should I compare the mean or the median?
Either, as long as you compare the same measure for both sets. If one set is skewed or has an outlier, the medians give the fairer comparison.
How do you compare groups of different sizes?
Compare their means, not their totals. A bigger group can have a bigger total and still have a lower mean.