Comparing Data Sets
Comparing Data Sets in NSW Year 10 Core maths brings together the tools for comparing two groups (NESA outcome MA5‑DAT‑C‑01): the mean, median, range, interquartile range and standard deviation, along with box plots and back-to-back stem-and-leaf plots.
You will find and compare summary statistics, use a calculator to find a standard deviation, and decide which measure of centre suits data with an outlier, such as pocket money that includes a birthday week. You will also compare golf scores, two cricket batters, bakeries with the same mean but a different spread, and two classes' test results, and explain what a larger standard deviation tells you. It is part of Year 10 Core Statistics.
Every question with a fully worked solution.
- Comparing Data Sets - Video - Back to Back Stem Plots Watch
Theory
To compare data sets, compare their centre (mean or median), their spread (range, IQR or standard deviation) and their shape. This Year 10 maths page covers back-to-back stem-and-leaf plots, finding the standard deviation on a calculator, how outliers affect each statistic, and which statistics to trust for skewed data.
A fair comparison of two groups looks at the centre (which group is generally higher), the spread (which group is more consistent) and the shape (symmetrical or skewed, outliers).
Measures of centre: the mean (sum \(\div\) number of scores) and the median (middle score). Measures of spread: the range, the interquartile range and the standard deviation.
The standard deviation, written \(\sigma\), measures how far the scores typically are from the mean. A small \(\sigma\) means the scores are bunched close to the mean (consistent); a large \(\sigma\) means they are spread out. Find it with a calculator in statistics mode.
The mean, range and standard deviation use every score, so an outlier or skew can change them a lot. The median and IQR are resistant: they barely move. For symmetrical data compare means and standard deviations; for skewed data or data with outliers, compare medians and IQRs.
A back-to-back stem-and-leaf plot shows two groups on one shared stem, one on each side, so their shapes can be compared directly.
Population standard deviation (what the calculator works out as \(\sigma x\)):
| Data | Centre | Spread |
|---|---|---|
| roughly symmetrical, no outliers | mean | standard deviation |
| skewed or with outliers | median | IQR |
How to compare two data sets
- Organise each set: order it, or read it from the graph or table (left leaves of a back-to-back plot read outwards from the stem).
- Calculate a centre and a spread for each: median and IQR, or mean and standard deviation.
- Describe the shape and check for outliers; switch to median and IQR if the data is skewed.
- Compare in a sentence that names the statistic and quotes both values.
- Conclude in context: which group is higher, which is more consistent.
| Group A leaves | Stem | Group B leaves |
|---|---|---|
| 4 | 5 8 | |
| 5 2 | 5 | 3 7 8 |
| 8 4 3 1 | 6 | 0 2 2 6 |
| 5 2 0 | 7 | 1 3 |
| 6 1 | 8 |
Key: \(8\,|\,6\,|\,0\) means \(68\) in A and \(60\) in B
Group A in order: \(52, 55, 61, 63, 64, \mathbf{68}, 70, 72, 75, 81, 86\). Group B: \(45, 48, 53, 57, 58, \mathbf{60}, 62, 62, 66, 71, 73\). The median is the 6th score; the quartiles are the middles of the 5 scores on each side.
| \(\text{A}\) | \(:\) | \(\text{median } 68,\ \text{IQR } 75 - 61 = 14,\ \text{range } 34\) |
| \(\text{B}\) | \(:\) | \(\text{median } 60,\ \text{IQR } 66 - 53 = 13,\ \text{range } 28\) |
Group A scored higher: its median is 8 points above B's. The spreads are similar (IQRs \(14\) and \(13\)); A's range is a little larger.
Calculator: statistics mode (1-variable), enter the data, read \(\bar{x}\) and \(\sigma x\).
| \(\text{A: } \bar{x}\) | \(=\) | \(\dfrac{30}{5} = \mathbf{6},\quad \sigma = \mathbf{2.0}\) |
| \(\text{B: } \bar{x}\) | \(=\) | \(\dfrac{30}{5} = \mathbf{6},\quad \sigma \approx \mathbf{4.1}\) |
The means are equal, so the typical delay is the same. Route A has the smaller standard deviation, so its delays stay closer to \(6\) minutes: Route A is more reliable.
| Before (8) | After (9) | |
|---|---|---|
| Mean | \(16.25\) | \(\dfrac{175}{9} \approx 19.4\) |
| Median | \(\dfrac{15 + 17}{2} = 16\) | \(17\) (5th) |
| IQR | \(18.5 - 14.5 = 4\) | \(19.5 - 14.5 = 5\) |
| Range | \(20 - 12 = 8\) | \(45 - 12 = 33\) |
| \(\text{mean change}\) | \(=\) | \(19.4 - 16.25 \approx 3.2\) |
| \(\text{range change}\) | \(=\) | \(33 - 8 = 25\) |
The mean and the range jump; the median and IQR move by only \(1\). With an outlier, compare groups using the median and IQR.
(i) Which team is more consistent?
(ii) The Hawks' mean is above their median. What does that suggest?
(iii) Which team scores more in a typical game?
(i) The Jets have the smaller standard deviation and IQR: Jets are more consistent.
| \(\text{Hawks: mean} - \text{median}\) | \(=\) | \(66.0 - 61 = 5 > 0\) |
(ii) A mean well above the median suggests positive skew: a few very high-scoring games pull the Hawks' mean up.
(iii) Because the Hawks' data is skewed, the median is the fairer centre. The Jets' median (\(64\)) is higher than the Hawks' (\(61\)), so in a typical game the Jets score more, even though the Hawks' mean is higher.
Common pitfalls
Frequently asked questions
What does standard deviation tell you?
It tells you how spread out the scores are from the mean. A small standard deviation means the scores are close to the mean; a large one means they are spread out.
How do you compare two sets of data?
Compare a measure of centre, such as the median or mean, and a measure of spread, such as the IQR or standard deviation. Describe the shapes too, and quote the numbers.
Should I use the mean or the median to compare?
Use the mean for roughly symmetrical data without outliers. Use the median when the data is skewed or has outliers, because the median is not pulled by extreme scores.
How do you read a back-to-back stem-and-leaf plot?
The stem is in the middle. Leaves on the right read normally; leaves on the left read outwards from the stem, so a \(6\) stem with an \(8\) leaf on the left is \(68\).
Which measure of spread is affected by outliers?
The range and the standard deviation are affected strongly, because they use the extreme scores. The interquartile range is hardly affected, because it ignores the top and bottom quarters.