Summarising and comparing data sets: QCE Essential Mathematics Unit 4
“Calculate and interpret the mean, median, mode and range of a data set, identify outliers and their effect, display and compare two or more data sets using back-to-back stem plots, dot plots and column graphs, and make decisions supported by the data”
Summarise data with the mean, median and mode (centre) and the range (spread). Outliers distort the mean and range, so use the median for typical values when they are present. Compare data sets with back-to-back stem plots, dot plots or column graphs, discussing both centre and spread, and use the comparison to justify a decision.
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What this dot point is asking
You need to summarise data with measures of centre and spread, recognise when outliers distort those measures, display and compare data sets, and use the comparison to make a decision. Our heading here follows the Unit 4 title; check your school's unit plan for the exact subject matter.
The answer
Measures of centre
- Mean: add all values and divide by the number of values. Affected by outliers.
- Median: the middle value when data is ordered (the average of the two middle values when there is an even number). Not affected much by outliers.
- Mode: the most common value. Useful for categories (most popular shoe size).
Measure of spread
- Range = highest value minus lowest value. Shows how spread out the data is, but depends only on the two extremes.
Outliers
An outlier is a value far from the rest. Check whether it is an error; if it is real, report it. Outliers pull the mean and stretch the range, so the median is usually the better "typical" value when there are outliers.
Displaying and comparing data sets
| Display | Use |
|---|---|
| Back-to-back stem plot | Compare two numerical data sets on a shared stem |
| Parallel dot plots | Compare small data sets on the same scale |
| Side-by-side column graphs | Compare categories or grouped frequencies |
When comparing, always discuss both centre and spread, use the data values, and link the comparison to the decision.
"Group A's median (28 minutes) is similar to Group B's (27 minutes), but A's range (13 minutes) is much smaller than B's (37 minutes), so A is more consistent."
Two netball teams' scores over six games: Team X: 32, 35, 36, 38, 40, 41. Team Y: 20, 30, 42, 44, 45, 47.
- Team X: mean = 222/6 = 37; median = (36 + 38)/2 = 37; range = 9.
- Team Y: mean = 228/6 = 38; median = (42 + 44)/2 = 43; range = 27.
- Comparison: Y has the higher mean and median, so it usually scores more, but its larger range shows it is less consistent (one very low score of 20).
- Decision: for a single important game, a coach wanting predictable performance might value X's consistency; for highest typical scoring, Y is stronger.
Forgetting to order data before finding the median.
Comparing only averages. Spread matters too.
Using the mean when there is an outlier without commenting on its effect.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation4 marksThe weekly pay (in dollars) of eight casual café staff is 220, 250, 250, 280, 300, 310, 340, 1450. Find the mean, median, mode and range.Show worked solution →
Sum = 220 + 250 + 250 + 280 + 300 + 310 + 340 + 1450 = 3400.
Mean = 3400 divided by 8 = $425.
Median = average of the 4th and 5th values = (280 + 300) divided by 2 = $290.
Mode = $250.
Range = 1450 minus 220 = $1230.
Marking guide: 1 mark each.
core3 marksUsing the pay data above, explain which measure best describes the typical weekly pay, and why.Show worked solution →
The median ($290) best describes typical pay. The value 1450 is an outlier (perhaps the manager), which pulls the mean up to $425, higher than seven of the eight staff earn. The median is not affected by the outlier, so it represents the typical casual worker better.
Marking guide: 1 mark for choosing the median, 1 mark for identifying the outlier, 1 mark for explaining its effect on the mean.
exam6 marksTwo delivery companies record delivery times (in minutes) for 10 deliveries each. Company A: 22, 25, 25, 27, 28, 29, 30, 31, 33, 35. Company B: 15, 18, 20, 24, 26, 28, 34, 38, 45, 52. A business wants reliable delivery. Compare the two companies using suitable statistics and recommend one.Show worked solution →
- Company A
- mean = 285/10 = 28.5 min; median = (28 + 29)/2 = 28.5 min; range = 35 minus 22 = 13 min.
- Company B
- mean = 300/10 = 30 min; median = (26 + 28)/2 = 27 min; range = 52 minus 15 = 37 min.
- Comparison
- The typical (median) times are similar: B is slightly faster at 27 minutes compared with 28.5. However, B's times are far more spread out (range 37 minutes against 13), with some deliveries taking 45 and 52 minutes. A is much more consistent.
- Recommendation
- For reliable delivery, choose Company A. Its times are consistent and never exceed 35 minutes, whereas B is unpredictable even though its median is slightly lower.
Marking guide: 2 marks for correct statistics for both companies, 2 marks for comparing centre and spread, 2 marks for a recommendation justified with the data.