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Two classes both have a mean test score of 65%. Class A scored between 60% and 70%. Class B scored between 20% and 100%. Which class had more consistent results — and how could you express this difference with a single number?
Two data sets can have the same mean but behave completely differently. The range tells you how spread out the data is — the difference between the largest and smallest values.
Range only uses two values. A single outlier completely controls the range — it becomes the maximum or minimum. This makes the range an unreliable measure of spread when outliers are present.
Example: Scores 50, 52, 54, 56, 58 have a range of 8. Add one outlier of 2 and the range becomes 56 — even though 9 of the 10 values are still tightly clustered between 50 and 58.
Always identify outliers before reporting the range, and state whether they are included.
$$\text{Range} = \text{Maximum} - \text{Minimum}$$
Example 1: Data: 45, 52, 58, 61, 67, 73, 95.
Maximum = 95. Minimum = 45. Range = 95 − 45 = 50.
Example 2 — Comparing two classes:
Class A is far more consistent — all students performed similarly. Class B has much greater spread — performance varied widely. Both classes have the same mean (65), but the range reveals this crucial difference.
Large range → data is spread out → results are inconsistent → harder to describe with a "typical" value.
Small range → data is clustered together → results are consistent → the mean/median is a reliable summary.
Limitations of the range:
An outlier is a value that stands apart from the main cluster of data — much higher or lower than the others.
How to identify an outlier:
Example: Sprint times: 11.2, 11.4, 11.5, 11.6, 11.8, 12.0, 15.3 seconds. The value 15.3 s is an outlier — it is nearly 3 seconds slower than the next value (12.0 s) and is isolated from the cluster of 11.2–12.0 s.
Using sprint times: 11.2, 11.4, 11.5, 11.6, 11.8, 12.0, 15.3 (n = 7).
| Statistic | With 15.3 | Without 15.3 | Effect of outlier |
|---|---|---|---|
| Mean | 11.97 s | 11.58 s | Increased by 0.39 s — strongly affected |
| Median | 11.6 s | 11.5 s | Changed by 0.1 s — barely affected |
| Range | 4.1 s | 0.8 s | Increased by 3.3 s — heavily affected |
Summary: The mean and range are heavily affected by outliers. The median is resistant. Always check for outliers before reporting statistics, and state whether they were included or excluded.
$$\text{Range} = \text{Maximum} - \text{Minimum}$$
Range measures spread (consistency): small range = consistent; large range = spread out.
Outlier = value much larger or smaller than the rest. Outliers heavily affect the mean and range but barely affect the median.
Always state whether outliers are included when reporting statistics, and justify any exclusion.
Calculate the range of: 25, 13, 48, 31, 22, 40, 17.
Set A: 45, 60, 70, 80, 95. Set B: 60, 63, 66, 68, 70. Which data set has the larger range?
Identify the outlier in: 47, 50, 2, 53, 51, 48, 55.
A data set has an outlier that is also the maximum value. When this outlier is removed, which statistic changes the most?
Which statement about the range is correct?
Q6. Two athletes' long jump distances (metres) over 8 attempts:
Athlete A: 6.1, 6.2, 6.2, 6.3, 6.3, 6.4, 6.4, 6.5
Athlete B: 5.2, 5.8, 6.1, 6.4, 6.5, 6.7, 7.0, 7.3
(a) Calculate the range for each athlete. (b) Which athlete is more consistent? Justify your answer using the range.
Q7. A data set: 52, 55, 57, 58, 60, 62, 98. (a) Identify the outlier. (b) Calculate the mean and median with the outlier included. (c) Calculate the mean and median without the outlier. (d) Compare your results — which statistic changes more, and why?
Q8. A sports coach records 100m sprint times (seconds) for 7 athletes: 11.2, 11.4, 11.5, 11.6, 11.8, 12.0, 15.3. (a) Identify the outlier. (b) Calculate the range with and without the outlier. (c) The coach suspects the 15.3 s was due to the athlete tripping. Should this time be included or excluded from the data? Justify your decision.
(a) Athlete A: range = 6.5 − 6.1 = 0.4 m. Athlete B: range = 7.3 − 5.2 = 2.1 m.
(b) Athlete A is more consistent — range of only 0.4 m shows their jumps are very close together, varying by less than half a metre. Athlete B's range of 2.1 m shows their performance varies much more widely from attempt to attempt.
(a) Outlier = 98 (isolated far above the cluster of 52–62).
(b) With 98: sum = 52+55+57+58+60+62+98 = 442. Mean = 442÷7 ≈ 63.1. Median = 4th value = 58.
(c) Without 98: sum = 442−98 = 344. n = 6. Mean = 344÷6 ≈ 57.3. Median = average of 3rd and 4th = (57+58)÷2 = 57.5.
(d) The mean changes far more (from 63.1 to 57.3 — a difference of 5.8). The median barely changes (from 58 to 57.5 — only 0.5). This confirms that the mean is strongly affected by the outlier, while the median is resistant.
(a) Outlier = 15.3 s — isolated nearly 3 seconds above the next value (12.0 s).
(b) With 15.3: range = 15.3 − 11.2 = 4.1 s. Without 15.3: range = 12.0 − 11.2 = 0.8 s.
(c) The 15.3 s time should be excluded from the analysis of typical performance, because it was caused by an external event (tripping) rather than the athlete's actual sprint ability. It should still be recorded in the raw data but labelled as an anomaly. Reporting with it included gives a misleading range of 4.1 s — it looks like the group is highly inconsistent, when in fact 6 of 7 athletes ran within a 0.8 s range.
Two factories produce bolts with a target diameter of 10 mm.
(a) A bolt is rejected if its diameter is more than 0.5 mm from the target. How many of Factory B's bolts would be rejected?
(b) Should Factory B include or exclude the outliers in their quality report? Argue both sides.
(c) What does the range tell a manufacturer about the reliability of each factory?