Averages and Range: Comparing Distributions
Section: Statistics | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
The Four Measures Recap
- Mode: the most frequently occurring value.
- Median: the middle value when data is ordered.
- Mean: the sum of all values ÷ the number of values.
- Range: highest value − lowest value (a measure of spread, not an average).
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Question: Find the mode, median, mean, and range of: 4, 7, 4, 9, 6, 4, 8.
- Step 1: Order the data: 4, 4, 4, 6, 7, 8, 9
- Mode: 4 (appears 3 times)
- Median: the 4th of 7 values = 6
- Mean: (4+4+4+6+7+8+9) ÷ 7 = 42 ÷ 7 = 6
- Range: 9 − 4 = 5
Comparing Two Distributions Using Averages and Range
To compare two data sets fairly, use an average (mean, median, or mode) together with the range - a higher average alone doesn't show how consistent the data is.
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Question: Two classes take the same test. Class A: mean 68, range 20. Class B: mean 72, range 8. Compare the two classes.
- Class B has a higher mean, so on average Class B performed better.
- Class B has a smaller range, so Class B's scores were more consistent (less spread out) than Class A's.
Estimating the Mean from Grouped Data
For grouped data, exact values aren't known, so the mean is estimated using each class's midpoint: estimated mean = Σ(midpoint × frequency) ÷ Σfrequency.
Estimating the mean using each class's midpoint
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Question: Homework hours: 0-2 (freq 5), 2-4 (freq 8), 4-6 (freq 4), 6-8 (freq 3). Estimate the mean.
- Step 1: Midpoints: 1, 3, 5, 7
- Step 2: Σ(midpoint × freq) = 1×5 + 3×8 + 5×4 + 7×3 = 5 + 24 + 20 + 21 = 70
- Step 3: Σfreq = 5 + 8 + 4 + 3 = 20
- Step 4: Estimated mean = 70 ÷ 20
- Answer: 3.5 hours
Cambridge exams often give you a table with "midpoint" and "midpoint × frequency" columns to fill in directly - lay your working out the same way, since it's exactly what a marker expects to see:
| Hours | Frequency | Midpoint | Midpoint × Frequency |
|---|---|---|---|
| 0-2 | 5 | 1 | 5 |
| 2-4 | 8 | 3 | 24 |
| 4-6 | 4 | 5 | 20 |
| 6-8 | 3 | 7 | 21 |
| Total | 20 | 70 |
Estimated mean = 70 ÷ 20 = 3.5 hours
Finding the Modal Class and Median Class
The modal class is the class interval with the highest frequency. The median class is the class interval containing the middle value, found using cumulative frequency.
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Question: Using the homework table above (total 20), find the modal class and the median class.
- Modal class: 2-4 has the highest frequency (8)
- Cumulative frequencies: 0-2 → 5, 2-4 → 13, 4-6 → 17, 6-8 → 20
- Median position: the 10th value (half of 20) falls within the 2-4 class (cumulative frequency reaches 13 by its end, having started at 5)
- Answer: Modal class = 2-4, Median class = 2-4
Comparing Grouped Distributions
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Question: Company X's delivery times: estimated mean 25 minutes, modal class 20-30. Company Y's delivery times: estimated mean 32 minutes, modal class 30-40. Compare the two companies.
- Company X has a lower estimated mean, so on average deliveries are faster.
- Company X's modal class (20-30) is also lower than Company Y's (30-40), consistent with typically faster deliveries.
Real-World Applications
Comparing distributions is used across many fields:
- Sports statistics: comparing players' average performance and consistency.
- Business: comparing sales distributions between stores or regions.
- Education: comparing test score distributions between classes or schools.
- Weather: comparing temperature averages and variability between cities.
- Quality control: comparing measurement consistency between production lines.
Exam Tips
Common Mistakes
MistakeUsing the class boundaries instead of midpoints when estimating a grouped mean
Fixfor grouped data, always use the MIDPOINT of each class, since exact values aren't known
MistakeForgetting the mean of grouped data is an ESTIMATE, not an exact value
Fixstate your answer as an "estimated mean," since it assumes values are evenly spread within each class
MistakeConfusing the modal class with the class containing the median
Fixthe modal class has the highest FREQUENCY; the median class contains the MIDDLE value
MistakeComparing only the mean (or only the range) between two distributions
Fixcompare an average AND a measure of spread together for a fuller picture
MistakeDividing by the number of classes instead of the total frequency when estimating a grouped mean
Fixalways divide by the TOTAL FREQUENCY (Σf), not the number of class intervals
For Exams
- Learn all four measures: mode (most common), median (middle), mean (average), range (spread).
- For grouped data: estimate the mean using Σ(midpoint × frequency) ÷ Σfrequency.
- Use cumulative frequency to find the median class.
- When comparing distributions, always discuss BOTH an average and the range.
- State grouped-data means as "estimates," since exact original values are unknown.
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