Close the gaps between classes
Show on a big screenA leaf is 17.4 mm long. The classes are 11-17 and 18-24. Which class holds it? Neither.
- The classes are 11-17, 18-24 and 25-31. Whole numbers are written.
- Between 17 and 18 is a gap of 1. Half of it is 0.5.
- Move each lower limit down by 0.5. Move each upper limit up by 0.5.
- The classes are 10.5-17.5, 17.5-24.5 and 24.5-31.5. They touch.
The rule
Half the gap goes to each side. Lower limits go down. Upper limits go up.
A class has two ends, a lower limit and an upper limit. Limit and bound mean the same. The leaf is 17.4 mm. It lies between 17 and 18, in the gap. These classes have a gap of 1 between them. After closing it, 17.4 falls in 10.5-17.5.
The median formula needs classes that touch. Closing the gaps does not change any frequency. It changes the limits and the width.
| Class before | Class after | Frequency |
|---|---|---|
| 11-17 | 10.5-17.5 | 6 |
| 18-24 | 17.5-24.5 | 10 |
| 25-31 | 24.5-31.5 | 4 |
Example. Find the median of these 20 readings. Half of n is 10. The totals are 6, 16 and 20. The first total to reach 10 is 16. So the median class is 17.5-24.5.
Here l = 17.5, cf = 6, f = 10. The new width is h = 24.5 − 17.5 = 7. Median = 17.5 + ((10 − 6) ÷ 10) × 7.
Step by step: 10 − 6 = 4. Then 4 ÷ 10 = 0.4. Then 0.4 × 7 = 2.8.
That is 17.5 + 2.8 = 20.3.
The gap can be another size. Classes 1.0-1.9 and 2.0-2.9 have a gap of 0.1. Half is 0.05. They become 0.95-1.95 and 1.95-2.95.
Now try it. Slide until the classes touch.
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Try the lab to finish this lesson.