No gaps between classes
Mode and median need classes that touch each other.
A mark of 19.5
A teacher groups marks in the classes 10–19, 20–29, 30–39, 40–49. Half marks are allowed, and one student scores 19.5.
Look at the end of one class and the start of the next: 19 and 20.
Which class does 19.5 belong to?
What this lesson covers
The idea
Before the formulae for the mode and median of grouped data are applied, the class intervals must be made continuous.
A mark of 19.5
A teacher groups marks in the classes 10–19, 20–29, 30–39, 40–49. Half marks are allowed, and one student scores 19.5.
Look at the end of one class and the start of the next: 19 and 20.
Which class does 19.5 belong to?
- 10–19
- 20–29
- Neither of them
Close the gap
The dial widens every class by the same amount at both ends. The purple dot is the mark 19.5. Turn the dial until there is no gap and no overlap.
Continuous classes
Before the formulae for the mode and the median of grouped data are used, the class intervals must be made continuous: each upper limit equals the next lower limit.
The formulae use the lower limit l, the class size h, and the totals of the classes before. They assume the classes touch. If there is a gap (here from 19 to 20), move the limits so that it closes: each upper limit goes up, and each lower limit goes down, by half the gap. Here the gap is 1, so we move by 0.5, and 10–19, 20–29 become 9.5–19.5, 19.5–29.5. The class size becomes 10, not 9.
Notes
Before using the mode or median formula, make the class intervals continuous: each upper limit equals the next lower limit.
Check yourself
Which set of classes is continuous?
The classes 10–19, 20–29, 30–39 are made continuous. What is the new lower limit of the class 20–29?
Answer: 19.5
The gap is 20 − 19 = 1, and half of it is 0.5. So 20 − 0.5 = 19.5 (and the upper limit of 10–19 becomes 19 + 0.5 = 19.5).
After the classes 10–19, 20–29, 30–39 are made continuous, what is the class size h of the class 30–39?
Answer: 10
The new class is 29.5–39.5, so h = 39.5 − 29.5 = 10.
The classes 1–4 and 6–9 have a gap of 2 (from 4 to 6). What is the new upper limit of 1–4 after making them continuous?
Answer: 5
The gap is 6 − 4 = 2, and half of it is 1. The upper limit of 1–4 becomes 4 + 1 = 5, and the lower limit of 6–9 becomes 6 − 1 = 5.
- 0–10, 10–20, 20–30 — correct. Yes! Each upper limit (10, 20) is the next lower limit.
- 0–9, 10–19, 20–29. There are gaps between 9 and 10, and between 19 and 20.
- 0–10, 11–20, 21–30. There are gaps between 10 and 11, and between 20 and 21.