Statistics with Grouped Data
Senior 2 · Mathematics
Statistics with Grouped Data
Lesson 2 of 5
18 min
The head teacher's real question is not "how many students take 25 to 34 minutes?" but "how many students take less than 25 minutes?" or "how many would arrive late if they left home 35 minutes before the bell?" Questions like these are about running totals, and a cumulative frequency column answers them instantly.
A frequency distribution table lists each class with its frequency (the number of values in it). Useful extra columns are the midpoint, the relative frequency (frequency ÷ total, the fraction of the data in that class) and the cumulative frequency (the running total of the frequencies).
| Time (min) | Midpoint | Frequency f | Relative frequency | Cumulative frequency |
|---|---|---|---|---|
| 5–14 | 9.5 | 5 | 5 ÷ 40 = 0.125 | 5 |
| 15–24 | 19.5 | 12 | 12 ÷ 40 = 0.3 | 17 |
| 25–34 | 29.5 | 13 | 13 ÷ 40 = 0.325 | 30 |
| 35–44 | 39.5 | 7 | 7 ÷ 40 = 0.175 | 37 |
| 45–54 | 49.5 | 3 | 3 ÷ 40 = 0.075 | 40 |
Start with the first frequency, then add each new frequency to the total above it: 5; 5 + 12 = 17; 17 + 13 = 30; 30 + 7 = 37; 37 + 3 = 40. The last cumulative frequency always equals the total number of values, which is a built-in check.
The relative frequencies always add up to 1: 0.125 + 0.3 + 0.325 + 0.175 + 0.075 = 1.
Worked example 1
How many students take at most 24 minutes? Read the cumulative frequency at the end of the class 15–24: 17.
Worked example 2
How many take 35 minutes or more? Total − (cumulative frequency up to 34) = 40 − 30 = 10.
Worked example 3
What percentage take from 25 to 34 minutes? Relative frequency 0.325 = 32.5%.
Each cumulative frequency belongs to the upper boundary of its class: "17 students take less than 24.5 minutes". You will use this in the next lessons, when you find the median.
Tip
Running total, top to bottom. The cumulative column can only go up (or stay the same), and its last entry must equal the total. If either check fails, you have added wrongly.
Marks out of 50 of 30 students: 0–9: 2; 10–19: 5; 20–29: 9; 30–39: 10; 40–49: 4.
Write the cumulative frequencies.
Solution: 2, 7, 16, 26, 30.
How many students scored 29 or less?
Solution: the cumulative frequency at the end of 20–29: 16.
How many scored 20 or more?
Solution: 30 − 7 = 23.
What is the relative frequency of the class 30–39?
Solution: 10 ÷ 30 = 13, about 0.33 (33%).
What percentage scored 40 or more, to 1 decimal place?
Solution: 4 ÷ 30 × 100 = 13.3%.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
