Measures of central tendency · Ages 14–15
Averages of grouped data
Class marks for the mean; the most frequent class is the modal class.
How to do it
Method
Class marks for the mean; the most frequent class is the modal class
(14.5 × 2 + 24.5 × 5 + 34.5 × 3) ÷ 10 = 25.5
Another way
Number the classes
Class numbers average 1.1: 14.5 + 1.1 × 10 = 25.5
In full
Once data are grouped the exact values are lost, so the averages can only be estimated. The estimated mean takes every value in a class to be at its class mark: mean ≈ Σ(class mark × frequency) ÷ total frequency. Classes 10–19, 20–29, 30–39 with frequencies 2, 5, 3 have class marks 14.5, 24.5, 34.5: (14.5 × 2 + 24.5 × 5 + 34.5 × 3) ÷ 10 = 25.5. The modal class is the class with the highest frequency. The median class holds the (n + 1) ÷ 2-th value: cumulative frequencies find it quickest. The median can also be estimated from a cumulative frequency polygon: across from half the total frequency to the curve, then down to the value. Another way to the estimated mean, with equal class widths: number the classes 0, 1, 2 and average those ((0 × 2 + 1 × 5 + 2 × 3) ÷ 10 = 1.1), then change back: 14.5 + 1.1 × 10 = 25.5.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The table shows a clinic's patients each day for a month (classes on the left, days on the right). With the class marks, Σ(class mark × frequency) =
Show the answer and working
Answer: 685
The class marks are 44.5, 54.5, 64.5, 74.5, 84.5; each times its frequency, added: 685.
Example 2
The table shows the audience at each showing in a cinema for a month (classes on the left, showings on the right). Which class holds the median?
- A. 50–59
- B. 80–89
- C. 90–99
- D. 70–79
Choose the right answer
Show the answer and working
Answer: D. 70–79
31 showings, so the median is the (31 + 1) ÷ 2-th; adding the frequencies class by class, it falls in 70–79.
Example 3
The table is the cumulative frequency table of how many minutes 25 members of a gym spend on a workout (classes on the left, cumulative frequencies on the right). What is the estimated mean workout time, in minutes?
Show the answer and working
Answer: 42 minutes
First subtract neighbouring cumulative frequencies to get the frequencies back, 4, 4, 8, 6, 3; the class marks are 32, 37, 42, 47, 52: Σ(class mark × frequency) = 1050, and 1050 ÷ 25 = 42 minutes.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The table shows how many minutes each visitor stayed in a park one day (classes on the left, visitors on the right). Which is the modal class?
- A. 60–64
- B. 55–59
- C. 65–69
- D. 70–74
Choose the right answer
Answer:
2.The table shows how many seconds 25 customers queued at a supermarket checkout (classes on the left, customers on the right). What is the estimated mean queuing time, in seconds?
Answer:
3.The table shows the audience at each showing in a cinema for a month (classes on the left, showings on the right). Which class holds the median?
- A. 80–89
- B. 50–59
- C. 70–79
- D. 90–99
Choose the right answer
Answer:
4.The table is the cumulative frequency table of how many minutes 20 members of a gym spend on a workout (classes on the left, cumulative frequencies on the right). What is the estimated mean workout time, in minutes?
Answer:
5.The cumulative frequency polygon shows how many minutes a group of runners took for 10 km. Reading across at half the total frequency, what is the estimated median, to 1 decimal place?
Answer:
All the answers
- 1. B. 55–59
- 2. 73.8 s
- 3. C. 70–79
- 4. 65 minutes
- 5. 34.9 minutes
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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