In a grouped frequency table, each row is a class of values, so the exact values are unknown. The modal class is the class with the highest frequency. To estimate the mean, use the midpoint of each class: multiply each midpoint by its frequency, add these, and divide by the total frequency. The median can only be placed in a class, found from the running total of frequencies.
Grouped data
- Midpoint of a class = (lower boundary + upper boundary) ÷ 2
- Estimated mean = Σ(f × midpoint) ÷ Σf
- Modal class: the class with the highest frequency
- Median class: the class containing position (Σf + 1) ÷ 2
Why grouped data needs midpoints
When data is grouped into classes such as 10 ≤ t < 20, the individual values are lost. To estimate the mean, assume every value in a class is at its midpoint. The midpoint of 10 ≤ t < 20 is 15. Because the true values are unknown, the result is only an estimate.
Estimating the mean
Add a midpoint column and an f × midpoint column. Total the f × midpoint column and divide by the total frequency. For journey times with frequencies 5, 11, 9 and 5 in the classes 0–10, 10–20, 20–30 and 30–40, Σfx = 590 and Σf = 30, so the estimated mean is 19.7 minutes.
The modal class
The modal class is the class with the highest frequency. Give the class itself, written as an interval such as 10 ≤ t < 20, not its frequency. Classes should normally be the same width for the modal class to be meaningful.
The class containing the median
Find the median position, (Σf + 1) ÷ 2, then add up the frequencies class by class. The median lies in the class where the running total first reaches that position. For 30 values the position is 15.5; running totals 5 and 16 show it lies in the 10–20 class.
Why it is only an estimate
Exam questions often ask you to explain why the mean is an estimate. The answer is that the exact data values are not known, so midpoints are used instead, and the real values in each class may not average to the midpoint.
Worked examples
Example 1. Journey times (minutes): 0–10 (5), 10–20 (11), 20–30 (9), 30–40 (5). Write down the modal class.
- The highest frequency is 11.
Answer: 10 ≤ t < 20.
Example 2. Estimate the mean journey time.
- Midpoints: 5, 15, 25, 35.
- f × midpoint: 25, 165, 225, 175, so Σfx = 590 and Σf = 30.
- 590 ÷ 30 = 19.66…
Answer: 19.7 minutes (1 d.p.).
Example 3. Which class contains the median?
- Median position: (30 + 1) ÷ 2 = 15.5.
- Running totals: 5, 16, so the 15.5th value is in the second class.
Answer: 10 ≤ t < 20.
Example 4. Explain why the mean found above is only an estimate.
- The exact times are not known.
- Each time was assumed to be at its class midpoint.
Answer: Because midpoints are used instead of the actual values.
Example 5. Write down the midpoint of the class 1.5 ≤ m < 2.5.
- (1.5 + 2.5) ÷ 2.
Answer: 2.
Common mistakes
- Using the class width instead of the midpoint. Multiply each frequency by the class midpoint.
- Dividing by the number of classes. Divide by the total frequency Σf.
- Giving the frequency as the modal class. State the class interval itself.
- Saying the median is a single value. With grouped data you can only state the class that contains it.
- Calling the result exact. A mean from grouped data is always an estimate.
How grouped frequency tables are examined
Estimating the mean from grouped data is one of the most frequent statistics questions on Foundation and Higher papers for 1MA1 and 4MA1, usually worth 3 or 4 marks. Questions ask you to:
- write down the modal class
- work out an estimate for the mean
- find the class that contains the median
- explain why the mean is only an estimate
Practise with the frequency tables worksheet, which includes grouped tables with answers, and the data and statistics revision engine.
For ungrouped tables, see averages from frequency tables. Grouped data is also shown in cumulative frequency graphs and histograms.
Frequently asked questions
How do you estimate the mean from grouped data?
Multiply each class midpoint by its frequency, add the results, and divide by the total frequency.
Why is the mean from grouped data only an estimate?
Because the exact values are unknown, so midpoints are used instead.
What is the modal class?
The class with the highest frequency.