Grouped Data and the Estimated Mean
When data is grouped into class intervals, such as 10 <= h < 20, the table records only which range each value falls into, so an exact mean cannot be calculated, only an estimate, found using the midpoint of each class interval and the same sum-of-frequency-times-value method used for an ungrouped frequency table. Grouped data also allows the modal class, the class with the highest frequency, and the class containing the median to be identified, since a single exact mode or median cannot be found from it alone.
Before you start
Make sure you're comfortable with these topics first:
Method
- Check the table uses class intervals (ranges), such as 10 <= t < 20, rather than single values.
- Find the midpoint of each class interval by adding its two boundary values and dividing by 2.
- Multiply each class's midpoint by its frequency to estimate the total for that class, then add these products for every class.
- Divide this total by the sum of all the frequencies to find the estimated mean.
- To find the modal class, identify the class interval with the highest frequency - this is a class, not a single value.
- To find the class containing the median, use a cumulative frequency running total to locate the position (total frequency + 1) / 2, then state which class interval that position falls into.
Worked example
The table shows the time, t minutes, taken by 40 pupils to complete a puzzle: 0 <= t < 10 (frequency 6), 10 <= t < 20 (frequency 14), 20 <= t < 30 (frequency 12), 30 <= t < 40 (frequency 8). Calculate an estimate for the mean time taken.
- Find the midpoint of each class: 0-10 gives 5, 10-20 gives 15, 20-30 gives 25, 30-40 gives 35.
- Multiply each midpoint by its frequency: 5 x 6 = 30, 15 x 14 = 210, 25 x 12 = 300, 35 x 8 = 280.
- Add these products together: 30 + 210 + 300 + 280 = 820.
- Add the frequencies to find the total number of pupils: 6 + 14 + 12 + 8 = 40.
- Divide the total by the number of pupils: 820 / 40 = 20.5 minutes.
Practice questions
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Q1State the midpoint of the class interval 20 <= m < 30.Show answer
Answer: 25
Q2State the midpoint of the class interval 15 <= h < 25.Show answer
Answer: 20
Q3A table of the heights of 30 plants gives these class frequencies: 0-10 cm (frequency 4), 10-20 cm (frequency 7), 20-30 cm (frequency 13), 30-40 cm (frequency 6). State the modal class.Show answer
Answer: 20-30 cm (it has the highest frequency, 13)
Q4Using the same plant-height table (frequencies 4, 7, 13, 6 for classes 0-10, 10-20, 20-30, 30-40), state which class interval contains the median.Show answer
Answer: 20-30 cm (the cumulative frequencies are 4, 11, 24, 30, so the middle two values, the 15th and 16th out of 30, both fall after a cumulative total of 11 and before 24, i.e. in this class)
Q5The table shows the mass, m kg, of 25 parcels: 0 <= m < 2 (frequency 5), 2 <= m < 4 (frequency 11), 4 <= m < 6 (frequency 9). Calculate an estimate for the mean mass of a parcel.Show answer
Answer: 3.32 kg (midpoints 1, 3, 5; sum of f x midpoint = 5 + 33 + 45 = 83, divided by 25)
Q6The table shows the amount, in pounds, spent by 60 customers in a shop: 0 <= p < 10 (frequency 18), 10 <= p < 20 (frequency 24), 20 <= p < 30 (frequency 12), 30 <= p < 40 (frequency 6). Calculate an estimate for the mean amount spent.Show answer
Answer: 16 pounds (midpoints 5, 15, 25, 35; sum of f x midpoint = 90 + 360 + 300 + 210 = 960, divided by 60)
Q7Explain why it is only possible to find an ESTIMATE of the mean from a grouped frequency table, rather than the exact mean.Show answer
Answer: The table only shows which class interval each value falls into, not the exact individual values, so the midpoint used for each class is only an approximation of the true values in that class - the actual mean could be slightly different from the estimate
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
The table shows the distance, d km, travelled to work by 50 employees: 0 <= d < 5 (frequency 8), 5 <= d < 10 (frequency 19), 10 <= d < 15 (frequency 15), 15 <= d < 20 (frequency 8). (a) Write down the modal class. (b) Calculate an estimate for the mean distance travelled to work, giving your answer to 1 decimal place.
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The table shows the time, t hours, spent on homework each week by 45 pupils: 0 <= t < 2 (frequency 6), 2 <= t < 4 (frequency 16), 4 <= t < 6 (frequency 14), 6 <= t < 8 (frequency 9). (a) State the class interval that contains the median. (b) Calculate an estimate for the mean time spent on homework, giving your answer to 1 decimal place.
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The table shows the number of pages read by 32 pupils during a reading session: 0 <= p < 10 (frequency 5), 10 <= p < 20 (frequency n), 20 <= p < 30 (frequency 9), 30 <= p < 40 (frequency 4). (a) Find the value of n. (b) Calculate an estimate for the mean number of pages read.
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Free printable worksheet
Want more practice on paper? Download the grouped data and the estimated mean worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 36 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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