Statistics
Grade 5-6Averages from Grouped Data
When data is grouped into class intervals rather than listed individually, the exact values are no longer known, so the mode and median can only be given as a class, and the mean can only be estimated. This lesson covers finding the modal class, finding the class containing the median, estimating the mean using midpoints, and finding a missing frequency from a known estimated mean.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding the modal class
The modal class is the class interval with the highest frequency. Since the individual values are not known, only the class itself can be given, not a single value.
A table shows the heights of 40 tomato plants grouped into 5 cm class intervals, and the 25 ≤ h < 30 class has the highest frequency of 12. The modal class is 25 ≤ h < 30.
Finding the class that contains the median
Find the total frequency, then use cumulative frequencies to locate which class contains the middle value, found at position n ÷ 2 for a total of n values.
With a total frequency of 40, the median lies at the 20th value. Adding frequencies in order, the cumulative frequency reaches 21 by the end of the 20 ≤ h < 25 class, so the median class is 20 ≤ h < 25.
Estimating the mean using midpoints
Find the midpoint of each class interval, multiply each midpoint by its frequency to find Σfx, add these together, then divide by the total frequency, Σf. This gives an estimate, since the exact values within each class are unknown.
For the tomato plant heights, the midpoints are 12.5, 17.5, 22.5, 27.5 and 32.5. Multiplying each by its frequency and summing gives Σfx = 965, and dividing by the total frequency of 40 gives an estimated mean of 24.1 cm.
Finding a missing frequency from a known mean
Write Σfx and Σf as expressions involving the unknown frequency, using the class midpoints, set their ratio equal to the given estimated mean, then solve the resulting equation.
With frequencies 6, 10, x, 9 and 3 for classes with midpoints 5, 15, 25, 35 and 45, and an estimated mean of 23, Σfx = 630 + 25x and Σf = 28 + x. Setting (630 + 25x) ÷ (28 + x) = 23 gives 630 + 25x = 23(28 + x), which solves to x = 7.
Worked Examples
Three exam-style questions, fully solved.
The table shows the heights of 40 tomato plants, grouped into 5 cm classes from 10 to 35 cm, with frequencies 3, 7, 11, 12 and 7. Write down the modal class.
Easy- 1.Identify the class with the highest frequency: 12, for the 25 ≤ h < 30 class
Answer: 25 ≤ h < 30
The same table shows the heights of 40 tomato plants, with midpoints 12.5, 17.5, 22.5, 27.5 and 32.5 for the five classes. Calculate an estimate for the mean height, correct to 1 decimal place.
Medium- 1.Find Σfx using the midpoints: 3(12.5) + 7(17.5) + 11(22.5) + 12(27.5) + 7(32.5) = 965
- 2.Find the total frequency, Σf: 3 + 7 + 11 + 12 + 7 = 40
- 3.Divide Σfx by Σf: 965 ÷ 40
Answer: 24.1 cm
The table shows the heights of 35 tomato plants with frequencies 6, 10, x, 9 and 3 for classes with midpoints 5, 15, 25, 35 and 45. The estimated mean height is 23 cm. Find the value of x.
Hard- 1.Write Σfx in terms of x: 6(5) + 10(15) + x(25) + 9(35) + 3(45) = 630 + 25x
- 2.Write Σf in terms of x: 6 + 10 + x + 9 + 3 = 28 + x
- 3.Set the mean equation: (630 + 25x) ÷ (28 + x) = 23, so 630 + 25x = 23(28 + x)
Answer: x = 7
Avoid These
The most common mistakes students make.
Giving the frequency of the modal class instead of the class interval itself as the answer.
Using the class boundary instead of the midpoint when estimating the mean, which gives an incorrect Σfx.
Forgetting that the mean from grouped data is only an estimate, since the exact values within each class are unknown.
Miscounting the position needed to find the median class, or misreading the cumulative frequencies when locating it.
When finding a missing frequency, forgetting to include it correctly in both the Σfx expression, multiplied by the right midpoint, and in the total Σf.
FAQ
Questions parents and students ask.
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