Mean from grouped frequency tables

When data is in grouped classes, you cannot find the exact mean. Instead, you use the midpoint of each class as an estimate. This gives an estimated mean.

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Key facts to remember

1Use the midpoint of each class interval as the representative value.
2Midpoint = (lower class boundary + upper class boundary) ÷ 2.
3Estimated mean = Σ(midpoint × frequency) ÷ Σfrequency.
4The modal class is the class with the highest frequency.
5The median class is found using the cumulative frequency — locate the n/2 th value.
6Answers are estimates because the exact data values are unknown.

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Formulas

Estimated mean

Mean ≈ Σ(m × f) / Σf

m = midpoint of class, f = frequency

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Worked examples

Example 1

Heights (cm): 150-160 (f=5), 160-170 (f=12), 170-180 (f=8), 180-190 (f=5). Estimate the mean.

Working

Midpoints: 155, 165, 175, 185
Σ(m × f) = 155×5 + 165×12 + 175×8 + 185×5
= 775 + 1980 + 1400 + 925 = 5080
Σf = 5 + 12 + 8 + 5 = 30
Estimated mean = 5080 ÷ 30 ≈ 169.3 cm

Answer≈ 169.3 cm

Example 2

State the modal class from the table above.

Working

The class with the highest frequency is 160-170 (f = 12)

AnswerModal class: 160-170 cm

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Common mistakes

✗Using the class boundaries instead of the midpoints.

✗Dividing by the number of classes instead of the total frequency.

✗Giving an exact mean rather than an estimate (use "≈" and the word "estimate").

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Exam tips

✓Always add a midpoint column to your table before calculating — it makes the working much clearer.

✓State clearly that your answer is an estimate, not an exact value.

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