Why grouped data needs new methods
When data is grouped into class intervals, we no longer know the individual values. We assume every observation in a class is at its class mark (midpoint) and work from there.
Class mark xᵢ = (lower limit + upper limit) ÷ 2.
Mean
Direct method
Mean = Σfᵢxᵢ ÷ Σfᵢ
Example. Marks of 30 students:
Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
Students (fᵢ) | 3 | 5 | 9 | 8 | 5 |
Class mark (xᵢ) | 5 | 15 | 25 | 35 | 45 |
fᵢxᵢ | 15 | 75 | 225 | 280 | 225 |
Σfᵢ = 30 and Σfᵢxᵢ = 820, so the mean = 820 ÷ 30 ≈ 27.33.
Assumed mean method
When the numbers are large, choose an assumed mean a (usually a central class mark) and use deviations dᵢ = xᵢ − a:
Mean = a + Σfᵢdᵢ ÷ Σfᵢ
With a = 25 above: dᵢ = −20, −10, 0, 10, 20 and fᵢdᵢ = −60, −50, 0, 80, 100. Σfᵢdᵢ = 70, so the mean = 25 + 70/30 ≈ 27.33. The same answer, with smaller numbers.
Step-deviation method
If all deviations share a common factor h (usually the class width), use uᵢ = (xᵢ − a)/h:
Mean = a + h × (Σfᵢuᵢ ÷ Σfᵢ)
Here h = 10, uᵢ = −2, −1, 0, 1, 2, and Σfᵢuᵢ = −6 − 5 + 0 + 8 + 10 = 7. Mean = 25 + 10 × 7/30 ≈ 27.33.
All three methods give the same result; the last two simply reduce arithmetic.
Mode
The class with the highest frequency is the modal class. The mode lies inside it:
Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h
where l is the lower limit of the modal class, h the class width, f₁ the frequency of the modal class, f₀ the frequency of the class before it and f₂ of the class after it.
Example. In the table above, the modal class is 20–30 (f₁ = 9), with f₀ = 5, f₂ = 8, l = 20, h = 10.
Mode = 20 + (9 − 5)/(18 − 5 − 8) × 10 = 20 + 4/5 × 10 = 28.
Median
First build the cumulative frequency (cf) column: the running total of frequencies.
Marks | f | cf |
|---|---|---|
0–10 | 3 | 3 |
10–20 | 5 | 8 |
20–30 | 9 | 17 |
30–40 | 8 | 25 |
40–50 | 5 | 30 |
n = 30, so n/2 = 15. The median class is the first class whose cf is greater than or equal to n/2. Here that is 20–30 (cf 17).
Median = l + [(n/2 − cf) ÷ f] × h
where l is the lower limit of the median class, cf the cumulative frequency of the class before it, f the frequency of the median class and h the class width.
Median = 20 + (15 − 8)/9 × 10 = 20 + 70/9 ≈ 27.78.
The empirical relationship
For moderately skewed data, the three measures are connected by
3 × Median = Mode + 2 × Mean
If you know two, you can estimate the third. With mean 27.33 and median 27.78, mode ≈ 3(27.78) − 2(27.33) ≈ 28.7, close to the value 28 found directly.
Which measure to use?
Mean uses every observation; it is good for comparing groups but affected by extreme values.
Median is better when there are a few extreme values, such as incomes.
Mode answers "what is most common?", such as the most popular shoe size.
Summary
Mean: direct (Σfx/Σf), assumed mean (a + Σfd/Σf) or step deviation (a + h·Σfu/Σf).
Mode = l + (f₁ − f₀)h/(2f₁ − f₀ − f₂), using the modal class.
Median = l + (n/2 − cf)h/f, using the cumulative frequency.
3 Median ≈ Mode + 2 Mean.