Mean, Mode, Median of grouped data, and Ogive curves
In Class 9, we studied raw data (ungrouped). In Class 10, we deal with grouped data presented in frequency distribution tables. We find three measures of central tendency: Mean, Mode, and Median.
x̄ = Σfᵢxᵢ / Σfᵢ
where xᵢ = class mark (midpoint) = (upper limit + lower limit)/2
fᵢ = frequency of each class
x̄ = a + (Σfᵢdᵢ / Σfᵢ)
where a = assumed mean (any class mark), dᵢ = xᵢ – a
x̄ = a + (Σfᵢuᵢ / Σfᵢ) × h
where uᵢ = (xᵢ – a)/h, h = class size (width)
Best used when class widths are equal and numbers are large.
Find mean of the following data:
| Class | Frequency (f) | Midpoint (x) | fx |
|---|---|---|---|
| 0–10 | 5 | 5 | 25 |
| 10–20 | 8 | 15 | 120 |
| 20–30 | 15 | 25 | 375 |
| 30–40 | 10 | 35 | 350 |
| 40–50 | 7 | 45 | 315 |
| Total | 45 | 1185 |
Mean = Σfx / Σf = 1185 / 45 = 26.33
The modal class is the class with the highest frequency.
Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)] × h
where:
l = lower boundary of modal class
f₁ = frequency of modal class
f₀ = frequency of class before modal class
f₂ = frequency of class after modal class
h = class width
Using the above data: Modal class = 20–30 (highest f = 15)
l = 20, f₁ = 15, f₀ = 8, f₂ = 10, h = 10
Mode = 20 + [(15–8)/(2×15–8–10)] × 10
= 20 + [7/(30–18)] × 10 = 20 + (7/12) × 10 = 20 + 5.83 = 25.83
First, create a cumulative frequency table. The median class is the class where cumulative frequency first exceeds n/2 (n = total frequency).
Median = l + [(n/2 – cf) / f] × h
where:
l = lower boundary of median class
n = total frequency
cf = cumulative frequency before the median class
f = frequency of median class
h = class width
| Class | f | Cumulative f (cf) |
|---|---|---|
| 0–10 | 5 | 5 |
| 10–20 | 8 | 13 |
| 20–30 | 15 | 28 |
| 30–40 | 10 | 38 |
| 40–50 | 7 | 45 |
n = 45, n/2 = 22.5
Median class: cf first exceeds 22.5 → class 20–30 (cf = 28 > 22.5)
l = 20, cf = 13, f = 15, h = 10
Median = 20 + [(22.5 – 13)/15] × 10 = 20 + [9.5/15] × 10 = 20 + 6.33 = 26.33
An ogive is a graph drawn by plotting cumulative frequencies against class boundaries.
Less Than Ogive: Plot (upper boundary, cumulative frequency). Curve goes upward.
More Than Ogive: Plot (lower boundary, total cf – previous cf). Curve goes downward.
The intersection of both ogives gives the Median.
Mode = 3 × Median – 2 × Mean
This formula is approximate and holds for moderately skewed distributions. It's useful for finding one measure when the other two are known.
x̄ = Σfᵢxᵢ / Σfᵢ
x̄ = a + (Σfᵢuᵢ/Σfᵢ) × h
uᵢ = (xᵢ – a) / h
l + [(f₁–f₀)/(2f₁–f₀–f₂)] × h
l + [(n/2 – cf)/f] × h
Mode = 3×Median – 2×Mean