When data is large, instead of writing down every observation, we often organize it into a frequency distribution table. The measures of central tendency (Mean, Median, Mode) can then be calculated using special formulas for grouped data.
In this article, we’ll explain each measure in detail with examples.
1. Mean of Grouped Data
Formula (Step-Deviation Method)
xˉ=a+∑fiui∑fi⋅h\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \cdot hxˉ=a+∑fi∑fiui⋅h
Where:
- aaa = Assumed mean (any mid-point chosen for convenience)
- fif_ifi = frequency of the ithi^\text{th}ith class
- xix_ixi = mid-point of the ithi^\text{th}ith class
- hhh = class size (width of each class)
- ui=xi−ahu_i = \frac{x_i – a}{h}ui=hxi−a
Example
The marks of 50 students are given in the following table:
| Class Interval | Frequency (fif_ifi) |
|---|---|
| 0–10 | 5 |
| 10–20 | 8 |
| 20–30 | 12 |
| 30–40 | 15 |
| 40–50 | 10 |
Step 1: Find mid-points (xix_ixi) xi=Lower limit + Upper limit2x_i = \frac{\text{Lower limit + Upper limit}}{2}xi=2Lower limit + Upper limit
| Class | fif_ifi | xix_ixi | ui=xi−ahu_i = \frac{x_i – a}{h}ui=hxi−a | fiuif_i u_ifiui |
|---|---|---|---|---|
| 0–10 | 5 | 5 | -2 | -10 |
| 10–20 | 8 | 15 | -1 | -8 |
| 20–30 | 12 | 25 | 0 | 0 |
| 30–40 | 15 | 35 | 1 | 15 |
| 40–50 | 10 | 45 | 2 | 20 |
Here, a=25a = 25a=25, h=10h = 10h=10.
Step 2: Apply formula xˉ=a+∑fiui∑fi⋅h\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \cdot hxˉ=a+∑fi∑fiui⋅h xˉ=25+1750⋅10\bar{x} = 25 + \frac{17}{50} \cdot 10xˉ=25+5017⋅10 xˉ=25+3.4=28.4\bar{x} = 25 + 3.4 = 28.4xˉ=25+3.4=28.4
Answer: Mean = 28.4
2. Median of Grouped Data
Formula
Median=L+(N2−CFf)⋅h\text{Median} = L + \left( \frac{\frac{N}{2} – CF}{f} \right) \cdot hMedian=L+(f2N−CF)⋅h
Where:
- LLL = lower boundary of the median class
- NNN = total frequency (∑fi\sum f_i∑fi)
- CFCFCF = cumulative frequency before the median class
- fff = frequency of the median class
- hhh = class width
Example
Using the same table:
| Class | Frequency | Cumulative Frequency |
|---|---|---|
| 0–10 | 5 | 5 |
| 10–20 | 8 | 13 |
| 20–30 | 12 | 25 |
| 30–40 | 15 | 40 |
| 40–50 | 10 | 50 |
Step 1: Find N=50N = 50N=50.
Median class = Class where N/2=25N/2 = 25N/2=25.
Here, the 20–30 class contains the 25th value → Median class = 20–30.
Step 2: Apply formula L=20, CF=13, f=12, h=10L = 20, \; CF = 13, \; f = 12, \; h = 10L=20,CF=13,f=12,h=10 Median=20+(25−1312)⋅10\text{Median} = 20 + \left( \frac{25 – 13}{12} \right) \cdot 10Median=20+(1225−13)⋅10 Median=20+(1212)⋅10\text{Median} = 20 + \left( \frac{12}{12} \right) \cdot 10Median=20+(1212)⋅10 Median=30\text{Median} = 30Median=30
Answer: Median = 30
3. Mode of Grouped Data
Formula
Mode=L+(f1−f02f1−f0−f2)⋅h\text{Mode} = L + \left( \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \right) \cdot hMode=L+(2f1−f0−f2f1−f0)⋅h
Where:
- LLL = lower boundary of the modal class
- f1f_1f1 = frequency of the modal class
- f0f_0f0 = frequency of the class before the modal class
- f2f_2f2 = frequency of the class after the modal class
- hhh = class width
Example
From the same data:
The highest frequency = 15 (class 30–40).
So, modal class = 30–40. L=30, f1=15, f0=12, f2=10, h=10L = 30, \; f_1 = 15, \; f_0 = 12, \; f_2 = 10, \; h = 10L=30,f1=15,f0=12,f2=10,h=10 Mode=30+(15−122(15)−12−10)⋅10\text{Mode} = 30 + \left( \frac{15 – 12}{2(15) – 12 – 10} \right) \cdot 10Mode=30+(2(15)−12−1015−12)⋅10 =30+(330−22)⋅10= 30 + \left( \frac{3}{30 – 22} \right) \cdot 10=30+(30−223)⋅10 =30+(38)⋅10= 30 + \left( \frac{3}{8} \right) \cdot 10=30+(83)⋅10 =30+3.75=33.75= 30 + 3.75 = 33.75=30+3.75=33.75
Answer: Mode = 33.75
Quick Recap
| Measure | Formula | Example Result |
|---|---|---|
| Mean | xˉ=a+∑fiui∑fi⋅h\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \cdot hxˉ=a+∑fi∑fiui⋅h | 28.4 |
| Median | L+(N2−CFf)⋅hL + \left( \frac{\frac{N}{2} – CF}{f} \right) \cdot hL+(f2N−CF)⋅h | 30 |
| Mode | L+(f1−f02f1−f0−f2)⋅hL + \left( \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \right) \cdot hL+(2f1−f0−f2f1−f0)⋅h | 33.75 |
Final Notes
- Mean uses all values but is influenced by extreme classes.
- Median divides the data into two equal halves and is resistant to extreme values.
- Mode shows the most frequent class and is useful in practical life (e.g., most common shoe size).
👉 Together, they provide a complete picture of the central tendency of grouped data.
