Median & Partition Values: Calculation, Merits & Utility

1. What Is the Median?

The Median (denoted by M) is a positional average. It is the value of the middle-most item when all data points are arranged in ascending or descending order of magnitude.

It divides a series into two equal parts:

50% of the observations lie below the Median.
50% of the observations lie above the Median.

2. Calculation of Median Across All Series

A. Individual Series

Mandatory First Step: Arrange the data in ascending order!
If N is Odd:
Median = Size of [(N + 1) ÷ 2]th item
- Example: Marks: 15, 22, 28, 35, 42 (N = 5).
  Item = (5 + 1) ÷ 2 = 3rd item → Median = 28.
If N is Even:
Median = [Size of (N ÷ 2)th item + Size of ((N ÷ 2) + 1)th item] ÷ 2
- Example: Marks: 10, 18, 25, 30, 36, 44 (N = 6).
  Item 1 = 6 ÷ 2 = 3rd item (25)
  Item 2 = 4th item (30)
  Median = (25 + 30) ÷ 2 = 27.5.

B. Continuous Series (The Golden Formula)

3 Steps to Compute Median in Continuous Grouped Series

Step 1

Cumulative Frequency

Construct the 'cf' column by adding frequencies cumulatively

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Step 2

Locate Median Class

Find class where 'cf' is equal to or just greater than (N ÷ 2)

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Step 3

Interpolation Formula

Apply M = L1 + [((N ÷ 2) - cf) ÷ f] × i

Master Formula:

Median = L1 + [ ((N ÷ 2) - cf) ÷ f ] × i

Where:

L1 = Lower limit of the median class
N = Total frequency (Σf)
cf = Cumulative frequency of the class preceding (just above) the median class
f = Simple frequency of the median class itself
i = Class interval width (L2 - L1)

Solved Numerical Problem:

Calculate Median for the following wage distribution:

Wages (₹)	100 - 200	200 - 300	300 - 400	400 - 500	500 - 600
Workers (f)	6	14	20	12	8

Solution:

Construct Cumulative Frequency table:

Wages (₹)	Frequency (f)	Cumulative Frequency (cf)
100 - 200	6	6
200 - 300	14	20
300 - 400	20 (f)	40 (Contains N/2 = 30)
400 - 500	12	52
500 - 600	8	60 (N)

Locate Median Class:
N ÷ 2 = 60 ÷ 2 = 30th item.
Looking down the cf column, 30 falls in the class 300 - 400.
Therefore, Median Class = 300 - 400.
Identify Variables:
L1 = 300, cf = 20 (preceding class cf), f = 20, i = 100.
Calculate:
Median = 300 + [ (30 - 20) ÷ 20 ] × 100
Median = 300 + [ 10 ÷ 20 ] × 100 = 300 + 50 = ₹350

3. Partition Values: Quartiles, Deciles & Percentiles

Just as the Median divides a series into 2 equal parts, other partition values divide it into smaller equal sections:

Partition Values Family

Quartiles (Divide into 4 Parts)

Q1 (Lower Quartile): Covers 25% of data. Term = N ÷ 4
Q3 (Upper Quartile): Covers 75% of data. Term = 3N ÷ 4
(Note: Q2 = Median = 50%)

Deciles (Divide into 10 Parts)

Nine deciles (D1 to D9).
D5 = Median
Formula for Dk term = (k × N) ÷ 10

Percentiles (Divide into 100 Parts)

99 percentiles (P1 to P99).
P50 = Median, P25 = Q1, P75 = Q3
Used in CAT/GATE entrance exam scores!

Formulas for Continuous Series:

Lower Quartile (Q1):
Q1 = L1 + [ ((N ÷ 4) - cf) ÷ f ] × i
Upper Quartile (Q3):
Q3 = L1 + [ ((3N ÷ 4) - cf) ÷ f ] × i

4. Merits, Demerits & Business Utility of Median

Merits:

Resistant to Extreme Values (Robust Average): Unaffected by freak millionaires or zero-income earners.
Can Be Computed for Open-Ended Distributions: Since only the central item matters, classes like “Below ₹5,000” or “Above ₹50,000” do not obstruct calculation.
Can Be Determined Graphically: Readily found where “Less than” and “More than” Ogives intersect.

Demerits:

Not based on all observations (ignores values at both tails).
Incapable of algebraic manipulation (cannot calculate combined median of two groups).

Business Applications:

Income & Wealth Studies: Reserve Bank and NITI Aayog use Median income to define poverty thresholds.
Customer Satisfaction Ratings: Analyzing median app review stars.

Median & Partition Values: Calculation, Merits & Utility | BBA Notes