01 · MEDIAN CLASS
Cumulative Frequency & Median Class
To find the median of grouped data, construct the cumulative frequency (cf) column. Find $N = \sum f_i$. The Median Class is the class interval whose cumulative frequency is just greater than (or equal to) N / 2.
02 · MEDIAN FORMULA
Median Formula for Grouped Data
Median = l + [ ( (N/2) - cf ) / f ] × h
l
Lower limit of median class.
N
Number of observations (N = Σfᵢ).
cf
Cumulative frequency of class preceding the median class.
f
Frequency of the median class.
h
Class size (assuming equal class sizes).
03 · EMPIRICAL RULE
Empirical Relationship Between Measures of Central Tendency
3 × Median = Mode + 2 × Mean
04 · SOLVED NCERT EXAMPLE
Step-by-Step Problem Solving
EXAMPLE 1
A survey regarding heights (in cm) of 51 girls of Class X was conducted. Find the median height.
Class Interval | Frequency fᵢ | Cumulative Frequency cf
135 - 140 | 4 | 4
140 - 145 | 7 | 11 (cf of preceding class!)
145 - 150 | 18 | 29 ◄ MEDIAN CLASS (cf ≥ 25.5)
150 - 155 | 11 | 40
155 - 160 | 6 | 46
160 - 165 | 5 | 51 (N = 51)
135 - 140 | 4 | 4
140 - 145 | 7 | 11 (cf of preceding class!)
145 - 150 | 18 | 29 ◄ MEDIAN CLASS (cf ≥ 25.5)
150 - 155 | 11 | 40
155 - 160 | 6 | 46
160 - 165 | 5 | 51 (N = 51)
Identifying Parameters:
N = 51 ⟹ N/2 = 25.5
Median Class = 145 - 150
Lower Limit l = 145, Class Size h = 5
f = 18 (median class frequency)
cf = 11 (preceding cumulative frequency)
Median Class = 145 - 150
Lower Limit l = 145, Class Size h = 5
f = 18 (median class frequency)
cf = 11 (preceding cumulative frequency)
Calculating Median:
Median = 145 + [ (25.5 - 11) / 18 ] × 5
Median = 145 + [ 14.5 / 18 ] × 5 = 145 + (72.5 / 18) = 145 + 4.03 = 149.03 cm
Median = 145 + [ 14.5 / 18 ] × 5 = 145 + (72.5 / 18) = 145 + 4.03 = 149.03 cm
Median height of girls is 149.03 cm.
05 · QUICK SUMMARY
Key Takeaways
01
Median class is chosen where cumulative frequency cf ≥ N/2.
02
Median formula: Median = l + [ ( (N/2) - cf ) / f ] × h.
03
Empirical relationship formula: 3 Median = Mode + 2 Mean.