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Difficulty: Very hardMeasures of Central Tendency

The table below shows the frequency distribution of weekly cocoa bean output (in metric tonnes) produced by 40 agricultural cooperative societies in Southwestern Nigeria:

Output Range (Tonnes)Number of Cooperatives (ff)
10−1910 - 1944
20−2920 - 291010
30−3930 - 391616
40−4940 - 4988
50−5950 - 5922

What is the estimated median weekly cocoa output of the cooperatives?

  1. 33.2533.25 tonnesAnswer
  2. B
    33.7533.75 tonnes
  3. C
    32.8832.88 tonnes
  4. D
    34.5034.50 tonnes

Answer

The estimated median weekly cocoa output is 33.2533.25 tonnes.
The option stating 33.2533.25 tonnes is correct because the median class is 30−3930 - 39 with lower boundary L=29.5L = 29.5, cumulative frequency prior F=14F = 14, class frequency fm=16f_m = 16, and class width c=10c = 10. Substituting into Median=L+(N/2−Ffm)c\text{Median} = L + \left(\frac{N/2 - F}{f_m}\right)c yields 29.5+(20−1416)×10=33.2529.5 + \left(\frac{20 - 14}{16}\right) \times 10 = 33.25.

Step-by-Step Solution

1
Determine the median position and identify the median class
Total frequency N=4+10+16+8+2=40N = 4 + 10 + 16 + 8 + 2 = 40. Median position is N2=402=20th\frac{N}{2} = \frac{40}{2} = 20^{\text{th}} item. Cumulative frequencies are: 10−19:410-19: 4; 20−29:1420-29: 14; 30−39:3030-39: 30. The 20th20^{\text{th}} item lies within the 30−3930 - 39 interval.
The median class is the class interval containing the N2\frac{N}{2} position.
2
Identify the required statistical parameters for grouped median formula
Lower class boundary (LL) = 29.529.5; Cumulative frequency before median class (FF) = 1414; Frequency of median class (fmf_m) = 1616; Class width (cc) = 39.5−29.5=1039.5 - 29.5 = 10.
Grouped data estimation requires exact continuous boundaries rather than discrete limits.
3
Apply the grouped median interpolation formula
Median=L+(N2−Ffm)×c=29.5+(20−1416)×10=29.5+(616)×10=29.5+3.75=33.25\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times c = 29.5 + \left(\frac{20 - 14}{16}\right) \times 10 = 29.5 + \left(\frac{6}{16}\right) \times 10 = 29.5 + 3.75 = 33.25.
Interpolates the exact median position within the continuous interval of the median class.

Key Concept

Calculation of Median from Grouped Frequency Distribution
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Topics

Measures of Central Tendency

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