The cumulative frequency distribution table below shows the mass of cocoa beans (in kg) ha… — JAMB UTME

The cumulative frequency distribution table below shows the mass of cocoa beans (in kg) harvested by $40$ smallholder farmers in a agricultural cooperative:

Mass (kg)	Cumulative Frequency
$\leq 20$	$4$
$\leq 30$	$12$
$\leq 40$	$28$
$\leq 50$	$36$
$\leq 60$	$40$

Using linear interpolation from the cumulative frequency table, what is the median mass (in kg) of cocoa beans harvested by the farmers?

Answer: 35 kg

Answer

The median mass of cocoa beans harvested by the farmers is

35.0\text{ kg}

The total number of farmers is

N = 40

. The median corresponds to the

\frac{40}{2} = 20^{\text{th}}

position. From the cumulative frequency table, the

20^{\text{th}}

item lies within the

30 - 40\text{ kg}

class interval. Using the grouped median formula

\text{Median} = L + \left(\frac{\frac{N}{2} - c.f.}{f}\right) \times w

, where

L = 30

c.f. = 12

f = 28 - 12 = 16

, and

w = 10

, we obtain

\text{Median} = 30 + \left(\frac{20 - 12}{16}\right) \times 10 = 35.0\text{ kg}

Step-by-Step Solution

Find the median position in the cumulative frequency distribution

Median position

= \frac{N}{2} = \frac{40}{2} = 20^{\text{th}}

item

The median corresponds to the 50th percentile, which is half of the total cumulative frequency

N = 40

Locate the median class interval and extract its statistical parameters

Median class interval is

30 - 40\text{ kg}

, with lower limit

L = 30\text{ kg}

, preceding cumulative frequency

c.f. = 12

, class frequency

f = 28 - 12 = 16

, and width

w = 10\text{ kg}

The cumulative frequency just below

20

12

(at upper boundary

30

), and at upper boundary

40

it rises to

28

Substitute values into the grouped data median formula

\text{Median} = 30 + \left(\frac{20 - 12}{16}\right) \times 10 = 30 + 5 = 35.0\text{ kg}

Linear interpolation estimates the exact position of the median within the median class interval.

Key Concept

Calculation of Median from Cumulative Frequency Data

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