The cumulative frequency distribution below shows the monthly electricity consumption (in … — JAMB UTME

The cumulative frequency distribution below shows the monthly electricity consumption (in kWh) recorded for $50$ households in a residential estate:

Using linear interpolation, calculate the 60th percentile ( $P_{60}$ ) of electricity consumption in kWh.

Answer: 137.5 kWh

137.5 kWh

The 60th percentile corresponds to the 30th observation out of 50 (

0.60 \times 50 = 30

). From the cumulative frequency table, the 30th value falls in the class interval between

100

and

150

. Using the lower boundary of

100

, previous cumulative frequency of

15

, interval frequency of

20

, and interval width of

50

, linear interpolation gives

100 + \frac{30 - 15}{20} \times 50 = 137.5

kWh.

Determine the rank for the 60th percentile

Rank position = 30th household

The 60th percentile rank corresponds to

60\%

of the total sample size (

N = 50

), which is

0.60 \times 50 = 30

Identify the parameters of the percentile class interval

Lower boundary

L = 100

CF_{\text{prev}} = 15

, class frequency

f = 20

, width

w = 50

The cumulative frequency rises from 15 to 35 in the class interval

(100, 150]

, so the 30th value lies within this interval.

Calculate the value using linear interpolation

137.5 kWh

Substitute values into

P_{60} = 100 + \left(\frac{30 - 15}{20}\right) \times 50 = 100 + 37.5 = 137.5

kWh.

Linear interpolation for percentiles from a cumulative frequency distribution

Question