The table below shows the distribution of examination scores of 50 candidates in a selection test: | Score Class | Frequency (f) | | :---: | :---: | | 10 - 19 | 5 | | 20 - 29 | 9 | | 30 - 39 | 16 | | 40 - 49 | 14 | | 50 - 59 | 6 | Using the cumulative frequency distribution (ogive), what is the esti

The table below shows the distribution of examination scores of 50 candidates in a selecti… — JAMB UTME

The table below shows the distribution of examination scores of $50$ candidates in a selection test:

Score Class	Frequency ( $f$ )
$10 - 19$	$5$
$20 - 29$	$9$
$30 - 39$	$16$
$40 - 49$	$14$
$50 - 59$	$6$

Using the cumulative frequency distribution (ogive), what is the estimated $70^{\text{th}}$ percentile score?

A
$33.1$
$43.1$ Answer
C
$43.6$
D
$38.1$

Answer

The estimated

70^{\text{th}}

percentile score is

43.1

The

70^{\text{th}}

percentile corresponds to the score at the

35^{\text{th}}

candidate (

70\%

50

). This falls within the

40 - 49

score class, which has a lower boundary of

39.5

, a frequency of

14

, and a class width of

10

. Interpolating linearly gives

39.5 + \left(\frac{35 - 30}{14}\right) \times 10 = 43.07 \approx 43.1

Step-by-Step Solution

Construct the cumulative frequency table to find class boundaries and cumulative frequencies.

Cumulative frequencies:

10-19

(

CF = 5

20-29

(

CF = 14

30-39

(

CF = 30

40-49

(

CF = 44

50-59

(

CF = 50

Cumulative frequencies are required to locate the position of the desired percentile.

Determine the rank position of the

70^{\text{th}}

percentile (

P_{70}

Rank position =

\frac{70}{100} \times 50 = 35^{\text{th}}

position.

The

70^{\text{th}}

percentile corresponds to the score below which

70\%

of the total candidates fall.

Identify the class interval containing the

35^{\text{th}}

cumulative frequency and state its parameters.

The target class is

40 - 49

. Parameters: Lower class boundary

L = 39.5

, preceding cumulative frequency

CF_{\text{prev}} = 30

, class frequency

f = 14

, class width

c = 10

Since

30 < 35 \le 44

, the

35^{\text{th}}

entry falls within the

40 - 49

class.

Apply the linear interpolation formula for percentiles from an ogive.

P_{70} = L + \left(\frac{\frac{70N}{100} - CF_{\text{prev}}}{f}\right) \times c = 39.5 + \left(\frac{35 - 30}{14}\right) \times 10 = 39.5 + \frac{50}{14} \approx 43.07 \approx 43.1

This calculates the exact score estimate on the cumulative frequency curve.

Key Concept

Estimating Percentiles from Grouped Data and Ogives

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