The frequency distribution table below shows the marks obtained by 40 students in a mathematics quiz: | Mark Interval | Frequency (f) | | :---: | :---: | | 1 - 10 | 5 | | 11 - 20 | 8 | | 21 - 30 | 12 | | 31 - 40 | 10 | | 41 - 50 | 5 | From a cumulative frequency curve (ogive) constructed for this da

The frequency distribution table below shows the marks obtained by 40 students in a mathem… — JAMB UTME

The frequency distribution table below shows the marks obtained by $40$ students in a mathematics quiz:

From a cumulative frequency curve (ogive) constructed for this data, what is the score corresponding to the $40^{\text{th}}$ percentile?

23.0

To find the 40th percentile score, calculate

40\%

of the total

40

students, which gives the

16^{\text{th}}

cumulative student. The

16^{\text{th}}

student falls in the

21 - 30

mark interval, which has a lower class boundary of

20.5

. Interpolating gives

20.5 + \left(\frac{16 - 13}{12}\right) \times 10 = 20.5 + 2.5 = 23.0

Construct the cumulative frequency distribution table.

Cumulative frequencies (

CF

1-10 \rightarrow 5

;

11-20 \rightarrow 13

;

21-30 \rightarrow 25

;

31-40 \rightarrow 35

;

41-50 \rightarrow 40

. Total frequency

N = 40

Cumulative frequencies are needed to locate percentile positions on an ogive.

Determine the rank position for the 40th percentile (

P_{40}

Rank position

= 0.40 \times 40 = 16^{\text{th}}

position.

The 40th percentile represents the value below which

40\%

of the observations fall.

Identify the percentile class interval and its boundaries.

Since

13 < 16 \leq 25

, the

16^{\text{th}}

position lies in the interval

21 - 30

. Lower class boundary

L = 20.5

, cumulative frequency preceding the class

CF_b = 13

, class frequency

f = 12

, and class width

c = 10

Interpolation on an ogive relies on the lower class boundary of the target interval.

Apply the linear interpolation formula for percentiles.

P_{40} = L + \left( \frac{\text{Rank} - CF_b}{f} \right) \times c = 20.5 + \left( \frac{16 - 13}{12} \right) \times 10 = 20.5 + 2.5 = 23.0

This yields the exact mark corresponding to the 40th percentile reading from the ogive.

Calculating percentiles from a cumulative frequency distribution or ogive

Tahmini Süre:1m 30s