Calculate pass marks and top students marks from following series (Probabilty and Statistics)

Hello, it's me Dikpal. Today I am here to solve one of the probability and statistics problem which has been asked frequently in the competitive examination.

Suppose a marks obtained and frequency of students as follows:

Marks of student     students   Cumulative Frequency

0-20                                50            50
20-40                              100          50+100= 150
40-50                              150          150+150= 300
50-60                               90            390
60-80                               60            450
80-100                            50             500

Total students = 500

Questions are:

1.Find the pass marks if 20% failed the exam.
2. Find the marks of TOP 25% students.

Solution steps are here:

Any problems which are given in percentage are percentile. So the term can be calculated as iN/100 th term. (Similar like: in median iN/2, Quartile iN/4 etc.)

So we have to focus on the question: 

1. 20% Failed the exam means we have to find terms P20 as iN/20 th, where N= number of students and i= 20 (20% failed, above which student pass the exam.)

So we will calculate the marks at 20% failed above which students pass the exam.

P20= iN/100= 20*500/100= 100 th term or students of cumulative frequency (CF).

So from above table, pass marks lies in 20-40 series, whose frequency(f) is 100,  proceeding CF= 50 (look for previous series CF i.e. 0-20 series in above question)

So general formula for value Pi= L+ (iN/100- CF)* h/ F 

Where 
L = 20 (first number of series 20-40) where P20 lies.
i= 20 ( 20% failed exam, mentioned earlier above)
N= 500 (total number of students)
CF= 50 (Previous one CF than affectes series)
h= 20 (series difference)
F= 100 (Frequency of affected class)


Hence P20= 20+ (100-50)*20/100= 30.

So pass marks is =30.


2. Solution:

25% students top the examination. In this case we have to solve from 75% students who are not top. (Always proceed from less value to high value) So we have to find P75 but not p25. (It is because our class or series starts from 0 to 100) .

So to find P75, calculate term as iN/100= 75*500/100=375 th term or 375th students of CF.

From above table, 375th students lie in 50-60 class (Because in previous class only 300 students lie, while 50-60 class contains students from 301 to 390).

So to find marks of 25% top student: 
P75= L + (iN/100-CF)*h/F
L=50
iN/100= 375
CF= 30)
h=20
F= 90

Hence, P75= 50+ (375-300)*10/90= 66.67

Hence marks of top 25% students is 66.67.


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