1.

The following table shows the marks scored by 140 students in an examination of a certain paper: Marks: 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 Number of students: 20 24 40 36 20Calculate the average marks by using all the three methods: direct method, assumed mean deviation and shortcut method.

Answer»

From Direct method:

 Class interval Mid value (xi) fi fixi
 0 - 10 5 20 100
 10 - 20 1524 360
 20 - 30 25401000
 40 - 50 45 20 900
 
 N = 140\(\sum\)fixi = 3620

Mean = \(\frac{\sum f_ix_i}{N}\)

\(=\frac{3620}{140}=25.857\)

Assumed mean method: 

let assumed mean (A) = 25

Mean = A + \(\frac{\sum f_iu_i}{N}\)

 Class interval Mid value (xi) ui = xi - A fi fiui
 0 - 10 5-2020-400
 10 - 2015-1024-240
 20 - 30250400
 30 - 40351036360
 40 - 50452020400
 N = 140\(\sum\)fiui = 120


Mean = A + \(\frac{\sum f_ix_i}{N}\)

\(=25+\frac{120}{140}=25+0.857\)

\(=25.857\)

Step deviation method: Let the assumed mean (A) = 25

 Class interval Mid value (xi) di = xi - A = xi - 25 ui = \(\frac{xi-25}{10}\) Frequency (fi) fiui
 0 - 105-20-220-40
 10 - 2015-10-124-24
 20 - 302500400
 30 - 40351013636
 40 - 504520220 40
 N = 140\(\sum\)fiui = 12


Mean = A + \(\frac{\sum f_iu_i}{N}\times h\)

\(=25+0.857=25.857\)

\(=25+\frac{12}{140}\times10\)

\(=25+0.857=25.857\)


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