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
20
Calculate the average marks by using all the three methods: direct method, assume mean deviation and shortcut method.
Answers
DIRECT METHOD:
In this method find the class marks of class interval. These class marks would serve as the representative of whole class and are represented by xi. For each class interval we have the frequency fi corresponding to the class mark xi.
Class marks = ( lower limit + upper limit)/2
Then find the product of fi, & xi for each class interval. Find Σ fi & Σ fixi.
Use the formula :
MEAN = Σfixi/ Σfi
[‘Σ’ Sigma means ‘summation’ ]
FREQUENCY DISTRIBUTION TABLE IS IN THE ATTACHMENT
From the table : Σfixi = 3620 ,Σfi = 140
MEAN = Σfixi/ Σfi
Mean = 3620/140 = 362/14
Mean = 25.857
Hence, the mean is 25.857 .
ASSUMED MEAN METHOD :
In this method, first of all, one among xi 's is chosen as the assumed mean denoted by ‘A’. After that the difference ‘di’ between ‘A’ and each of the xi's i.e di = xi - A is calculated .
MEAN = A + Σfidi / Σfi
[‘Σ’ Sigma means ‘summation’ ]
★★ We may take Assumed mean 'A’ to be that xi which lies in the middle of x1 ,x2 …..xn.
FREQUENCY DISTRIBUTION TABLE IS IN THE ATTACHMENT
From the table : Σfidi = 120 ,Σfi = 140
Let the assumed mean, A = 25
MEAN = A + Σfidi / Σfi
MEAN = 25 + (120/140)
= 25 + 6/7
= 25 + 0.857
= 25.857
Hence, the mean is 25.857 .
STEP DEVIATION METHOD:
Step deviation method is used in the cases where the deviation from the assumed mean 'A' are multiples of a common number. If the values of ‘di’ for each class is a multiple of ‘h’ the calculation become simpler by taking ui= di/h = (xi - A )/h
Here, h is the class size of each class interval.
★★ Find the class marks of class interval. These class marks would serve as the representative of whole class and are represented by xi.
★★ Class marks (xi) = ( lower limit + upper limit) /2
★★ We may take Assumed mean 'A’ to be that xi which lies in the middle of x1 ,x2 …..xn
MEAN = A + h ×(Σfiui /Σfi) , where ui = (xi - A )/h
[‘Σ’ Sigma means ‘summation’ ]
FREQUENCY DISTRIBUTION TABLE IS IN THE ATTACHMENT
From the table : Σfiui = 12 , Σfi = 140
Let the assumed mean, A = 25, h = 10
MEAN = A + h ×(Σfiui /Σfi)
MEAN = 25 + 10(12/140)
= 25 + 6/7
= 25 + 0.857
= 25.857
Hence, the mean is 25.857 .
HOPE THIS ANSWER WILL HELP YOU….
Σfi = 140 , Σfixi = 3620
=> Mean=Σfixi/Σfi
= 3620/140
= 25.86
ii ) Assume mean method :
a = 25 , Σfidi = 120, Σfi = 140
Mean= a + Σfidi/Σfi
= 25 + (120/140)
= 25 + 0.86
= 25.86
iii ) Step - deviation method :
a = 25 , Σfiui = 12 , Σfi = 140 , h = 10
Mean = a +[ Σfiui/Σfi ]×h
= 25 + ( 12/140 ) × 10
= 25 + 0.86
= 25.86
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