Mean of Grouped Data

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Computation of Measures of Central Tendency - Mean
Direct Method of Mean
Assumed Mean Method
Step Deviation Method for Mean
Step-deviation method

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To calculate the mean of grouped data, the first step is to determine the midpoint (also called a class mark) of each interval, or class. These midpoints must then be multiplied by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean.

Average of any observation is known as mean. In this chapter there are three methods to find mean-

1) Direct method

2) Assumed mean method

3) Step deviation method

We will take a common example to understand these three methods

1) Direct method-

Class interval	10-25	25-40	40-55	55-70	70-85	85-100
No. of students	2	3	7	6	6	6

CI- class interval

fi- Number of students

xi= class mark= `"Upper class limit+ Lower class limit"/2`

∑ means summation, which means the total

CI	fi	xi	fixi
10-25	2	(10+25)/2= 17.5	35
25-40	3	32.5	97.5
40-55	7	47.5	332.5
55-70	6	62.5	375
70-85	6	77.5	465
85-100	6	92.5	555
	∑ fi= 30		∑ fixi= 1860

Mean through direct method= `bar(x)`= `(sum "fixi")/ (sum "fi")`

Mean through direct method= `1860/30= 62`

2) Assumed mean method-

Mean through assumed mean method= `bar(x)`= ` a+(sum "fidi")/(sum "fi")`

where a= assumed mean i.e any value of xi

di= deviation= xi-a

CI	fi	xi	fixi	di=xi-a	fidi
10-25	2	17.5	35	-30	-60
25-40	3	32.5	97.5	-15	-45
40-55	7	47.5	332.5	0	0
55-70	6	62.5	375	15	90
70-85	6	77.5	465	30	180
85-100	6	92.5	555	45	270
	∑ fi= 30		∑ fixi= 1860		∑ fidi= 435

Let a= 47.5

Mean through assumed mean method= `bar(x)`= ` a+(sum "fidi")/(sum "fi")`

= 47.5+ 435/30

= 47.5+ 14.5

Mean through assumed mean method= 62

3) Step deviation method-

Mean through step deviation method= `bar(x)`= `a+ (sum "fiui")/(sum "fi") xx h`

where, ui= modified class mark= `(di)/h`

h= Class size

CI	fi	xi	fixi	di=xi-a	fidi	ui=di/h	fiui
10-25	2	17.5	35	-30	-60	-2	-4
25-40	3	32.5	97.5	-15	-45	-1	-3
40-55	7	47.5	332.5	0	0	0	0
55-70	6	62.5	375	15	90	1	6
70-85	6	77.5	465	30	180	2	12
85-100	6	92.5	555	45	270	3	18
	∑fi= 30		∑ fixi= 1860		∑ fidi= 435		∑fiui= 29

Mean through step divation method= `bar(x)`= `a+ (sum "fiui")/(sum "fi") xx h`

=`47.5+ (29/30) xx 5`

=` 47.5+ 14.5`

Mean through step deviation= `bar(x)`= 62

As you can see, the mean obtained is same i.e 62 from any of the method.

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Related QuestionsVIEW ALL [8]

The following table gives the frequency distribution of trees planted by different Housing Societies in a particular locality:

No. of Trees	No. of Housing Societies
10-15	2
15-20	7
20-25	9
25-30	8
30-35	6
35-40	4

Find the mean number of trees planted by Housing Societies by using ‘Assumed Means Method’

The measurements (in mm) of the diameters of the head of the screws are given below:

Diameter (in mm)	No. of Screws
33 — 35	10
36 — 38	19
39 — 41	23
42 — 44	21
45 — 47	27

Calculate mean diameter of head of a screw by ‘Assumed Mean Method’.

The yield of soyabean per acre in the farm of Mukund for 7 years was 10,7,5,3,9,6,9 quintal. Find the mean of yield per acre.

There are three dealers A, B and C in Maharashtra. Suppose, the trade of each of them in september 2018 was as shown in the following table.
The rate of GST on each transaction was 5%.
Read the table and answer the questions below it.

Dealer	GST collected on the sale	GST paid at the time of purchase	ITC	Tax paid to the Govt.	Taxbalance with the Govt.
A	Rs.5000	Rs. 6000	Rs. 5000	Rs. 0	Rs. 1000
B	Rs 5000	Rs. 4000	Rs. 4000	Rs. 1000	Rs. 0
C	Rs.5000	Rs. 5000	Rs. 5000	Rs. 0	Rs. 0

(i) How much amount did the dealer A get by sale ?
(ii) For how much amount did the dealer B buy the articles ?
(iii) How much is the balance of CGST and SGST left with the government that was paid by A ?

The formula to find mean from a grouped frequency table is

\[X = A + \frac{\sum f_i u_i}{\sum f_i} \times hg\]

(A)

\[\frac{x_i + A}{g}\]

(B)

\[\left( x_i - A \right)\]

(C)

\[\frac{x_i - A}{g}\]

(D)

\[\frac{A - x_i}{g}\]

The following table shows the income of farmers in a grape season. Find the mean of their income.