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Steps in Calculating Mean

Dr. Anshul Singh Thapa

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – DIRECT METHOD

In case of discrete series, frequency against each observation is multiplied by the value of the observation. The values, so obtained, are summed up and divided by the total number of frequencies.

Symbolically: Mean = ΣfX

Σf

Where, ΣfX = sum of the product of variables and frequencies. Σf = sum of frequencies.

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Direct Method

Direct method of calculating mean of the discrete series involves the following steps:

Values of various items in the series are indicated by ‘X’, and their frequencies by ‘f’.
Each item is multiplied by its frequency to get ‘fX’. These multiples are added to get ΣfX.
Frequencies are added up to get Σf.
ΣfX is divided by Σf to obtain mean.

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – DIRECT METHOD

Farm Size in acres (x) (1)	No. of cultivating households (f) (2)
64	8
63	18
62	12
61	9
60	7
59	6

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – DIRECT METHOD

Farm Size in acres (x) (1)	No. of cultivating households (f) (2)	(x × f) (3)
64	8	512
63	18	1134
62	12	744
61	9	549
60	7	420
59	6	354

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – DIRECT METHOD

Farm Size in acres (x) (1)	No. of cultivating households (f) (2)	(x × f) (3)
64	8	512
63	18	1134
62	12	744
61	9	549
60	7	420
59	6	354
		ΣfX = 3713

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – DIRECT METHOD

Farm Size in acres (x) (1)	No. of cultivating households (f) (2)	(x × f) (3)
64	8	512
63	18	1134
62	12	744
61	9	549
60	7	420
59	6	354
	Σf = 60	ΣfX = 3713

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Step 1 - X = ΣfX

Σf

Step 2 - ΣfX = 3713, Σf = 60

Step 3 - X = 3713 =

60

Step 4 - X = 61.88 Ans

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – ASSUMED MEAN METHOD

As in case of individual series the calculations can be simplified by using assumed mean method, as described earlier, with a simple modification.
Since frequency (f) of each item is given here, we multiply each deviation (d) by the frequency to get fd. Then we get Σfd. The next step is to get the total of all frequencies i.e. Σf. Then find out Σfd/ Σf. Finally the arithmetic mean is calculated by

X = A + Σfd

Σf

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ASSUMED MEAN METHOD

Before calculating the actual average of the series, some value which lies in the middle of the series is taken as assumed mean. This may be indicated by ‘A’.
Deviation of each value of different items in the series is calculated from the assumed average. The deviation is noted against the concerned item of the series. The deviation may be positive (+) or negative (-).
Accordingly, the (+) or negative (-) sign is to be noted against each deviation. If the deviation is zero, no sign need to be specified.
Find the multiple of ‘d’ and its corresponding ‘f’, that is ‘fd’. Find out the difference between two to get Σfd. Add up, separately, the positive and negative values of all ‘fd’. Find out the difference between the two to get ‘Σfd’.
Add up all the frequencies, to get Σf.
Divide Σfd by Σf to get mean.

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – ASSUMED MEAN METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)
64	8
63	18
62	12
61	9
60	7
59	6

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – ASSUMED MEAN METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)
64	8
63	18
62	12
61	9
60	7
59	6
	Σf = 60

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – ASSUMED MEAN METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)	d (X - 62)
64	8	64 – 62 = +2
63	18	64 = 63 = +1
62	12	62 – 62 = 0
61	9	61 – 62 = -1
60	7	60 – 62 = -2
59	6	59 – 62 = -3
	Σf = 60

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – ASSUMED MEAN METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)	d (X - 62) (3)	Fd (4)
64	8	64 – 62 = +2	+16
63	18	64 = 63 = +1	+18
62	12	62 – 62 = 0	0
61	9	61 – 62 = -1	-9
60	7	60 – 62 = -2	-14
59	6	59 – 62 = -3	-18
	Σf = 60

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – ASSUMED MEAN METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)	d (X - 62) (3)	Fd (4)
64	8	64 – 62 = +2	+16
63	18	64 – 63 = +1	+18
62	12	62 – 62 = 0	0
61	9	61 – 62 = -1	-9
60	7	60 – 62 = -2	-14
59	6	59 – 62 = -3	-18
	Σf = 60		Σfd = -7

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Step 1 - X = A + Σfd

Σf

Step 2 - A = 62, Σfd = -7, Σf = 60

Step 3 - X = 62 + (-7) =

60

Step 4 - X = 61.88 Ans

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – STEP DEVIATION METHOD

In this case the deviations are divided by the common factor ‘c’ which simplifies the calculation. Here we estimate

d’ = d = X – A

c c

In order to reduce the size of numerical figures for easier calculation. Then get fd' and Σfd'. The formula for arithmetic mean using step deviation method is given as,

X = A + Σfd’ x c

Σf

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STEP DEVIATION METHOD

This method is variant of assumed mean method. it is adopted when deviation from the assumed mean have some common factor. The common factor may be indicated by ‘C’.
Step deviation is obtained by dividing the deviation (of the actual value from the assumed mean) by a common factor.
Each step deviation is multiplied with its frequency to get ‘fd’. Sum total of fd’ is obtained to get Σfd’.
Σfd’ is divided by Σf, and then multiplied by the common factor ‘C’. The resultant value is added to A to get the actual average of the series.

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – STEP DEVIATION METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)
64	8
63	18
62	12
61	9
60	7
59	6
	Σf = 60

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – STEP DEVIATION METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)	d (X – 62) (3)
64	8	64 – 62 = +2
63	18	64 – 63 = +1
62	12	62 – 62 = 0
61	9	61 – 62 = -1
60	7	60 – 62 = -2
59	6	59 – 62 = -3
	Σf = 60

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – STEP DEVIATION METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)	d (X – 62) (3)	Fd (4)
64	8	64 – 62 = +2	+16
63	18	64 – 63 = +1	+18
62	12	62 – 62 = 0	0
61	9	61 – 62 = -1	-9
60	7	60 – 62 = -2	-14
59	6	59 – 62 = -3	-18
	Σf = 60

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – STEP DEVIATION METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)	d (X – 62) (3)	Fd (4)
64	8	64 – 62 = +2	+16/10
63	18	64 – 63 = +1	+18/10
62	12	62 – 62 = 0	0/10
61	9	61 – 62 = -1	-9/10
60	7	60 – 62 = -2	-14/10
59	6	59 – 62 = -3	-18/10
	Σf = 60

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – STEP DEVIATION METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)	d (X – 62) (3)	Fd (4)	fd’ (5)
64	8	64 – 62 = +2	+16/10	1.6
63	18	64 – 62 = +1	+18/10	1.8
62	12	62 – 62 = 0	0/10	0
61	9	61 – 62 = -1	-9/10	-0.9
60	7	60 – 62 = -2	-14/10	-1.4
59	6	59 – 62 = -3	-18/10	-1.8
	Σf = 60

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ARITHMETIC MEAN FOR GROUPED DATA (DISCRETE SERIES) – STEP DEVIATION METHOD

Farm Size in acres (X) (1)	No. of cultivating households (f) (2)	d (X – 62) (3)	Fd (4)	fd’ (5)
64	8	64 – 62 = +2	+16/10	1.6
63	18	64 – 63 = +1	+18/10	1.8
62	12	62 – 62 = 0	0/10	0
61	9	61 – 62 = -1	-9/10	-0.9
60	7	60 – 62 = -2	-14/10	-1.4
59	6	59 – 62 = -3	-18/10	-1.8
	Σf = 60		Σfd = - 7	Σfd’ = - 0.7

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Step 1 - X = A + Σfd’ _x c

Σf

Step 2 - A = 62, Σfd’ = -0.7, Σf = 60, c = 10

Step 3 - X = 62 + (-0.7) _x 10 =

60

Step 4 - X = 61.88 Ans