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Steps in Calculating Mean

Dr. Anshul Singh Thapa

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Introduction

  • When scores or other measures have been tabulated into a frequency distribution, usually the next task is to calculate a measure of central tendency, or central position.
  • The value of a measure of central tendency is twofold. First, it is an average which represent all of the scores made by the group, and as such gives a concise description of the performance of the group as a whole; and second it enable us to compare two or more groups in terms of typical performance.
  • There are several statistical measures of central tendency or “averages”. The three most commonly used averages are:
    • Arithmetic Mean
    • Median
    • Mode

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Arithmetic mean

  • Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X .
  • In general, if there are N observations as X1, X2, X3, ..., XN, then the Arithmetic Mean is given by

X = X1 + X2 + X3 +…+ XN

N

= ΣX

N

  • Where, ΣX = sum of all observations and N = total number of observations.

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ARITHMETIC MEAN

ARITHMETIC MEAN FOR UNGROUPED DATA

ARITHMETIC MEAN FOR GROUPED DATA

DIRECT METHOD

ASSUMED MEAN METHOD

STEP DEVIATION METHOD

DISCRETE SERIES

CONTINUOUS SERIES

DIRECT METHOD

ASSUMED MEAN METHOD

STEP DEVIATION METHOD

DIRECT METHOD

STEP DEVIATION METHOD

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How Arithmetic Mean is Calculated

  • The calculation of arithmetic mean can be studied under two broad categories:
    • Arithmetic Mean for Ungrouped Data.
    • Arithmetic Mean for Grouped Data.

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    • Arithmetic mean by direct method is the sum of all observations in a series divided by the total number of observations. Following steps are involved in this method:
      • Add up values of all the items of a series, (ΣX)
      • find out total number of items in the series (N)
      • Divide the total of value (ΣX) of all the items with the number of items (N) thus,

X = ΣX

N

​

​

ARITHMETIC MEAN FOR UNGROUPED DATA – DIRECT METHOD

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ARITHMETIC MEAN FOR UNGROUPED DATA – DIRECT METHOD

Employee

Monthly Income (Rs.)

X

1

1780

2

1760

3

1690

4

1750

5

1840

6

1920

7

1100

8

1810

9

1050

10

1950

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ARITHMETIC MEAN FOR UNGROUPED DATA – DIRECT METHOD

Employee

Monthly Income (Rs.)

X

1

1780

2

1760

3

1690

4

1750

5

1840

6

1920

7

1100

8

1810

9

1050

10

1950

N = 10

ΣX = 16,650

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Step 1 - X= ΣX

N

Step 2 - ΣX = 16650, N=10

​

​

Step 3 - X = 16650 = 1665

10

Step 4 - X = 1665 Ans

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ARITHMETIC MEAN FOR UNGROUPED DATA – ASSUMED MEAN METHOD

Employee

Income

(X)

1

1780

2

1760

3

1690

4

1750

5

1840

6

1920

7

1100

8

1810

9

1050

10

1950

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ARITHMETIC MEAN FOR UNGROUPED DATA – ASSUMED MEAN METHOD

Employee

Income

(X)

1

1780

2

1760

3

1690

4

1750

5

1840

6

1920

7

1100

8

1810

9

1050

10

1950

N = 10

​

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ARITHMETIC MEAN FOR UNGROUPED DATA – ASSUMED MEAN METHOD

Employee

Income

(X)

Deviation (D = X – A)

(X - 1800)

1

1780

​

2

1760

​

3

1690

​

4

1750

​

5

1840

​

6

1920

​

7

1100

​

8

1810

​

9

1050

​

10

1950

​

N = 10

​

​

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ARITHMETIC MEAN FOR UNGROUPED DATA – ASSUMED MEAN METHOD

Employee

Income

(X)

Deviation (D = X – A)

(X - 1800)

1

1780

1780 – 1800 = -20

2

1760

1760 – 1800 = -40

3

1690

1690 – 1800 = -110

4

1750

1750 – 1800 = -50

5

1840

1840 – 1800 = +40

6

1920

1920 – 1800 = +120

7

1100

1100 – 1800 = -700

8

1810

1810 – 1800 = +10

9

1050

1050 – 1800 = -750

10

1950

1950 – 1800 = +150

N = 10

​

​

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ARITHMETIC MEAN FOR UNGROUPED DATA – ASSUMED MEAN METHOD

Employee

Income

(X)

Deviation (D = X – A)

(X - 1800)

1

1780

1780 – 1800 = -20

2

1760

1760 – 1800 = -40

3

1690

1690 – 1800 = -110

4

1750

1750 – 1800 = -50

5

1840

1840 – 1800 = +40

6

1920

1920 – 1800 = +120

7

1100

1100 – 1800 = -700

8

1810

1810 – 1800 = +10

9

1050

1050 – 1800 = -750

10

1950

1950 – 1800 = +150

N = 10

​

Σd = -1350

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Step 1 - X = A + (Σd)

N

Step 2 - A = 1800, Σd = - 1350, N = 10

​

Step 3 - X = 1800 – 1350 = 1800 – 135 =

10

Step 4 - X = 1665 Ans

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ARITHMETIC MEAN FOR UNGROUPED DATA – STEP DEVIATION METHOD

Employee

Income

(X)

1

1780

2

1760

3

1690

4

1750

5

1840

6

1920

7

1100

8

1810

9

1050

10

1950

N = 10

​

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ARITHMETIC MEAN FOR UNGROUPED DATA – STEP DEVIATION METHOD

Employee

Income

(X)

Deviation (D = X – A)

(X - 1800)

1

1780

1780 – 1800 = -20

2

1760

1760 – 1800 = -40

3

1690

1690 – 1800 = -110

4

1750

1750 – 1800 = -50

5

1840

1840 – 1800 = +40

6

1920

1920 – 1800 = +120

7

1100

1100 – 1800 = -700

8

1810

1810 – 1800 = +10

9

1050

1050 – 1800 = -750

10

1950

1950 – 1800 = +150

N = 10

​

​

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ARITHMETIC MEAN FOR UNGROUPED DATA – STEP DEVIATION METHOD

Employee

Income

(X)

Deviation (D = X – A)

(X - 1800)

(d’ = X – A)

C

1

1780

1780 – 1800 = -20

-20/ 10 = -2

2

1760

1760 – 1800 = -40

-40/10 = -4

3

1690

1690 – 1800 = -110

-110/10 = -11

4

1750

1750 – 1800 = -50

-50/10 = -5

5

1840

1840 – 1800 = +40

+40/10 = 4

6

1920

1920 – 1800 = +120

+120/10 = 12

7

1100

1100 – 1800 = -700

-700/10 = -70

8

1810

1810 – 1800 = +10

+10/10 = 1

9

1050

1050 – 1800 = -750

-750/ 10 = -75

10

1950

1950 – 1800 = +150

+150/10 = 15

N = 10

​

​

​

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ARITHMETIC MEAN FOR UNGROUPED DATA – STEP DEVIATION METHOD

Employee

Income

(X)

Deviation (D = X – A)

(X - 1800)

(d’ = X – A)

C

1

1780

1780 – 1800 = -20

-20/ 10 = -2

2

1760

1760 – 1800 = -40

-40/10 = -4

3

1690

1690 – 1800 = -110

-110/10 = -11

4

1750

1750 – 1800 = -50

-50/10 = -5

5

1840

1840 – 1800 = +40

+40/10 = 4

6

1920

1920 – 1800 = +120

+120/10 = 12

7

1100

1100 – 1800 = -700

-700/10 = -70

8

1810

1810 – 1800 = +10

+10/10 = 1

9

1050

1050 – 1800 = -750

-750/ 10 = -75

10

1950

1950 – 1800 = +150

+150/10 = 15

N = 10

​

Σd = -1350

Σd’ = -135

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Step 1 - X = A + Σd’ x c

N

Step 2 - A = 1800, Σd’ = - 135, N = 10, c = 10

​

Step 3 - X = 1800 – 135 x 10 = 1800 – 135 =

10

Step 4 - X = 1665 Ans