Central Tendency

Measures of Central Tendency are the methods to obtain the numerical values which are used to represent a large collection of numerical data. These obtained numerical values are called central or average values. A central or average value of any statistical data or series is the value of that variable that is representative of the entire data or its associated frequency distribution. Such a value is of great significance because it depicts the nature or characteristics of the entire data, which is otherwise very difficult to observe.

Some of the most commonly used measures of central tendency are:

• Arithmetic mean (AM)
• Geometric mean (GM)
• Harmonic mean (HM)
• Median
• Mode

However, for class 9 (CBSE) we shall study only the arithmetic mean, median, and mode for raw (ungrouped) data.

Arithmetic Mean (Mean)

(i) For Individual or Ungrouped data: Arithmetic mean (X’) is defined as the sum of the individual observations (x_i) divided by the total number of observations N.

X’ = (∑ x_i) ÷ N

Example: If there are 5 observations, which are 27, 11, 17, 19, 21 then Arithmetic mean (X’) is given by

X = (27 + 11 + 17 + 19 + 21) ÷ 5

X = 95 ÷ 5

X = 19

(ii) For Grouped data or Direct Method: Arithmetic mean (X’) is defined as the sum of the product of observations (x_i) and their corresponding frequencies (f_i) divided by the sum of all the frequencies (f_i).

X’ = (∑ x_i × f_i) ÷ (∑f_i)

Example: If the values (x_i) of the observations and their frequencies (f_i) are given as follows:

xi:	4	6	15	10	9
fi:	5	10	8	7	10

then Arithmetic mean (X’) of the above distribution is given by

X’ = (4×5 + 6×10 + 15×8 + 10×7 + 9×10) ÷ (5 + 10 + 8 + 7 + 10)

X’ = (20 + 60 + 120 + 70 + 90) ÷ 40

X’ = 360 ÷ 40

X’ = 9

Properties of Arithmetic mean

1. It can’t be calculated graphically.
2. The algebraic sum of deviations from arithmetic mean is zero; ∑ (x_i – X’) = 0.
3. If X is the arithmetic mean of observations and a is added to each of the observations, then the new arithmetic mean is given by X’ = X + a.
4. If X is the arithmetic mean of observations and a is subtracted to each of the observations, then the new arithmetic mean is given by X’ = X − a.
5. If X is the arithmetic mean of observations and a is multiplied to each of the observations, then the new arithmetic mean is given by X’ = X × a.
6. If X is the arithmetic mean of observations and each of the observations is divided by a, then the new arithmetic mean is given by X’ = X ÷ a.

Median

The Median of any distribution is that value that divides the distribution into two equal parts such that the number of observations above it is equal to the number of observations below it. To calculate the Median, the observations must be arranged in ascending or descending order. If the total number of the observations is N then

(i) Median = Value of observation at [(n + 1) ÷ 2]^th position; if N is an odd.

(ii) Median = Arithmetic mean of Values of observations at (n ÷ 2)^th and [(n ÷ 2) + 1]^th position; if N is an even.

Example 1: If the observations are 25, 36, 31, 23, 22, 26, 38, 28, 20, 32 then Median is given by 

Arranging the data in ascending order: 20, 22, 23, 25, 26, 28, 31, 32, 36, 38

N = 10 which is even then

Median = Arithmetic mean of values at (10 ÷ 2)th and [(10 ÷ 2) + 1]th position

Median = (Value at 5th position + Value at 6th position) ÷ 2

Median = (26 + 28) ÷ 2

Median = 27

Example 2: If the observations are 25, 36, 31, 23, 22, 26, 38, 28, 20 then Median is given by

Arranging the data in ascending order: 20, 22, 23, 25, 26, 28, 31, 36, 38

N = 9 which is odd then

Median = Value at [(9 + 1) ÷ 2]th position

Median = Value at 5th position

Median = 26

Properties of Median

1. It can be calculated graphically.
2. It is not affected by the extreme values i.e. Maximum or Minimum values in the data.

Mode

The Mode is the value of that observation which has a maximum frequency corresponding to it. It is calculated by simply inspecting the given ungrouped data.

Example: If the observations are 5, 3, 4, 3, 7, 3, 5, 4, 3 then Mode is given by

Xi	5	3	4	7
fi	2	4	2	1

Since 3 has occurred a maximum number of times i.e. 4 times in the given data;

Hence Mode = 3

Properties of Mode

1. It can also be calculated graphically.
2. It is also not affected by the extreme values i.e. Maximum or Minimum values in the data.

Notes

• For Symmetric Distributions (Mean = Median = Mode)
• For Asymmetric or Moderately Symmetric Distributions (Mode + 2 × Mean = 3 × Median)

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