Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce PDF Download

Measures of Central Tendency

Generally, average value of a distribution in the middle part of the distribution, such type of values are known as measures of central tendency.

The following are the five measures of central tendency

1. Arithmetic Mean

2. Geometric Mean

3. Harmonic Mean

4. Median

5. Mode

Arithmetic Mean

The arithmetic mean is the amount secured by dividing the sum of values of the items in a series by the number.

1. Arithmetic Mean for Unclassified Data

If n numbers, x₁, x₂, x₃,….., x_n, then their arithmetic mean

2. Arithmetic Mean for Frequency Distribution

Let f₁, f₂ , f_n be corresponding frequencies of x₁, x₂,…, x_n. Then,

3. Arithmetic Mean for Classified Data

Class mark of the class interval a-b, then x = a + b / 2

For a classified data, we take the class marks x₁, x₂,…, x_n of the classes as variables, then arithmetic mean

Step Deviation Method

where, A₁ = assumed mean

u_i = x_i – A₁ / h

f_i = frequency

h = width of interval

4. Combined Mean

If x₁, x₂,…, x_r be r groups of observations, then arithmetic mean of the combined group x is called the combined mean of the observation

A = n₁ A₁ + n₂A₂ +….+ n_rA_r / n₁ + n₂ +…+ n_r

A_r = AM of collection x_r

n_r = total frequency of the collection x_r

5. Weighted Arithmetic Mean

If w be the weight of the variable x, then the weighted AM

A_w = Σ wx / Σ w

Shortcut Method

A_w = A_w‘ + Σ wd / Σ w, A_w‘ = assumed mean

Σ wd = sum of products of the deviations and weight

Properties of Arithmetic Mean

(i) Mean is dependent of change of origin and change of scale.

(ii) Algebraic sum of the deviations of a set of values from their arithmetic mean is zero.

(iii) The sum of the squares of the deviations of a set of values is minimum when taken about mean.

Geometric Mean

If x₁, x₂,…, x_n be n values of the variable, then

G = ⁿ√x₁, x₂,…, x_n

or G = antilog [log x₁ + log x₂ + … + log x_n / n]

For Frequency Distribution

or 

Harmonic Mean (HM)

The harmonic mean of n items x₁, x₂,…, x_n is defined as

If their corresponding frequencies f₁, f₂,…, f_n respectively, then