**Central Tendency: -**The tendency of the individual item of a statistical series to cluster around the central value is called the Central Tendency. Sometimes it is called the measure of location or a measure of representation.

Several types of Central Tendency can be defined: The commons are

- The Arithmetic Mean
- The Median
- The Mode
- The Geometric Mean
- The Harmonic Mean

**The Arithmetic Mean: -** The Arithmetic Mean of a grouped frequency distribution is defined as

A = any guessed or assumed class mark.

f = Frequency of each class interval.

n = Sum of total frequency.

i = Range of class interval.

d = Deviation of the assumed class mark from each class interval by the range of class interval.

d = (Xi – A) / i

**The Median: -** The Median of a grouped is defined as

Where,

M_{e} = Median of the total class.

f_{c} = Previous cumulative frequency of all classes above the media class.

f_{m} = Frequency of the corresponding class interval.

i = range of class interval.

L = Lower class boundary of median class.

n = Sum of total frequency.

**The Mode: -** The Mode of a set of number is that value which occurs with the greatest frequency.

The Mode for a grouped data/frequency distribution is denoted by

Where,

L = Lower limit of modal class interval.

∆_{1} = Difference between modal and pre-modal group.

∆_{2} = Difference between modal and post-modal group

i = range of class interval.

**The Geometric Mean: -** The Geometric Mean is for a grouped frequency distribution is denoted by

Where,

G = Geometric Mean

n = sum of total frequency.

f_{i} = Frequency corresponding each class interval.

x_{i} = Class Mark.

**The Harmonic Mean: - **The Harmonic Mean H for a grouped frequency distribution is

Where,

H = Harmonic Mean.

n = sum of total frequency.

f_{i} = Frequency corresponding each class interval.

x_{i} = Class Mark.

**Problem: - **

The given frequency is the efficiency score of 115 students in their 70% marks. Find the Arithmetic Mean, Median, Mode, Geometric Mean and Harmonic Mean.

Solution:

**
**

**Some special measurements following any section of Central Tendency:**

- Quartiles
- Deciles and
- Percentile

**Quartiles: -** The Quartiles are those values in a series which divide the total frequency into *four *equal parts. It is denoted by Q where

Where,

r = 1, 2, 3,…….

L_{r} = Lower limit of the Quartiles class,

n = Sum of the total frequency,

r = Position of Quartiles,

F_{r} = Cumulative frequency of the pre-rth Quartiles class,

f_{r} = Corresponding frequency,

i = Range of class interval.

**Deciles: -** The Deciles are those values in a series which divide the total frequency into *ten* equal parts. It is denoted by D where

Where,

r = 1, 2, 3,…….

L_{r} = Lower limit of the Deciles class,

n = Sum of the total frequency,

r = Position of Deciles,

F_{r} = Cumulative frequency of the pre-rth Deciles class,

f_{r} = Corresponding frequency,

i = Range of class interval.

**Percentiles: -** The Percentiles are those values in a series which divide the total frequency into *100 *equal parts. It is denoted by P where

Where,

r = 1, 2, 3,…….

L_{r} = Lower limit of the Percentiles class,

n = Sum of the total frequency,

r = Position of Percentiles,

F_{r} = Cumulative frequency of the pre-rth Percentiles class,

f_{r} = Corresponding frequency,

i = Range of class interval.

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