Friday, January 29, 2010

Measures of Central Tendency

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

  1. The Arithmetic Mean
  2. The Median
  3. The Mode
  4. The Geometric Mean
  5. 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,

Me = Median of the total class.

fc = Previous cumulative frequency of all classes above the media class.

fm = 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.

fi = Frequency corresponding each class interval.

xi = Class Mark.

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




Where,

H = Harmonic Mean.

n = sum of total frequency.

fi = Frequency corresponding each class interval.

xi = 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:

Justify Full






Some special measurements following any section of Central Tendency:

  1. Quartiles
  2. Deciles and
  3. 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,…….

Lr = Lower limit of the Quartiles class,

n = Sum of the total frequency,

r = Position of Quartiles,

Fr = Cumulative frequency of the pre-rth Quartiles class,

fr = 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,…….

Lr = Lower limit of the Deciles class,

n = Sum of the total frequency,

r = Position of Deciles,

Fr = Cumulative frequency of the pre-rth Deciles class,

fr = 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,…….

Lr = Lower limit of the Percentiles class,

n = Sum of the total frequency,

r = Position of Percentiles,

Fr = Cumulative frequency of the pre-rth Percentiles class,

fr = Corresponding frequency,

i = Range of class interval.

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