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Define:

The Measurement of central tendency is a summary of data that represents the center point or typical value of a dataset. These measures indicate where most values in a distribution fall and are also referred to as the central location of a distribution.

In statistics, there are three most common measures of central tendency named as mean, median, and mode. Each of these measures calculates the location of the central point using a different process.

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Measures of Central Tendency

(i) It is an average of a distribution , having single value.

(ii) It represents whole distribution .

(iii) Measures of central tendency exhibit a tendency to cluster or centre around a specific value.

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Objective of Measures of Central Tendency

(i) To present vast data in a summarized form .

(ii) Helpful to compare and to establish relationship .

(iii) Very helpful in decision making .

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Requisites of a Good Statistical Average

(i) Defined firmly

(ii) Simple in computation.

(iii) Easy to understand.

(iv) Based on all observations

(v) Allow further algebraic treatment.

(vi)There is no affect of extreme observations.

(vii)The least affect of sampling fluctuations.(sampling stability)

(viii)Able to calculate in open-end class interval.

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Types of Averages

Classified into two categories .

(i) Mathematical Averages :

(a) Arithmetic Mean or Mean .

(b) Geometric Mean .

(c) Harmonic Mean .

(ii) Positional Averages :

( a) Median .

(b) Mode .

(c) Quantiles or fractiles or partition values (Quartiles, Deciles , Percentiles).

Calculation of Median:

(i) For raw data:

(a) Arrange the data in ascending order.

(b) Apply the formula (n + 1)/2 ^th term. Where n is number of terms.

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(ii) For ungrouped data:

(a) Find cumulative frequency

(b) Apply the formula (∑f + 1)/2 ^th term.

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(iii) For grouped data:

(a) calculate N/2, which give us median class.

(b) Find cumulative frequency

(c) Apply the formula: = L₁ + (

N/2

- p.c.f) x

h/f

L₁ = Lower limit of median class

N = total number of observations

p.c.f. = previous class cumulative frequency to median class .

h = class width

f = frequency of median class .

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Calculation of Mode:

(i) For raw data:

The data which is repeated many times will be the mode.

for example : x = 4,6,7,5,4,8,9,4,8,10

in this case the mode is 4.

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(ii) For ungrouped data:

In this case the value which has maximum frequency will be the mode

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(iii) For grouped data :

= L₁ +

(f₁ - f_o) x h/ (f₁ - f_o) + (f₁ - f₂)

= L₁ +

(f₁ - f_o) x h/ 2f₁ - f_o - f₂

In this case first find modal class , the class which has maximum frequency will be considered as modal class

L₁ = Lower limit of modal class

f₁ = frequency of modal class

f_o = frequency of previous class to modal class

f₂ = frequency of next class to modal class

h = class width.

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