MEASURE OF CENTRAL TENDENCY (QUARTILES AND DECILES)

QUARTILES

The three values which divide the distribution into four equal parts are called quartiles. These values are denoted by Q1, Q2 and Q3 respectively. Ql is called lower quartile, Q2 is also called median and Q3 is called upper quartile.

Q1 = (n/4)th value of the data

Q2= (n/2)th value of the data

Q3= (3n / 4)^th value of the data

It is important to note that n is the total number of frequency. i.e. n=∑f

DECILE AND PERCENTILE

Deciles divide the data into ten equal parts:

D1 = (n / 10)th value of the data

D7= (7n / 10)^th value of the data

Percentiles divide the data into hundred equal parts.

P10 = (10n/100)

th value of the data

P₇₃= (73n /100)th value of the data

Note: For odd Number of observations we use n + 1 in the place of n, above mentioned

Formulas.

Example

Data Set: 6, 47, 49, 15, 42, 41, 7, 39, 43, 40, 36

For finding

Median i.e. Q2, Q1 and Q3 first we have to arrange set of data into ascending order.
Ordered Data Set: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49

Q1 = (n+1/4)th value of the data ( n+1 because of odd number we add 1)

Q1=11+1/4

Q1=3^rd value that is 15

Q1=15

Q2= (n+1/2)th value of the data

Q2=11+1/2

Q2=6^th value that is 40

Q2=40

Q3= (3n +1/ 4)^th

Q3=43

THE RANGE

It is the simplest way to measure of dispersion. The range is defined as the difference b the largest and the smallest observations in a set of data.

Range = largest value - smallest value

THE INTER QUARTILE RANGE

The inter-quartile range is a measure of dispersion; define by the difference between the third and the first quartiles.

Inter Quartile Range=Q3-Q1

SEMI INTER-QUARTILE RANGE

The half difference between the third and first quartiles is called semi inter-quartile range.

SEMI INTER-QUARTILE RANGE= (Q3-Q1)/2

From the above Example:

Data Set: 6, 47, 49, 15, 42, 41, 7, 39, 43, 40, 36

For finding

median i.e. Q2, Q1 and Q3 first we have to arrange set of data into ascending order.
Ordered Data Set: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49

Q1=15, Q2=40, Q3=43

Range =largest-smallest

Range=49-6

Range=43

Inter Quartile Range=Q3-Q1

=43-15

=28

SEMI INTER-QUARTILE RANGE= (Q3-Q1)/2

=28/2

=14

GRAPHICAL METHOD

http://www.onlinemathlearning.com/image-files/percentile_clip_image002.gif

upper quartile=3n/4

upper quartile=30^th value

upper quartile=52

In the same way we will find Q2, Q1 and so interquartile range.