Tag: quartile deviation or semi-interquartile range

The third quartile (Q3) is the value below which 75% of the data falls, and it is commonly referred to as the upper quartile.

To divide a series into four equal parts, you need three partition points (Q1, Q2, and Q3).

A fundamental property of the arithmetic mean is that the sum of the deviations of all observations from the mean is always zero.

The arithmetic mean is calculated by summing all values in a dataset and dividing by the count, making it dependent on every single observation.

The cumulative frequency of the table is a given below:

Lower Quartile: 
We have $\displaystyle\frac{N}{4} = \displaystyle\frac{49}{4} = 12.25$
The cumulative frequency just greater than $\displaystyle\frac{N}{4}$ is $26$ and the corresponding value of the variable is $61$
$\therefore\quad Q _1 = Rs.\space 61$

Middle Quartile (median): 
We have $\displaystyle\frac{N}{2} = \displaystyle\frac{49}{2} = 24.5$
The cumulative frequency just greater than $\displaystyle\frac{N}{2}$ is $26$ and the corresponding value of the variable is $61$
$\therefore\quad Q _2 = Rs.\space 61$

Upper Quartile:
We have $\displaystyle\frac{3N}{4} = \displaystyle\frac{3\times49}{4} = 36.75$
The cumulative frequency just greater than $\displaystyle\frac{3N}{4}$ is $41$ and the corresponding value of the variable is $63$
$\therefore\quad Q _3 = Rs.\space 63$

$\therefore\quad \displaystyle\frac{Q _1+Q _3}{2} = \displaystyle\frac{61+63}{2} = 62$

$N=12$
${ Q } _{ 1 }=$ Size of $\displaystyle\frac { N+1 }{ 4 } th=3.25th$ item
$=$ size of $3rd$ item $+0.25$(size of $4th$ item)$-$(size of $3rd$ item)
$=151+0.25(151-151)=$Rs.$151$
${ Q } _{ 3 }=$ Size of $\displaystyle\frac { 3(N+1) }{ 4 } th=9.75th$ item
$=$ size of $9th$ item $+0.75$(size of $10th$ item)$-$(size of $9th$ item)
$=162+0.75(162-162)=$Rs.$162$
$\therefore$ Quartile Deviation $\displaystyle=\frac { 1 }{ 2 } \left( { Q } _{ 3 }-{ Q } _{ 1 } \right) =\frac { 11 }{ 2 } =$Rs.$5.5$

Writing the series in ascending order $7,10,12,15,17,19,25$
$\displaystyle{ Q } _{ 1 }=\frac { 15+7 }{ 2 } =11,{ Q } _{ 3 }=\frac { 25+15 }{ 2 } =20$
$\therefore$ Quartile Deviation
$\displaystyle=\frac { { Q } _{ 3 }-{ Q } _{ 1 } }{ 2 } =\frac { 20-11 }{ 2 } =4.5$

The cumulative frequency table is as given below

Lower Quartile: We have, $\displaystyle\frac{N}{4} = \displaystyle\frac{543}{4} = 135.75$
Cumulative frequency just greater than $N/4$ is $196$ and the corresponding value of the variable is $40$.
$\therefore\quad Q _1 = $ Lower Quartile = $40$ years
Middle Quartile (Medium):
We have, $\displaystyle\frac{N}{2} = \displaystyle\frac{543}{2} = 271.5$
Cumulative frequency just greater than $N/2$ is $349$ and the corresponding value of the variable is $50$.
$\therefore\quad Q _2 = $ Middle quartile = $50$ years
Upper Quartile:
We have $\displaystyle\frac{3N}{4} = \displaystyle\frac{3\times543}{4} = 407.25$
Cumulative frequency just greater than $407.25$ is $489$ and the corresponding value of the variable is $60$.
$\therefore\quad Q _3= $ Upper quartile  $=60$ years
$\therefore\quad Q _1 + Q _3 - Q _2 = 40+60-50 = 50 = Q _2$

Quartile deviation is the half of the inter quartile deviation.