Interquartile range - class-XII

Given,  lower quartile $=60$ and upper quartile $=80$ 
$\therefore$ median $=\frac{\mbox{
lower quartile
}+\mbox{
upper quartile
}}{2}=\dfrac{60+80}{2}=70$

The range is defined as the difference between the maximum and minimum values. It is limited because it ignores the distribution of intermediate values, is highly sensitive to outliers, and cannot be computed for open-ended distributions.

Range is the difference between the maximum and minimum values. Because it only considers these two extremes, it is highly sensitive to outliers and does not account for the distribution of all data points.

Quartile deviation is defined as half of the interquartile range, which is why it is also known as the semi-interquartile range.

Range refers to the value of variation between the largest and the smallest values in a particular data set. It can be calculated as the difference between the upper limit and lower limit of a particular set of data.

In a frequency distribution, the range is calculated as the difference between the upper limit of the highest class interval and the lower limit of the lowest class interval.

The median divides an ordered series into two equal halves. The 2nd quartile (Q2), 5th decile (D5), and 50th percentile (P50) all represent this same middle point.

Percentiles are values that divide a dataset into 100 equal parts, where each part represents 1% of the data.

Quartile deviation is based on the interquartile range (Q3 - Q1), which covers the middle 50% of the observations, effectively ignoring the extreme 25% at both ends.

The median is the 50th percentile, the 5th decile, and the 2nd quartile. Therefore, specific quartiles, deciles, and percentiles can indeed be the median.

Quartile deviation is calculated using only the middle 50% of the data, making it robust against extreme values at the tails, which allows it to be calculated for open-ended distributions.

The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1), which encompasses the middle 50% of the data.

Quartile deviation is defined as (Q3 - Q1) / 2, which is equivalent to half of the interquartile range.

The first quartile (Q1) represents the 25th percentile, meaning 25% of the data is below it and 75% is above it. The ratio of 25:75 simplifies to 1:3.

The arithmetic mean requires the exact values of all observations. In open-ended distributions, the exact values of the first or last classes are unknown, making the mean impossible to calculate.

Median is the most suitable measure for qualitative data (ordinal data) because it is based on the position of values rather than their magnitude, which cannot be averaged numerically.

The first quartile (Q1) is the value below which 25% of the data falls, and it is commonly referred to as the lower quartile.

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.

Given series of numbers is already arranged in ascending order.
Thus median is the centre value $ = 20 $
First Quartile $ {Q} _{1} $ will be the mid value of the observations to the left of $ 20 $ which is $ 9 $. 
Similarly, Third Quartile $ {Q} _{3} $ will be the mid value of the observations to the right of $ 20 $ which is $ 34 $.
And Quartile deviation will be $ \dfrac {{Q} _{3} - {Q} _{1}}{2} = \dfrac {34-9}{2} = \dfrac {25}{2} = 12.5 $.

use:  M.D.$=\displaystyle \frac{4}{4}\sigma ,Q.D=\frac{2}{3}\sigma $ $\displaystyle \Rightarrow \frac{M.D.}{Q.D}=\frac{6}{5}\Rightarrow Q.D.=\frac{5}{6}\left ( M.D. \right )$


$MD=\frac { 4 }{ 5 } \times \sigma \quad ,\quad QD=\frac { 2 }{ 3 } \times \sigma \quad ,\quad \ \frac { MD }{ QD } =\frac { 6 }{ 5 } ,\quad QD\quad =\quad \frac { 5\times 15 }{ 6 } =12.5$

Lower Quartile $ {Q} _{1} = 4 $

Upper Quartile $ {Q} _{3} = 12 $

And Quartile deviation will be $ \frac {{Q} _{3} - {Q} _{1}}{2} = \frac {12-4}{2} = \frac {8}{2} = 4 $

Range, mean deviation, and standard deviation are all measures of dispersion (spread). A quartile is a partition value, not a measure of dispersion itself.

For grouped data, the median is calculated using an interpolation formula based on the median class interval.

Range is the difference between the maximum and minimum values. If every value is decreased by 25, both the maximum and minimum decrease by 25, so the difference (max - min) remains unchanged.

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion.

The coefficient of quartile deviation is calculated as (Q3 - Q1) / (Q3 + Q1). Given Q3 = 64 and Q1 = 50, the calculation is (64 - 50) / (64 + 50) = 14 / 114, which is approximately 0.1228 or 12.28%.

To find Q3, calculate the cumulative frequency: 3, 9, 18, 36, 56, 70, 80, 90, 97, 99. The total N is 99, so the 3rd quartile position is 0.75 * (99 + 1) = 75th value, which corresponds to X = 8.

First, sort the data: 15, 18, 28, 29, 30, 34, 34, 35, 42, 45. The 8th decile (D8) is the 0.8 * (10 + 1) = 8.8th position. Interpolating between the 8th (35) and 9th (42) values: 35 + 0.8 * (42 - 35) = 35 + 5.6 = 40.6.

The intersection of the 'less than' ogive and the 'more than' ogive on a graph provides the median of the frequency distribution.

Median is the middle value, which corresponds to the 50th percentile (P50), the 5th decile (D5), and the 2nd quartile (Q2). D4 represents the 40th percentile, not the median.

Total spending = (2 * 8) + (3 * 10) + (5 * 6) = 16 + 30 + 30 = 76. Total students = 10. Average = 76 / 10 = 7.6.

Total frequency N = 99. The 82nd percentile position is 0.82 * (99 + 1) = 82nd value. Looking at the cumulative frequencies (3, 9, 18, 36, 56, 70, 80, 90, 97, 99), the 82nd value falls within the category X = 9.

Measure of dispersion is a statistical method to find a specific value from a dispersed series where the extent to which various values of the series tend to disperse from each other or from the average is measured. So all the measures of dispersion from range to standard deviation has positive value. 

Range is the most simple and commonly understandable measures of dispersion.Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. Therefore, it is the most affected measures of dispersion by the extreme values of the series. 

Semi inter-quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q₁ and the third quartile is denoted as Q₃ .

Semi inter-quartile deviation = (Q₃-Q₁) /2

                             = (29-17)/2

                             = 12/2

                             = 6

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most affected measures of dispersion by the extreme values of the series therefore it has the lowest degree of reliability. 

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion.

Quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q₁ and the third quartile is denoted as Q₃ .

The coefficient of quartile deviation is found by dividing the difference between the first quartile and the third quartile by the sum between the first quartile and the third quartile.

In the given series, 10, 15,17,18,20,22,25,28,29,32,34. 

Q₁= 17 and Q₃=29 

Coefficient of quartile deviation =(Q₃-Q₁)/ (Q₃+Q₁)

                                            = (29-17)/ (29+17)

                                            =
12/46 = 0.26 

Variance = sum((x_i - mean)^2) / n = 2. For data 1, 2, 4, 5, x, the mean is (12 + x) / 5. Solving for x where variance is 2 leads to x = 3.

Quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q₁ and the third quartile is denoted as Q₃ . Therefore, quartile deviation is not affected by the extreme values of the series. 

Quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q₁ and the third quartile is denoted as Q₃ .

In the given series, 10, 15,17,18,20,22,25,28,29,32,34 

Q₁= 17 and Q₃=29

Therefore, Quartile deviation = (Q₃-Q₁) /2

                                                 = (29-17)/2

                                                  = 12/2

                                                  = 6

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion. Therefore, it is the most affected measures of dispersion by the extreme values of the series. 

Range = H - L 

In the given series, H= 67 and L= 12 

Range = 67 -12 = 55 .

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion.

Range = 55

In the given series, H= x and L= 15

Range => x-15 = 55

            => x = 55+15 = 70 

An ogive is a graphical representation of a cumulative frequency distribution, where points are plotted at the upper limits of intervals.

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion. 

Range = 60 

In the given series, H= 80 and L= x 

Range => 80-x = 60

            => x = 80-60 = 20 

Quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q₁ and the third quartile is denoted as Q₃ .

Quartile deviation = (Q₃-Q₁) /2

                             = 74/2

                             = 37

Quartile deviation also known as semi inter-quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q₁ and the third quartile is denoted as Q₃ .

Quartile deviation or semi inter-quartile deviation = (Q₃-Q₁) /2

Sturges' rule and general statistical practice suggest that for most data sets, having between 5 and 20 classes provides a good balance between detail and clarity.

Quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q₁ and the third quartile is denoted as Q₃ . Since quartile deviation is not affected by the extreme values of the series. Therefore, it can be calculated in case of open end intervals also. 

In a normal distribution, the inter-quartile range (IQR) is approximately 1.3489 times the standard deviation. 1.3489 * 5 = 6.7445, which is closest to 6.67.

The coefficient of variation is a relative measure of dispersion, not an absolute one, because it is expressed as a percentage or ratio of the mean.

Class width is the difference between the upper boundary of a class and the lower boundary. For 0-9, 10-19, the width is 10 - 0 = 10.