Tag: histograms and frequency distribution diagrams

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.