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Quartile deviation equal to _______

Last updated date: 24th Jul 2024
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Hint: Quartile deviation is a measure of dispersion in statistics. Quartile deviation can be found for both ungrouped and grouped data.
Something that we have to know before getting into Quartile deviation are quartiles.
Quartiles are nothing but the values that divide the data into $3$ quarters, that is ${Q_1},{Q_2}$ and ${Q_3}$ .
${Q_2}$ is nothing but the median that is center value, ${Q_1}$ represents the part that lies above the median (upper quartile) and ${Q_1}$ represents the part that lies below the median (lower quartile).
Quartile deviation is half of the difference between the upper quartile and the lower quartile, which can be denoted as $QD$
For instance, ${Q_1}$ is the upper quartile, ${Q_2}$ is the center value and ${Q_3}$ is the lower quartile for the given data then quartile deviation is given by,
$QD = \dfrac{{({Q_3} - {Q_1})}}{2}$
where ${Q_1}$ is ${\left( {\dfrac{{n + 1}}{4}} \right)^{th}}$ term, ${Q_2}$ is ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$ term and ${Q_3}$ is ${\left( {\dfrac{{3\left( {n + 1} \right)}}{4}} \right)^{th}}$ term.
For a grouped data we first find the cumulative frequency if it is not given by adding the frequencies in each step. Then we will find the quartile class from the cumulative frequency, where the value of $\dfrac{N}{4}$ lies. Then we will apply the formula to find ${Q_1}$ and ${Q_3}$ .
${Q_r} = {l_1} + \dfrac{{r\left( {\dfrac{N}{4}} \right) - c}}{f}\left( {{l_2} - {l_1}} \right)$
where ${Q_r}$ is the ${r^{th}}$ quartile, ${l_1}$ is the lower limit of the quartile class, ${l_2}$ is the upper limit of the quartile class, $f$ is the frequency of the quartile class, $c$ is the cumulative frequency of the class preceding the quartile class and $N$ is the number of observations.
Then we will substitute ${Q_1}$ and ${Q_3}$ in the ungrouped deviation formula to get the required answer.