Question

From the following frequency distribution, find:

1. Median
2. lower quartile
3. upper quartile
4. interquartile range

Variate	15	18	20	22	25	27	30
Frequency	4	6	8	9	7	8	6

Answer 1

Quartiles
Observation variables contain many kinds of quartiles.
Median is defined as the central value (or middle observation) of a statistical data if it is arranged in ascending or descending order. We will assume n as the total number of observations, then
Median = \left\\{ \dfrac{{n + 1}}{2}th\;observation,\;if\;n\;is\;odd \\\ \dfrac{{\dfrac{n}{2}th + \left( {\dfrac{n}{2} + 1} \right)th\;observation}}{2},\;if\;n\;is\;even \\\ \right.
The second quartile, or median, is the value that cuts off the first fifty percent.
Lower{\text{ }}Quartile = \left\\{ \dfrac{{n + 1}}{4}th\;observation,\;if\;n\;is\;odd \\\ \dfrac{n}{4}th\;observation,\;if\;n\;is\;even \\\ \right.
The first quartile, or lower quartile, is the value that cuts off the first twenty five percent of the data when it is sorted in ascending order.
Upper{\text{ }}Quartile = \left\\{ \dfrac{{3\left( {n + 1} \right)}}{4}th\;observation,\;if\;n\;is\;odd \\\ \dfrac{{3n}}{4}th\;observation,\;if\;n\;is\;even \\\ \right.
The third quartile, or upper quartile, is the value that cuts off the first seventy five percent.
$Inter{\text{ }}quartile - range{\text{ }} = {\text{ }}upper{\text{ }}quartile{\text{ }} - lower{\text{ }}quartile$

Complete step by step solution:
We write the variates in a cumulative frequency table. A cumulative frequency distribution is defined as the sum of frequency distribution of that class and all classes below it. In easy language we can say that means is you’re adding up a value and all of the values that came before it

Variate	Frequency (f)	Cumulative frequency
15	4	4
18	6	10
20	8	18
22	9	27
25	7	34
27	8	42
30	6	48

Before we start solving the question, we need to observe Here the total number of observations, n = 48 which is even. So, all the formulas will even work here.
Formula for Median $Median = \dfrac{{\dfrac{n}{2}th + \left( {\dfrac{n}{2} + 1} \right)th\;observation}}{2}$So, $median = \dfrac{1}{2}{\text{ }}\left( {{\text{ }}{{\left( {\dfrac{n}{{2\;}}{\text{ }}} \right)}^{th}}\;term + {\text{ }}{{\left( {\left( {\dfrac{n}{2}} \right) + 1} \right)}^{th}}\;term} \right)$ \Rightarrow median = \dfrac{1}{2}{\text{ }}\left( {{\text{ }}{{\left( {\dfrac{{48}}{{2\;}}} \right)}^{th}}\;term{\text{ }} + {\text{ }}{{\left( {\left( {\dfrac{{48}}{2}} \right) + 1} \right)}^{th}}\;term} \right)$$$$ \Rightarrow median = \dfrac{1}{2}{\text{ }}\left( {{\text{ 2}}{{\text{4}}^{th}}\;term{\text{ }} + {\text{ }}{{25}^{th}}\;term} \right)
$\Rightarrow median = \dfrac{1}{2}{\text{ }}\left( {{\text{ 22 }} + {\text{ }}22} \right)$ [Here we can see from the table that all observations from 19th to 27th are 22]
$\Rightarrow median = \dfrac{1}{2}{\text{ }}\left( {{\text{ }}44} \right)$
$\Rightarrow median = 22$
Hence the median is 22.

Lower quartile, ${Q_1}\; = {\text{ }}{\left( {\dfrac{n}{4}} \right)^{\;th}}\;term$
$\Rightarrow {Q_1}\; = {\text{ }}{\left( {\dfrac{{48}}{4}} \right)^{\;th}}\;term$
$\Rightarrow {Q_1}\; = {\text{ }}{\left( {12} \right)^{\;th}}\;term$
$\Rightarrow {Q_1}\; = {\text{ }}20$
Hence the lower quartile is 20.

Upper quartile, ${Q_3}\; = {\text{ }}{\left( {\dfrac{{3n}}{4}} \right)^{\;th}}\;term$
$\Rightarrow {Q_3}\; = {\text{ }}{\left( {\dfrac{{3 \times 48}}{4}} \right)^{\;th}}\;term$
$\Rightarrow {Q_3}\; = {\text{ }}{\left( {3 \times 12} \right)^{\;th}}\;term$
$\Rightarrow {Q_3}\; = {\text{ }}{36^{\;th}}\;term$
$\Rightarrow {Q_3}\; = {\text{ }}27$
Hence the upper quartile is 27.

Interquartile range:
$Inter{\text{ }}quartile - range{\text{ }} = {\text{ }}upper{\text{ }}quartile{\text{ }} - lower{\text{ }}quartile$
$\Rightarrow Inter{\text{ }}quartile - range{\text{ }} = {\text{ }}{Q_3}{\text{ }} - {Q_1}$
$\Rightarrow Inter{\text{ }}quartile - range{\text{ }} = {\text{ }}27{\text{ }} - 20 = 7$
Therefore, the Interquartile range for above mentioned data is 7.

Note:
The Interquartile range has been used to measure the spread out of the data points in a data set from the mean of the same data set. The higher the IQR, it determines that the more spread out the data points; whereas, the smaller the IQR, it tells us that the more bunched up the data points are around the mean. The IQR range is the most important way to measure how spread out the data points in a data set are. It is best used with other methods of measurements such as the median and total range to get the whole image of a data set’s tendency to cluster around its mean.