## Quartiles: Measure of relative standing of an observation within data

Like Percentile and Deciles, Quartiles is a type of Quantile, which is a measure of the relative standing of observation within the data set. Quartiles are the values are three points that divide the data into four equal parts each group comprising a quarter of the data (the first quartile $Q_1$, second quartile $Q_2$ (also median) and the third quartile $Q_3$) in the order statistics. The first quartile, (also known as the lower quartile) is the value of order statistic that exceeds 1/4 of the observations and less than the remaining 3/4 observations. The third quartile is known as upper quartile is the value in the order statistic that exceeds 3/4 of the observations and is less than remaining 1/4 observations, while the second quartile is the median.

**Quartiles for Ungrouped Data**

For ungrouped data, the quartiles are calculated by splitting the order statistic at the median and then calculating the median of the two halves. If *n* is odd, the median can be included in both sides.

**Example:** Find the $Q_1, Q_2$ and $Q_3$ for the following ungrouped data set 88.03, 94.50, 94.90, 95.05, 84.60.Solution: We split the order statistic at the median and calculate the median of two halves. Since *n* is odd, we can include the median in both halves. The order statistic is 84.60, 88.03, 94.50, 94.90, 95.05.

\begin{align*}

Q_2&=median=Y_{(\frac{n+1}{2})}=Y_{(3)}\\

&=94.50 (\text{the third observation})\\

Q_1&=\text{Median of the first three value}=Y_{(\frac{3+1}{2})}\\&=Y_{(2)}=88.03 (\text{the second observation})\\

Q_3&=\text{Median of the last three values}=Y_{(\frac{3+5}{2})}\\

&=Y_{(4)}=94.90 (\text{the forth observation})

\end{align*}

**Quartiles for Grouped Data**

For the grouped data (in ascending order) the quartiles are calculated as:

\begin{align*}

Q_1&=l+\frac{h}{f}(\frac{n}{4}-c)\\

Q_2&=l+\frac{h}{f}(\frac{2n}{4}-c)\\

Q_3&=l+\frac{h}{f}(\frac{3n}{4}-c)

\end{align*}

where*l* is the lower class boundary of the class containing the $Q_1,Q_2$ or $Q_3$.*h* is the width of the class containing the $Q_1,Q_2$ or $Q_3$.*f* is the frequency of the class containing the $Q_1,Q_2$ or $Q_3$.*c* is the cumulative frequency of the class immediately preceding to the class containing $Q_1,Q_2$ or $Q_3, \left[\frac{n}{4},\frac{2n}{4} \text{or} \frac{3n}{4}\right]$ are used to locate $Q_1,Q_2$ or $Q_3$ group.

**Example:** Find the quartiles for the following grouped data

**Solution:**To locate the class containing $Q_1$, find $\frac{n}{4}$th observation which is here $\frac{30}{4}$th observation i.e. 7.5th observation. Note that 7.5th observation falls in the group ($Q_1$ group) 90.5–95.5.

\begin{align*}

Q_1&=l+\frac{h}{f}(\frac{n}{4}-c)\\

&=90.5+\frac{5}{4}(7.5-6)=90.3750

\end{align*}

For $Q_2$, the $\frac{2n}{4}$th observation=$\frac{2 \times 30}{4}$th observation = 15th observation falls in the group 95.5–100.5.

\begin{align*}

Q_2&=l+\frac{h}{f}(\frac{2n}{4}-c)\\

&=95.5+\frac{5}{10}(15-10)=98

\end{align*}

For $Q_3$, the $\frac{3n}{4}$th observation=$\frac{3\times 30}{4}$th = 22.5th observation. So

\begin{align*}

Q_3&=l+\frac{h}{f}(\frac{3n}{4}-c)\\

&=100.5+\frac{5}{6}(22.5-20)=102.5833

\end{align*}

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