Quartiles in Statistics
Like Percentiles and Deciles, Quartiles is a type of Quantile, which is a measure of the relative standing of observation within the data set. The Quartiles 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 $Q_1$) is the value of order statistic that exceeds 1/4 of the observations and less than the remaining 3/4 observations. The third quartile known as the upper quartile is the value in the order statistic that exceeds 3/4 of the observations and is less than the remaining 1/4 observations, while the second quartile is the median.
Quartiles in Statistics 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 on 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 calculated 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 fourth observation})
\end{align*}
Quartiles in Statistics 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 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.
Quartiles in Statistics 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 the 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*}
Application of Quartiles
By analyzing quartiles, one can get insights into the:
- Spread of the data: The distance between $Q_1$ and $Q_3$ (called the interquartile range or IQR) indicates how spread out the data is. A relatively large IQR indicates a wider distribution, while a small IQR shows that the data is more concentrated around the median ($Q_2$).
- Presence of outliers: If the data points are extremely far from the quartiles, they might be outliers that could skew the analysis of measures like the mean.
Reference:
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