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1.10 Quartiles

Quartiles involves dividing the given ordered data into four equal points

QmQLUiopdwpeerr QQuuaarrttiilele 13

Figure 9: The quartiles cut the ordered data into four equal parts, with the median in the middle.
1.10.1 Interquartile Range

This is a measure of dispersion which extends the ideal of the median. The median divides the data into two equal parts. If we, now find the middle value of each half, we have for the lower half \(Q_1=\) lower quartile and the upper half, we have the upper quartile \(=Q_3\). These divide the distribution into quartiles.\[\text {Interquartile Range}\ =Q_3-Q_1\] \[\text {Semi Interquartile Range}\ =\frac {1}{2}(Q_3-Q_1)\]
We can see that the quartiles are \(\frac {1}{4}(n+1)^{\text {th}}\) value and \(\frac {3}{4}(n+1)^{\text {th}}\) values.

Example 1.40. Find the interquartile range of the data below.

\(X\) 1 2 3 4 5 6 7 8 9 10
\(f\) 2 8 9 5 4 3 4 1 1 1

Solution. Quartiles are found by position, so first accumulate the frequencies. Here \(n=38\).

\(X\) 1 2 3 4 5 6 7 8 9 10
\(f\) 2 8 9 5 4 3 4 1 1 1
Cumulative \(f\) 2 10 19 24 28 31 35 36 37 38

Using the positions \(\tfrac 14(n + 1)\) and \(\tfrac 34(n + 1)\) given above, \(Q_1\) lies at position \(\tfrac {1}{4}(39)=9.75\), between the 9th and 10th ordered values. The cumulative frequency shows both are \(2\), so \(Q_1=2\).

\(Q_3\) lies at position \(\tfrac {3}{4}(39)=29.25\), between the 29th and 30th values. The cumulative frequency passes 28 at \(X=5\) and reaches 31 at \(X=6\), so both are \(6\) and \(Q_3=6\).

\[\text {IQR}=Q_3-Q_1=6-2=4.\]

The interquartile range covers the middle half of the data and, unlike the range, ignores both tails entirely. That is its purpose: the single observation at \(X=10\) stretches the range to 9, but moves the IQR not at all.

1.10.2 Interquartile Range for grouped data

For the data grouped into class interval we can estimate the quartiles from cumulative frequency curve

QQ133(1(nn ++ 11)t)thhQQ31 44

Figure 10: Reading \(Q_1\) and \(Q_3\) off the cumulative frequency curve, at the \(\tfrac {1}{4}(n+1)\)th and \(\tfrac {3}{4}(n+1)\)th values.

If \(\sum f=10\)
\(Q_1\) is the \(\frac {11}{4}^{\text {th}}\) observation \(\frac {11}{4}=2.75\implies Q_1\) lies between the 2\(^{\text {nd}}\) value and the \(3^{\text {nd}}\) value.
\(Q_3\) is the \(\frac {3}{4}\times 11^{\text {th}}\) observation \(\frac {3}{4}\times 11=\frac {33}{4}=8.25\implies Q_3\) lies between \(8^{\text {th}}\) value and \(9^{\text {th}}\) value.

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