Business Mathematics and Statistics · Ch 9 — Measures of Dispersion

Quartiles and Quartile Deviation

Quartiles divide an array of data (arranged in ascending order) into four equal parts. $Q_1$ (the lower quartile) is the value below which a quarter of the observations lie; $Q_2$ is simply the median (half the observations lie below it); and $Q_3$ (the upper quartile) is the value below which three-quarters of the observations lie.

For individual or discrete data (arranged in ascending order, $n$ = number of observations), $Q_1$ and $Q_3$ are located using a position formula exactly analogous to the median's:

Note

Quartiles — Individual/Discrete Series

$Q_1 = \text{size of the } \left(\dfrac{n+1}{4}\right)\text{th item}, \qquad Q_3 = \text{size of the } \left(\dfrac{3(n+1)}{4}\right)\text{th item}$

If the position falls between two items, interpolate linearly between them, exactly as done for the median.

For continuous/grouped data, $Q_1$ and $Q_3$ are found by the same interpolation technique used for the median, applied at $N/4$ and $3N/4$ instead of $N/2$ :

Note

Quartiles — Continuous/Grouped Series

$Q_k = L + \left(\dfrac{\dfrac{kN}{4} - cf}{f}\right)\times h, \qquad k = 1, 3$

where $L$ = lower boundary of the class containing $Q_k$ (the first class whose cumulative frequency reaches or exceeds $kN/4$ ), $cf$ = cumulative frequency of the class before it, $f$ = frequency of that class, and $h$ = class width.

Once $Q_1$ and $Q_3$ are known, the Quartile Deviation (also called the semi-interquartile range) measures the spread of the middle 50% of the data — a genuine improvement over Range, since it is unaffected by extreme values at either tail:

Note

Quartile Deviation (QD)

$QD = \dfrac{Q_3 - Q_1}{2}$

Note

Coefficient of Quartile Deviation

$\text{Coefficient of QD} = \dfrac{Q_3 - Q_1}{Q_3 + Q_1}$ …

Definition 1Quartiles ($Q_1, Q_2, Q_3$)

The three values that divide an ascending-ordered data set into four equal parts; $Q_2 …

Definition 2Quartile Deviation (QD)

Half the inter-quartile range, $QD=(Q_3-Q_1)/2$ — a measure of the spread of the middle 50% of the data, unaffect …

Definition 3Coefficient of Quartile Deviation

The unit-free relative form of QD, $(Q_3-Q_1)/(Q_3+Q_1)$ , used to compare the middle-50% spread of two …

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