Measures Based Upon Spread of Values

The first two measures of dispersion — range and quartile deviation — describe dispersion by measuring the spread within which the values of a distribution lie.

Range

Range (R) is simply the difference between the largest value (L) and the smallest value (S) in a distribution:

A higher value of range implies higher dispersion, and vice-versa. For a frequency distribution, the range is taken as the difference between the upper limit of the highest class and the lower limit of the lowest class.

Range — a note on its usefulness. Range is unduly affected by extreme values; it is not based on all the values; as long as the minimum and maximum stay unaltered, a change in any other value leaves the range unchanged; and it cannot be calculated for an open-ended frequency distribution (one in which the lower limit of the lowest class, or the upper limit of the highest class, or both, are not specified). Despite these limitations, range is widely used because of its simplicity — for example, the daily maximum and minimum temperatures of different cities shown on television let us judge the temperature variation between them at a glance.

Quartile Deviation

The presence of even one extremely high or low value can badly reduce the usefulness of range. We therefore need a measure that is not unduly affected by outliers. If the entire data is divided into four equal parts, each containing 25% of the values, we obtain the quartiles and the median (studied in the chapter on central tendency). The upper quartile $Q_3$ and the lower quartile $Q_1$ are used to define the inter-quartile range:

Because the inter-quartile range is based on the middle 50% of the values, it is not affected by extreme values. Half of the inter-quartile range is called the quartile deviation (Q.D.):

For this reason Q.D. is also called the Semi-Inter-Quartile Range. Quartile deviation can generally be calculated even for open-ended distributions, and is not unduly affected by extreme values.

Locating the quartiles. In an individual or discrete series, $Q_1$ is the size of the $\dfrac{n+1}{4}\text{th}$ value and $Q_3$ is the size of the $\dfrac{3(n+1)}{4}\text{th}$ value. In a continuous distribution, however, $n$ is used in place of $n+1$ — so $Q_1$ is the size of the $\dfrac{n}{4}\text{th}$ value and $Q_3$ the size of the $\dfrac{3n}{4}\text{th}$ value (the same rule applies to the median). Having found the class that contains the quartile, its exact value is obtained from the interpolation formulae: …