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Other measures of dispersion

Measures of dispersion measure how spread out a data set is. For example, those are standard deviation, variance and coefficient of variation. In this lesson we will define some other measures of dispersion: range, interquartile range and coefficient of quartile deviation.

Range 

The range $R_{X}$ of numeric variable $X$ is the difference between the maximum and minimum value of $X$ (if they exist):

$$R_{X} = X_{max} – X_{min}.$$

 

Furthermore, range also makes sense for ordinal variable.

 

Example 1: The scores of ten pupils in an exam are: $16, 24, 28, 35, 48, 59, 60, 63, 75$.

Solution:

The highest score is $75$ and the lowest score is $16$. Therefore, the range is the difference between these two scores:

$$R_{X} = 75 – 16 = 59.$$

 

As we can see, range is calculated very easily. But, it is not based on all the observations. Since the range of a data set only depends on two values (minimum and maximum values), it is relatively poor summary of spread. Much better summary of the spread is the interquartile range.

 

Interquartile range

The interquartile range $I_{Q}$ of numeric or ordinal variable is the difference between the upper quartile and lower quartile:

$$I_{Q} = Q_{3} – Q_{1}.$$ 

We can say that the interquartile range is the range of the central $50 \%$ elements of an ordered sequence.

Similarly, the interdecile range $I_{D}$ is the difference between the ninth decile and first decile:

$$I_{D}= D_{9} – D_{1}.$$

The interdecile range is the range of the central $80 \%$ elements of an ordered sequence.

Furthermore, the interpercentile range $I_{P} = P_{99} – P_{1}$ is the range of  the central $98 \%$ elements of an ordered sequence.

 

Example 2: Calculate the interquartile range for the following numbers: $1, 3, 4, 5, 5, 6, 7, 11$.

Solution:

$$N = 8, n = 4$$

$$\frac{N}{4} = \frac{8}{4} = 2 \rightarrow Q_{1} = \frac{y_{2} + y_{3}}{2} = \frac{7}{2} = 3.5$$

$$3\frac{N}{4} = 3 \frac{8}{4} = 6 \rightarrow Q_{3} = \frac{y_{6} + y_{7}}{2} = \frac{13}{2} = 6.5$$

The interquartile range is

$$I_{Q} = 6.5 – 3.5 = 3.$$

 

Coefficient of quartile deviation

The coefficient of quartile deviation (or the quartile coefficient of dispersion) is used for comparing dispersion for two or more sets of data. It is calculated by the following formula:

$$V_{Q} = \frac{Q_{3} – Q_{1}}{Q_{1} + Q_{3}}.$$

Furthermore, $V_{Q}$ makes sense only if variables have positive values. Also, $0 \leq V_{Q} < 1$.

If one data set has a larger coefficient of quartile deviation than another data set, then that data set’s interquartile dispersion is greater.

 

More examples

Example 3: Calculate the range and coefficient of quartile deviation for the following sequence: $1, 2, 4, 4, 6, 9, 9, 9, 10, 100$.

Solution:

The range is:

$$R = 100 – 1 = 99.$$

Furthermore, since $N = 10, n = 4$, we have

$$ \frac{N}{4} = \frac{10}{4} = 2.5 \notin \mathbf{N} \rightarrow r = 3$$

$$2 \frac{N}{4} = 2 \frac{10}{4} = 5 \rightarrow r = 6$$

$$3 \frac{N}{4} = 3 \frac{10}{4} = 7.5 \notin \mathbf{N} \rightarrow r = 8$$

Therefore, $Q_{1} = y_{3} = 4, Q_{3} = y_{8} = 9$ and

$$V_{Q} = \frac{9 – 4}{4 + 9} = \frac{5}{13} = 0.38.$$

 

Example 4: In the following table the number of cars sold by some company was recorded for ten days. Calculate the interquartile range.

 

Solution:

First, we need to sort the frequency data: $6, 11, 16, 18, 19, 20, 21, 22, 26, 29$.

$$n = 4, N = 10$$

$$\frac{N}{4} = \frac{10}{4} = 2.5 \notin \mathbf{N} \rightarrow Q_{1} = y_{3} = 16$$

$$3\frac{N}{4} = 3 \frac{10}{4} = 7.5 \notin \mathbf{N} \rightarrow Q_{3} = y_{8} = 22$$

Therefore, $I_{Q} = 22 – 16 = 6$.