Measures of Dispersion
The degree to which numerical data tend to spread about an average value is called the dispersion of the data. The four measures of dispersion are
1. Range
2. Mean deviation
3. Standard deviation
4. Square Deviation

Range
The difference between the highest and the lowest element of a data called its range.
i.e., Range =x_max–x_min
∴ The coefficient of range

It is widely used in statistical series relating to quality control in production.
(i) Inter quartile range =Q₃–Q₁
(ii) Semi-inter quartile range (Quartile deviation)
Q_d=½(Q₃–Q₁)
and coefficient of quartile deviation

Mean Deviation (MD)
The arithmetic mean of the absolute deviations of the values of the variable from a measure of their average (mean, median, mode) is called their average (mean, median, mode) is called Mean Deviation (MD). It is denoted by δ.
(i) For simple (discrete) distribution

where, n= number of terms, z equals either mean (A) or median (Md) or modus (Mo)
(ii) For unclassified frequency distribution

(iii) For classified distribution

Coefficient of Mean Deviation
It is the ratio of MD and the mean from which the deviation is measured. Thus, the coefficient of MD

Question 1:
Find the mean deviation about the mean for the data
4, 7, 8, 9, 10, 12, 13, 17
Answer:
The given data is
4, 7, 8, 9, 10, 12, 13, 17
Mean of the data, x̄=(4+7+8+9+10+12+13+17)÷8=80÷8=10
The deviations of the respective observations from the mean x̄, i.e x_i–x̄, are
-6, -3, -2, -1, 0, 2, 3, 7
The absolute values of the deviations, i.e. |x_i–x̄|, are
6, 3, 2, 1, 0, 2, 3, 7
The required mean deviation about the mean is

Question 2:
Find the mean deviation about the mean for the data
38, 70, 48, 40, 42, 55, 63, 46, 54, 44
Answer:
The given data is
38, 70, 48, 40, 42, 55, 63, 46, 54, 44
Mean of the given data, x̄=(38+70+48+40+42+55+63+46+54+44)÷10=500÷10=50
The deviations of the respective observations from the mean x̄, i.e x_i–x̄, are
-12, 20, -2, -10, -8, 5, 13, -4, 4, -6
The absolute values of the deviations, i.e. |x_i–x̄| are
12, 20, 2, 10, 8, 5, 13, 4, 4, 6
The required mean deviation about the mean is

Question 3:
Find the mean deviation about the median for the data.
13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
Answer:
The given data is
13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
Here, the numbers of observations are 12, which is even.
Arranging the data in ascending order, we obtain
10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18

The deviations of the respective observations from the median, i.e. x_i–M are
-3.5, -2.5, -2.5, -1.5, -0.5, -0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5
The absolute values of the deviations, |x_i–M|, are
3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5
The required mean deviation about the median is

=(3.5+2.5+2.5+1.5+0.5+0.5+0.5+2.5+2.5+3.5+3.5+4.5)÷12=28÷12=2.33

Question 4.
Find the mean deviation about the median for the data

36, 72, 46, 42, 60, 45, 53, 46, 51, 49
Answer:
The given data is
36, 72, 46, 42, 60, 45, 53, 46, 51, 49
Here, the number of observations is 10, which is even.
Arranging the data in ascending order, we obtain
36, 42, 45, 46, 46, 49, 51, 53, 60, 72

The deviations of the respective observations from the median, i.e. x_i–M are
-11.5, -5.5, -2.5, -1.5, -1.5, 1.5, 3.5, 5.5, 12.5, 24.5
The absolute values of the deviations, |x_i–M|, are
11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5
Thus, the required mean deviation about the median is

=(11.5+5.5+2.5+1.5+1.5+1.5+3.5+5.5+12.5+24.5)÷10=70÷10=7

Question 5:
Find the mean deviation about the mean for the data.

Question 6:
Find the mean deviation about the mean for the data

Question 7:
Find the mean deviation about the median for the data.

Answer:
The given observations are already in ascending order.
Adding a column corresponding to cumulative frequencies of the given data, we obtain the following table.

Here, N=26, which is even.
Median is the mean of 13th and 14th observations. Both of these observations lie in the cumulative frequency 14, for which the corresponding observation is 7.

The absolute values of the deviations from median, |x_i–M|, are

Question 8:
Find the mean deviation about the median for the data

Answer
The given observations are already in ascending order.
Adding a column corresponding to cumulative frequencies of the given data, we obtain the following table.

Here, N=29, which is odd.
∴ Median=((29+1)/2)th observation = 15th observation
This observation lies in the cumulative frequency 21, for which the corresponding observation is 30.
∴ Median =30
The absolute values of the deviations from median, i.e. |x_i–M|, are