# Mean Deviation

Suppose a hospital is proposed for students of five towns A, B, C, D and E which lie in that order along a road. Distances of towns in miles from town A and number of students in these towns are given below:

 Town Distance from town A No. of students A 0 90 B 2 150 C 6 100 D 14 200 E 18 80 620

Now, if the hospital is situated in town A, 150 students from town B will have to travel 2 miles each (a total of 300 miles) to reach the hospital. The objective is to find a location so that the average distance travelled by students is minimum. Suppose that the students will have to travel more than on an average, if the hospital is situated at town A or E. Otherwise, if it is somewhere in the middle, they are likely to travel less. The average distance travelled is calculated by Mean Deviation which is simply the arithmetic mean of the differences of the values from their average. The average used is either the arithmetic mean or median. (Since the mode is not a stable average, it is not used to calculate Mean Deviation.)

# Calculation of Mean Deviation from Arithmetic Mean for ungrouped data

Direct Method
Steps:
1. The A.M. of the values is calculated
2. Difference between each value and the A.M. is calculated. All differences are considered positive. These are denoted as |d|
3. The A.M. of these differences (called deviations) is the Mean Deviation. i.e. MD =
Example 3
Calculate the Mean Deviation of the following values; 2, 4, 7, 8 and 9.
The A M = = 6
 X {D} 2 4 4 2 7 1 8 2 9 3 12

Assumed Mean Method
Mean Deviation can also be calculated by calculating deviations from an assumed mean. This method is adopted especially when the actual mean is a fractional number. (Take care that the assumed mean is close to the true mean). For the values in example 3, suppose value 7 is taken as assumed mean, M.D. can be calculated as under:

Example 4
 X {D} 2 5 4 3 7 0 8 1 9 2 11

In such cases, the following formula is used,
Where is the sum of absolute deviations taken from the assumed mean. x is the actual mean. A x is the assumed mean used to calculate deviations.
is the number of values below the actual mean including the actual mean.
is the number of values above the actual mean. Substituting the values in the above formula:
= 2.4

# Mean Deviation from median for ungrouped data

Direct Method
Using the values in example 3, M.D. from the Median can be calculated as follows,
1. Calculate the median which is 7.
2. Calculate the absolute deviations from median, denote them as |d|.
3. Find the average of these absolute deviations. It is the Mean Deviation.
Example 5

 X {D} 2 5 4 3 7 0 8 1 9 2 11

M. D. from Median is thus,

Short-cut method
To calculate Mean Deviation by short cut method a value (A) is used to calculate the deviations and the following formula is applied.

Where, A = the constant from which deviations are calculated. (Other notations are the same as given in the assumed mean method).
Mean Deviation from Mean for Continuous distribution

TABLE 6.2

 Profits of companies (Rs.in Lakhs)CI No.of companies Frequencies 10-20 5 20-30 8 30-50 16 50-70 8 70-80 3 40

Steps:
1. Calculate the mean of the distribution.
2. Calculate the absolute deviations |d| of the class midpoints from the mean.
3. Multiply each |d| value with its corresponding frequency to get f|d| values. Sum them up to get Î£ f|d|.
4. Apply the following formula,
Mean Deviation of the distribution in Table 6.2 can be calculated as follows:

Example 6

 C.I f M.p [d] F[d] 10-20 5 15 25.5 127.5 20-30 8 25 15.5 124.0 30-50 16 40 0.5 8.0 50-70 8 60 19.5 156 70-80 3 75 34.5 103.5

= = 12.975

Mean Deviation from Median

TABLE 6.3

 CI Frequencies 20-30 5 30-40 10 40-60 20 60-80 9 80-90 6 50

The procedure to calculate Mean Deviation from the median is the same as it is in case of M.D. from Mean, except that deviations are to be taken from the median as given below:

Example 7

 C.I f M.p [d] F[d] 20-30 5 25 25 125 30-40 10 35 15 150 40-60 20 50 0 0 60-80 9 70 20 180 80-90 6 85 35 210 50 665

= = 13.3