Handbook of Civil Engineering Calculations

TABLE 22

(2)

(1) Number of (3) (4) (5) (6)

Month units soldX d = X-41 d^2 dA=X-40 d^2 A

1 32 -11 121 -8 64

2 49 6 36 9 81

3 51 8 64 11 121

4 44 1 1 4 16

5 37 -6 36 -3 9

6 41 -2 4 1 1

7 _47 _4 _16 __7 _49

Total 301 O 278 21 341

Calculation Procedure:

1. Compute the arithmetic mean
   Let X= number of units_sold monthly. Find the sum of the values of X 9 which is 301.
   Then setX = QX)In, or X = 301/7 = 43.


2. Find the median
   Consider that all values of X are arranged in ascending order of magnitude. If n is odd, the
   value that occupies the central position in this array is called the median. If n is even, the
   median is taken as the arithmetic mean of the two values that occupy the central positions.
   In either case, the total frequency of values below the median equals the total frequency
   of values above the median. The median is useful as an average because the arithmetic
   mean can be strongly influenced by an extreme value at one end of the array and thereby
   offer a misleading view of the data.
   In the present instance, the array is 32, 37, 41, 44, 47, 49, 51. The fourth value in the
   array is 44; then Xmed = 44.


3. Compute the standard deviation
   Compute the deviations of the X values from X, and record the results in column 3 of
   Table 22. The sum of the deviations must be O. Now square the deviations, and record the
   results in column 4. Find the sum of the squared deviations, which is 278. Set the variance
   s^2 = &d^2 )/n = 278/7. Then set the standard deviation s = V278/7 = 6.30.


4. Compute the arithmetic mean by using an assumed
   arithmetic mean
   SQtA = 40. Compute the deviations of the Jf values from A 9 recordjthe results in column 5
   of Table 22, and find the sum of the deviations, which is 21. Set X = A + (2dA)/n, or X =
   40 + 21/7 = 43.


5. Compute the standard deviation by using the arithmetic mean
   assumed in step 3
   Square the deviations from A 9 record the results in column 6 of Table 22, and find their
   sum, which is 341. Set s^2 = (ZdJ)In - [&dA)/n]^2 = 341/7 - (21/7)^2 = 341/7 - 9 = 278/7.
   Thens = V278/7 = 6.30.
   Related Calculations: Note that the equation applied in step 5 does not contain
   the true mean X. This equation serves to emphasize that the standard deviation is purely a
   measure of dispersion and thus is independent of the arithmetic mean. For example, if all