Handbook of Civil Engineering Calculations

values of Xincrease by a constant /z, then X increases by h, but s remains constant. Where
X has a nonintegral value, the use of an assumed arithmetic mean A of integral value can
result in a faster and more accurate calculation of s.

DETERMINATION OFARITHMETIC

MEAN AND STANDARD DEVIATION

OF GROUPED DATA

In testing a new industrial process, a firm assigned a standard operation to 24 employees

in its factory and recorded the time required by each employee to complete the operation.

The results are presented in columns 1 and 2 of Table 23. Find the arithmetic mean and

standard deviation of the time of completion.

Calculation Procedure:

1. Record the class midpoints
   Where the number of values assumed by a variable is very large, a comprehensive listing
   of these values becomes too cumbersome. Therefore, the data are presented by grouping
   the values in classes and showing the frequency of each class. The range of values of a
   given class is its class interval, and the end values of the interval are the class limits. The
   difference between the upper and lower limits is the class width, or class size. Thus, in
   Table 23, all classes have a width of 4 min. The arithmetic mean of the class limits is the
   midpoint, or mark. In analyzing grouped data, all values that fall within a given class are
   replaced with the class midpont. The midpoints are recorded in column 3 of Table 23, and
   they are denoted by X.


2. Compute the arithmetic mean
   Set ^ = (S/9/W, or X = (3 x 22 + 9 x 26 + 7 x 30 + 5 x 34)724 = 680/24 = 28.33 min.


3. Compute the standard deviation
   Set s^2 J=GfJ^2 Vn, ors^2 = [3(-6.33)^2 + 9(-2.33)^2 + 7(1.67)^2 + 5(5.67)^2 ]/24 = 14.5556. Then
   j = Vl4.5556 = 3.82min.


4. Compute the arithmetic mean by the coding method
   This method simplifies the analysis of grouped data where all classes are of uniform
   width, as in the present case. Arbitrarily selecting the third class, assign the integer O to

TABLE 23

(1) (2)

Time of Number of (3) (4)

completion, min employees Midpoint Code

(class interval) (frequency/) X c

20 to less than 24 3 22 -2

24 to less than 28 9 26-1

28 to less than 32 7 30 O

32 to less than 36 _5 34 1

Total 24