The standard deviation s is found from 
s = 
f. 
[uu] 
where [vv] = [ft 2 ] — 
n 
The class limit between the class intervals i— 1 and i is denoted b r Next the 
expressions can be computed for all class limits. This expression means 
the deviation between the class limit and the average x of the sample ex 
pressed in units of the standard deviation s. The expression is denoted stand 
ardized class limit. In the practical computation of the standardized class 
limits, at least two decimals are usually required. The standardized class limits 
are then used for the determination of the ideal number of values which should 
fall within the class intervals if a theoretically strict normal distribution were 
present. Consequently, the mathematical, expression for the normal distribu 
tion must be used for further calculations. 
If the class limits of the class interval i are denoted a and b, the number of 
values which theoretically should fall within this interval can be expressed as 
the product np t where n is the total number of values in the sample and p t is 
a probability which can be determined from the normal distribution function 
as follows: 
b (t ~* )2 1 
e " 2s 2 dt = 
y 271 
a 
s 
This function is tabulated in most textbooks on statistics and theory of errors. 
From the tables, the values of p t can be determined for each class interval as 
the differences between the upper and the lower limits. The theoretically 
correct number of values which should fall within the actual class interval for 
a strict normal distribution is then found as np { . 
Next the differences between the theoretical class frequencies and the actual 
class frequencies are computed as fi — np r 
npi 
standardized sum of the squares of the differences between the actual distri 
bution and a theoretical normal distribution and will have a chi-square distri 
bution if the assumed normal distribution of the sample is present. The sum 
is, therefore, compared with the corresponding tabular value of the c/zi-square 
distribution for a specific number of degrees of freedom and on a specific 
level. In other words, the quality of agreement between the ideal normal