preface to fifteenth edition

GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS 2.123
The standard deviation may be estimated by calculating the standard deviation s drawn from a
small sample set as follows:
q
N
(xi x) 2
2 2 2
i1 1 2 1 2
q
x x ··· [(x x ···) ]/N
s or s
N N 1
(2.12)
where xi
x represents the deviation of each number in the array from the arithmetic mean. Since
two pieces of information, namely s and x, have been extracted from the data, we are left with
N 1 degrees of freedom (df); that is, independent data points available for measurement of precision.
If a relatively large sample of data corresponding to N 30 is available, its mean can be
taken as a measure of , and s as equal to .
So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek
letters for the parameters and Latin letters for the estimates. For many purposes, one uses the
variance, which for the sample is s 2 and for the entire populations is 2 . The variance s 2 of a finite
sample is an unbiased estimate of 2 , whereas the standard deviation s is not an unbiased estimate
of .
Because the standard deviation for the universe is a characteristic of the measuring procedure,
it is possible to get a good estimate not only from a long series of repeated analyses of the same
sample, but also by taking together several short series measured with slightly different samples of
the same type. When a series of observations can be logically arranged into k subgroups, the variance
is calculated by summing the squares of the deviations for each subgroup, and then adding all the
k sums and dividing by N k because one degree of freedom is lost in each subgroup. It is not
required that the number of repeated analyses in the different groups be the same. For two groups
of observations consisting of N A and N B members of standard deviations s A and s B , respectively, the
variance is given by:
2 2
(NA 1)sA (NB 1)s
2
B
s (2.13)
N N 2
A
Another measure of dispersion is the coefficient of variation, which is merely the standard deviation
expressed as a fraction of the arithmetic mean, viz., s/x. It is useful mainly to show whether
the relative or the absolute spread of values is constant as the values are changed.
B
2.3.6 Student’s Distribution or t Test
In the next several sections, the theoretical distributions and tests of significance will be examined
beginning with Student’s distribution or t test. If the data contained only random (or chance) errors,
the cumulative estimates x and s would gradually approach the limits and . The distribution of
results would be normally distributed with mean and standard deviation . Were the true mean
of the infinite population known, it would also have some symmetrical type of distribution centered
around . However, it would be expected that the dispersion or spread of this dispersion about the
mean would depend on the sample size.
The standard deviation of the distribution of means equals /N 1/2 . Since is not usually known,
its approximation for a finite number of measurements is overcome by the Student t test. It is a
measure of error between and x. The Student t takes into account both the possible variation of
the value of x from on the basis of the expected variance 2 /N 1/2 and the reliability of using s in
place of . The distribution of the statistic is:
x
ts
t or x (2.14)
s/ pN pN