preface to fifteenth edition

2.120 SECTION 2
The breadth or spread of the curve indicates the precision of the measurements and is determined
by and related to the standard deviation, a relationship that is expressed in the equation for the
normal curve (which is continuous and infinite in extent):
2
1 1 x
Y exp (2.9)
p2
2
where is the standard deviation of the infinite population. The population mean expresses the
magnitude of the quantity being measured. In a sense, measures the width of the distribution, and
thereby also expresses the scatter or dispersion of replicate analytical results. When (x )/ is
replaced by the standardized variable z, then:
Y
1
(1/2)
e z2
(2.10)
p2
The standardized variable (the z statistic) requires only the probability level to be specified. It measures
the deviation from the population mean in units of standard deviation. Y is 0.399 for the most
probable value, . In the absence of any other information, the normal distribution is assumed to
apply whenever repetitive measurements are made on a sample, or a similar measurement is made
on different samples.
Table 2.26a lists the height of an ordinate (Y) as a distance z from the mean, and Table 2.26b
the area under the normal curve at a distance z from the mean, expressed as fractions of the total
area, 1.000. Returning to Fig. 2.10, we note that 68.27% of the area of the normal distribution curve
lies within 1 standard deviation of the center or mean value. Therefore, 31.73% lies outside those
limits and 15.86% on each side. Ninety-five percent (actually 95.43%) of the area lies within 2
standard deviations, and 99.73% lies within 3 standard deviations of the mean. Often the last two
areas are stated slightly different; viz. 95% of the area lies within 1.96 (approximately 2) and
99% lies within approximately 2.5. The mean falls at exactly the 50% point for symmetric normal
distributions.
Example 5 The true value of a quantity is 30.00, and for the method of measurement is 0.30.
What is the probability that a single measurement will have a deviation from the mean greater than
0.45; that is, what percentage of results will fall outside the range 30.00 0.45?
x 0.45
z 1.5
0.30
From Table 2.26b the area under the normal curve from 1.5 to 1.5 is 0.866, meaning that
86.6% of the measurements will fall within the range 30.00 0.45 and 13.4% will lie outside this
range. Half of these measurements, 6.7%, will be less than 29.55; and a similar percentage will
exceed 30.45. In actuality the uncertainty in z is about 1 in 15; therefore, the value of z could lie
between 1.4 and 1.6; the corresponding areas under the curve could lie between 84% and 89%.
Example 6 If the mean value of 500 determinations is 151 and 15, how many results lie
between 120 and 155 (actually any value between 119.5 and 155.5)?
119.5 151
z 2.10 Area: 0.482
15
155.5 151
z 0.30 0.118
15
500(0.600) 300 results
Total area: 0.600