2.3.4 The Normal Distribution of Measurements

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

p2

x

Y

1

exp

1

2

(2.9)

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

e (1/2) z2

(2.10)

p

2

The standardized variable (the

z statistic) requires only the probability level to be specified. It mea-

sures 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 2.26

a lists the height of an ordinate (

Y ) as a distance

z from the mean, and Table 2.26

b

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?

z

x

0.45

1.5

0.30

From

Table 2.26

b 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)?

z

119.5

151

2.10

Area: 0.482

15

z

155.5

151

0.30

0.118

15

Total area: 0.600

500(0.600)

300 results