Page 27 - Quality control of pharmaceuticals (07-PA 704)

For example, we can say that the chances are 68.3 in 100 that the random
error of any single measurement is no more than ±1σ. Similarly, the
chances are 95.4 in 100 that the error is less than ±2σ, and so forth.
The calculation of areas under the Gaussian curve is described later.

For a normal distribution with known mean, μ, and standard deviation, σ,
the exact proportion of values which lie within any interval can be
found from tables, provided that the values are first standardised so
as to give z-values. This is done by expressing any value of x in terms
of its deviation from the mean in units of the standard deviation, σ.
That is:

Example
Consider a normal distribution with µ = 20 and σ = 2. Determine the

probability that a measurement will be in the interval from 20 to 23.

z = = (23 - 20) / 2 = 1.5

Thus y = 23 lies 1.5 standard deviations above µ = 20.
Referring to Table 1, we find the area corresponding to z =1.5 to
be 0.4332. This is the probability that a measurement falls in the
interval from 20 to 23.