Page 28 - Mechanical Engineers' Handbook (Volume 2)

3 Statistics in the Measurement Process 17

The height of any rectangle of the histogram shown is denoted as the relative number m /n
j
and is equal to the statistical probability F (x) that a measured value will have the size x
j
j
(
x /2). The area of each rectangle can be made equal to the relative number by transforming
j
the ordinate of the histogram in the following way: Area relative number F (x) m /n
j
j
p (x)
x. Thus
j
m j
p (x) (34)
j
n
x
The shape of the histogram is preserved in this transformation since the ordinate is merely
changed by a constant scale factor (1/
x). The resulting diagram is called the probability
density diagram. The sum of the areas underneath all rectangles is then equal to 1 [i.e.,
p (x)
x 1]. In the limit, as the number of data approaches infinity and the least count
j
j
becomes very small, a smooth curve called the probability density function is obtained. For
this smooth curve we note that

p(x) dx 1 (35)
and that the probability of any measurement, x, having values between x and x is found
a b
from
x b
P(x x x ) p(x) dx (36)
a b
x a
To integrate this expression, the exact probability density function p(x) is required. Based
on the assumptions made, several forms of frequency distribution laws have been obtained.
The distribution of a proposed set of measurements is usually unknown in advance. However,
the Gaussian (or normal) distribution fits observed data distributions in a large number of
cases. The Gaussian probability density function is given by the expression

2 2
p(x) (1/ 2 )e (x x)/ 2 (37)
where is the standard deviation. The standard deviation is a measure of dispersion and is
defined by the relation
xp(x) dx (x ) 2
2
i (38)

p(x) dx n

3.9 Determination of Confidence Limits on

If a set of measurements is given by a random variable x, then the central limit theorem 13
x
states that the distribution of means, , of the samples of size n is Gaussian (normal) with
2
2
mean and variance /n , that is, x
G( , /n). [Also, the random variable z
2
x
(x )/( / n) is Gaussian with a mean of zero and a variance of unity, that is, z
G(0,1).]
The random variable z is used to determine the confidence limits on due to random error
of the measurements when is known.
The confidence limit is determined from the following probabilistic statement and the
Gaussian distribution for a desired confidence level :