Page 20 - Mechanical Engineers' Handbook (Volume 2)

3 Statistics in the Measurement Process 9

From Eqs. (1)–(3)
y a bx cx 2
xy ax bx cx 3 (10)
2
2
xy ax bx cx 4
3
2
A simultaneous solution of the above equations provides the desired regression coefficients:
a 0.1959 b 0.0205 c 0.0266 (11)

As was mentioned previously, a measurement process can only give you the best estimate
of the measurand. In addition, engineering measurements taken repeatedly under seemingly
identical conditions normally show variations in measured values. A statistical treatment of
measurement data is therefore a necessity.

3 STATISTICS IN THE MEASUREMENT PROCESS
3.1 Unbiased Estimates
Data sets typically have two very important characteristics: central tendency (or most rep-
resentative value) and dispersion (or scatter). Other characteristics such as skewness and
kurtosis (or peakedness) may also be of importance but will not be considered here. 10
A basic problem in every quantitative experiment is that of obtaining an unbiased es-
timate of the true value of a quantity as well as an unbiased measure of the dispersion or
uncertainty in the measured variable. Philosophically, in any measurement process a deter-
ministic event is observed through a ‘‘foggy’’ window. If so, ultimate refinement of the
measuring system would result in all values of measurements to be the true value . Because
errors occur in all measurements, one can never exactly measure the true value of any
quantity. Continued refinement of the methods used in any measurement will yield closer
and closer approximations, but there is always a limit beyond which refinements cannot be
made. To determine the relation that a measured value has with the true value, we must
specify the unbiased estimate x of the true value of a measurement and its uncertainty (or
precision) interval W based on a desired confidence level (or probability of occurrence).
x
9
An unbiased estimator exists if the mean of its distribution is the same as the quantity
being estimated. Thus, for sample mean x to be an unbiased estimator of population mean
, the mean of the distribution of sample means, , must be equal to the population mean.
x
3.2 Sampling
Unbiased estimates for determining population mean, population variance, and variance of
the sample mean depend on the type of sampling procedure used.
Sampling with Replacement (Random Sampling)
ˆ x (12)

where x is the sample mean and ˆ is the unbiased estimate of the population mean,
;