engineers need to perform experiments that consist of measuring the
value of an output response y of a normal process for a given setting of
an input variable x. In such experiments, we know from
experience that even if we can precisely set the input variable x to the
same value, we'll probably get a different value for the output response
y every time we measure it. This is because of the occurrence
a reality of engineering life that we all need to contend with.
several measurements of the value of output y for a given setting of x
are taken, we can compute for the mean
and the variance
sample measurements. These sample mean and variance values, however, are
different from the actual values of the mean
normal distribution of y values that can be taken given the set value of
simply sample estimates for the actual population mean
of the process output y, respectively.
In most real life
problems, it is not always possible to know the actual values of µ
with absolute certainty, so as engineers we have nothing to work with
which we can get from the finite number of measurements that we can do.
The question now is, how close is
To answer this question, let
us define a quantity, t, as follows:
t = (Y - µ) / (s / sqrt(n))
is the number of
random measurements taken.
A population of t-values forms a t-distribution, which also looks like a
normal distribution. It is, however, a bit wider than a normal
distribution, given that it uses s to define its dispersion.
As the number of
degrees of freedom
if n is the number of independent measurements taken)
of the t-distribution increases, it becomes more and more identical to
the normal distribution. In fact, at a high enough number of
degrees of freedom (say, > 100), the t-distribution becomes practically
indistinguishable from the normal distribution.
The t-distribution can
therefore be used to analyze normal distributions in cases where the
true value of σ can
be obtained, such as when the sample size is limited. Increasing the number of measurements
taken for y increases the degrees of freedom used to estimate σ, bringing the values
of Y and s closer to µ and σ of the entire normal population,
One of the
uses for the t-distribution is in determining the probability that the
actual population mean
of a group of measurements will fall between two values, provided
that the experiment was performed in a
manner. This is done by calculating the sample population mean Y
and variance s from measurements taken from an experiment with a number
of runs. The following equation is then applied:
fall at a given probability
= Y +/- [ρ(1-a,
d.f.)] [s / sqrt(n)] with the value of ρ (1-a,
d.f.) coming from the
t-table (Table 1).
Table 1 shows the
values instead of the probability values, so these have to be subtracted
from 1 in order to obtain their corresponding probability levels, e.g.,
95% probability level corresponds to the column where
example, suppose that you obtained 5
of the output of your process for the same set of inputs, from which you
calculated a sample mean Y = 4.5 and a sample variance s = 0.1. If
you're interested in the range of values between which there's a
probability of 95% that the real population mean µ will lie, then
(referring to the t-Table) you'd
have to use a value of 2.776
for ρ (for d.f. = 4 and
= 0.05). Thus, given
your data, there's 95% probability that µ
will lie between 4.5 - [2.776 (0.1/2.236)] and 4.5 + [2.776 (0.1/2.236)],
i.e., 4.376 < µ < 4.624.
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