S. Shapiro writes:
>I have a set of six numbers, as follows:
>
>6.77597
>7.04532
>7.17026
>7.13235
>7.56820
>6.97272
>
>which represent results from six different measurements of
>the same thing in six different trials, one measurement
>per trial. (As a consequence of measurement the samples
>are destroyed, so it is not possible to measure the same
>sample six different times. Therefore, I had to set up
>six separate, independent experiments and measure my
>parameter of interest once in each experiment.)
>
>The question I seek to answer is: are the 6 values
>obtained in the measuring process reproducible within
>statistically meaningful boundaries? I suppose another
>way of asking the same question is: is the null hypothesis
>Ho satisfied with respect to this series of measured
>values?
Your question is a bit vague, but let me try to answer it. First the phrase
"statistically meaningful boundaries" is an indication that you are using
statistics as a substitute for careful intellectual analysis. What you want
instead of statistical boundaries is to get a scientist or engineer to
specify practical boundaries that have relevance to your business or
industry. For example, an expert in your area might consider a measurement
process as reproducible if the range is less than 2 units or if the
coefficient of variation (standard deviation divided by the mean) is less
than 0.25.
Statistics can tell you nothing about what is important from a practical
perspective. In medicine, we might tolerate a large amount of deviation when
we are measuring the body temperature of an adult, but we would want far
more precision when measuring the body temperature of a pre-term infant.
Only a doctor could tell you this, though. A mere statistician like me is
clueless in deciding what is important from a medical perspective.
Although you give no context for your data, I suspect that the same is true
for your situation. No statistical summary is going to be useful until you
first define what a reasonable amount of variation might be from a
scientific or engineering perspective. Talk to the subject matter experts
before you compute any statistics.
If these measurements are for a product that you sell, you might also try
asking your customers to specify what is important.
Furthermore, although you have not stated in precise terms what your null
hypothesis is, I suspect that there is no reasonable null hypothesis worth
testing on this data.
If this data is part of an ongoing evaluation program, you might consider
using control charts. Wheeler's book has a good explanation of how to use
control charts (the voice of the data) and how to compare them to practical
boundaries (the voice of the customer). But don't bother with anything
involving statistically meaningful boundaries or testing hypotheses.
I'm sorry if these comments seem critical. One of the hardest things in
Statistics is deciding what your goal is when you start to collect some
data. Since you only have a vague idea what your goal is, you need to get
some outside advice from experts in your area. I hope this helps.
Wheeler, Donald J. (1993). Understanding Variation. The Key to Managing
Chaos. Knoxvile TN: SPC Press, Inc. (ISBN: 0-945320-35-3). For the beginning
student. An insightful introduction about variation in business processes,
how to identify it and how to control it. A must read for anyone working on
improving quality in work processes.
Steve Simon, [EMAIL PROTECTED], Standard Disclaimer.
STATS - Steve's Attempt to Teach Statistics: http://www.cmh.edu/stats
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