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CHAPTER 4
STATISTICS
When we make a measurement several different things may be true:
(1)
The parameter we are measuring is random, e.g. flow noise.
(2)
The parameter we are trying to measure is deterministic but other parameters affect the
measurement: e.g. environmental factors, electronic noise, quantization noise.
Many of
these parameters are random.
The result is our measurement is random, and the result of the next measurement cannot be
predicted exactly.
A deterministic measurement is one that can be predicted exactly.
So how do
we describe the measurements if they are random?
How does the description relate to what
we
are trying to measure
as opposed to what
we do measure
?
Let's consider the following example
as an illustration of some the things we should be thinking about when we make measurements
of parameters that are random in nature.
EXAMPLE
Suppose we wish to measure the average age of US citizens in Indiana, and we decide to do this
by going into a high school class, asking the ages of the students there and averaging the
answers.
It turns out that on this day only 5 students are in class.
Let's make some comments
about this.
(a)
The Process of Collecting the Data
The average age of students in a high school class is much lower than the average of the US
population.
So even if we had thousands of students in our high school class and averaged
their ages, the result would be much less than the US population average age.
We say that
there is a "bias" in our estimation procedure.
Also the variation of the ages in the high
school class would also be much lower than the variation of ages in the US population.
Had
we conducted the test at a location peopled by all age groups we might have wished to take
many samples of people's ages, so that we would feel "happier" that the calculated average
was closer to the true average age of the US population.
(b)
The Data
Suppose we wanted to know the average age of students in Grade 11, and the class we
sampled was Grade 11.
We would now feel more comfortable that we were collecting a
more appropriate range of samples.
However, we only have 5 samples.
If we calculated the
average age when more students were in class, the answer would be different, and perhaps
having more sample ages to average we would feel "better" about the calculation.
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In this example we have talked about:
the average value of the data samples,
the spread of the data samples,
the idea that somehow more data is "better",
the idea that the bigger the variation the more desirable it is to take more samples, and
finally,
the idea of bias where you are consistently "off" in your estimates due to some problem
with the way you are calculating what you wish
to measure.
In this chapter we will try to formalize these concepts with a mathematical description of the
random data.
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 Fall '07
 MERKLE

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