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chapter4 - CHAPTER 4 STATISTICS When we make a measurement...

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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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4-2 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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