Mean and Standard Deviation

1215 Words Jun 29th, 2012 5 Pages
Mean and Standard Deviation
The mean, indicated by μ (a lower case Greek mu), is the statistician 's jargon for the average value of a signal. It is found just as you would expect: add all of the samples together, and divide by N. It looks like this in mathematical form: In words, sum the values in the signal, xi, by letting the index, i, run from 0 to N-1. Then finish the calculation by dividing the sum by N. This is identical to the equation: μ =(x0 + x1 + x2 + ... + xN-1)/N. If you are not already familiar with Σ (upper case Greek sigma) being used to indicate summation, study these equations carefully, and compare them with the computer program in Table 2-1. Summations of this type are abundant in DSP, and you need to understand
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Think of these programs as an alternative way of understanding the equations used in DSP. If you can 't grasp one, maybe the other will help. In BASIC, the % character at the end of a variable name indicates it is an integer. All other variables are floating point. Chapter 4 discusses these variable types in detail.
This method of calculating the mean and standard deviation is adequate for many applications; however, it has two limitations. First, if the mean is much larger than the standard deviation, Eq. 2-2 involves subtracting two numbers that are very close in value. This can result in excessive round-off error in the calculations, a topic discussed in more detail in Chapter 4. Second, it is often desirable to recalculate the mean and standard deviation as new samples are acquired and added to the signal. We will call this type of calculation: running statistics. While the method of Eqs. 2-1 and 2-2 can be used for running statistics, it requires that all of the samples be involved in each new calculation. This is a very inefficient use of computational power and memory.
A solution to these problems can be found by manipulating Eqs. 2-1 and 2-2 to provide another equation for calculating the standard deviation: While moving through the signal, a running tally is kept of three parameters: (1) the number of samples already processed, (2) the sum of these samples, and (3) the sum of the squares of the
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