Can you give me an estimate?
Your score was merely average
In this new series, I expect to cover a lot of ground, with math that's going to get heavier and heavier as we go along. The last thing I want to do is to leave some of you behind in the starting blocks. So in the spirit of No Reader Left Behind, I'm going to begin with the most ridiculously simple example I can think of. If things go as planned, I'll continue to escalate in such small steps that, like the proverbial frog simmering in the proverbial stew pot, you'll be cooking along with state estimation without really noticing how you got there. I'll start by defining a set of 10 numbers:
Your challenge is to compute their average value. Hey, you know how to do this. First, add all the numbers:
Then divide by the number of terms:
There. That wasn't so hard, was it?
Note the bar over the y, which is the symbol usually used to indicate an average value. It's not the only symbol for the average. Another one is 〈y〉, which is a little harder to miss. I actually prefer 〈y〉, precisely because it's harder to miss. But we'll stick with the “bar” notation here. Note also that, while y is a set of values, y is only a single scalar number.
We can generalize the process with a little math notation (stay with me, now):
or, more generally, for a set of N values:
Even better, the shorthand notation of the summation symbol:
Careful, now. I can see some of your eyes starting to glaze over, and your pupils shrinking down to pinpoints. This isn't rocket science (yet). Equation 6 says nothing at all that isn't said by Equation 5. It's just a shorter way of saying it. Because mathematicians and physicists are a lazy lot, we tend to go for the shorter way when given the chance. Stated in words, the summation sign of Equation 6 says “let the index i take on all the integer values from 1 through N. For each i, add the term yi to the sum.”
If the summation sign makes you nervous, just remember that you can always expand it back out into the explicit sum of Equation 5. It just takes a little longer to write, that's all.
One last point: The average value of any set is often called its mean. Even more specifically, its arithmetic mean (because there are other kinds of means).
Why introduce another term? Can it be that we're so lazy we'd rather write four letters than seven? I guess it must be so. Deal with it.
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