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SD Part 2

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We saw before the equation for the standard deviation. We used it as an example when we were learning notation. Now we’re going to learn what that equation actually says, and where it comes from. Imagine we have the following scores. A 3 a 4 and a 5. We know the average is 4, the mean. How far do scores differ from 4? What is the average distance from that mean? That is the standard deviation. That is our definition of a standard deviation. Let’s go ahead and develop this and see what we get. Well, we know the distance from the mean is just every score minus the mean.
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If I take every score and I subtract the mean from it, I’ve essentially just rephrased that score in terms of how far it is from the mean. And I could take the average of that. That would kind of be our definition, right? So when we take every score, subtract the mean, and just average that sum and divide by n. Let’s try that. Let’s try that with our variable and see what happens. So here I’ve got my equation. So I take every score minus the mean. So 3 minus 4 plus 4 minus 4 plus 5 minus 4 and divide by 3. And you’ll notice something interesting happens. It comes out to zero. Hm. Why does it come out to zero?
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Well, look at the top of this fraction. I end up having a negative value, a score below the mean, and a positive value, a score above the mean. And they’re kind of cancelling each other out. This actually always happens and as a result of that, we can’t use this equation. Unfortunately you will always get 0. We need a method that makes the negatives essentially go away. So what we’ve chosen to do with standard deviation is we simply take those distances and we square them. When you square them everything becomes positive, and then at the end, we’re going to have to unsquare them, or square root. So we go back to our normal units.
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This is what we do in standard deviation. So you see here I have the exact same equation on the screen as I had a moment ago, but now I take every one of those distances and I’ve squared them. So 3 minus 4 squared is now a positive number. And when I do that, I get an end score of 0.81. This is our standard deviation. It is the average distance from the mean and it is 0.81 in this data. There’s two versions of this equation, and I just briefly want to point this out.
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If we’re dealing with the entire population, if we have all the scores of interest, we’re going to use the formula I just showed you and our symbol for it is going to be– it’s called lowercase sigma, and it’s just a little circle with a little shuopdedoo off to the side. I know that’s not the technical name, but that’s what I call it. And this is the formula we would use. If you are dealing with a sample, then we actually– our symbol becomes a little sigma with a hat on it. You can say sigma hat. Sometimes we use the letter S. And we just modify the formula a little bit we put n minus 1 on the bottom.
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I’m not going to go into why we do that if we’re dealing with a sample, but suffice to say we want our sample to be a good estimate of the population, and this ensures that it does that. Any more detail than that would go beyond the scope of this lesson. But for now we know that we have an equation. It takes the average distance from the mean, and I can use it in my sample with a little n minus 1, or if I’m dealing with a population just n. And there you go. That’s standard deviation.
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Essential Mathematics for Data Analysis in Microsoft Excel

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