Sigma Notation Part 4

Now I want to walk through another example of the principle I’ve just been saying, which is that we apply order of operations within those parentheses for your sigma notation. So I have two examples here that are very similar but are, in fact, a little bit different. On the top, I have the sum of 2x squared. On the bottom, I have the sum of 2x, in parentheses, squared. So let’s walk through how we would apply order of operations for these two examples. For the sum of 2x squared, I have to follow my order of operations within that parenthesis, so exponents then multiplication. So in other words, I have to square every age first and then multiply by 2.

And then I can sum. So it would look like this– 2 times 30 squared, 2 times 21 squared, et cetera. I have to do that, because I have to exponent before I can multiply. So that’s going to then turn into this, which will then sum to 13,694. This is a basic application of the order of operations within sigma notation. But let’s see what happens if I nest some other parentheses inside my parentheses. In this example, I have 2x parentheses squared. I’ve added those inner parentheses to tell you, you have to do the 2x first. So I’m forcing you to multiply every age by 2 first. And then you can apply that exponent.

So now I’m going to say you have to do 2 times 30. And then whatever that is, square it. And I’m going to do that for all of my ages. So keep in mind, we can have nested parentheses. And again, it might look intimidating. But those nested parentheses are just an instruction to do whatever that thing is first. So in this case, now I’m going to multiply everything times 2 and square it and add it up. It gives you a different result, which is exactly the point. I’m going to give you specific instructions to do whatever it is I need you to do.

From here, we have the basic things we need to do to build our statistical formulas with sigma notation. So let’s go ahead and see how we might apply this for some statistical formulas.