 In algebra, when we have a list of subscripted variables, we can add them directly. This is great if we have a few variables. But if we have ten items, that expression gets a lot longer. If we had fifty items, we'd have a ridiculously long series of additions. As in all things algebraic, we'd like to be able to generalize this to adding up sum number N of related items, numbered zero through N minus one. We need a compact notation to express this addition. Here it is. The big symbol that looks like a capital M on its side is the Greek letter sigma. And mathematicians use it to represent the word sum. That's S-U-M, not S-O-M-E. And this is how you read that mathematical expression. Sum from i equals zero to N minus one of X sub i. This translates nicely to a Java for loop. What does this expression mean when we expand it algebraically? I've left off the limits above and below the sigma. Here's the expansion. And here's what it evaluates to when we put in the numbers that you see at the top of the screen. Here's something important to note. These are not equivalent. The left side is the total of the squared items, which works out to fifty-four. The right side gets the sum of all the items and squares that result, which works out to a hundred and ninety-six. Here's one other bit of useful notation. We read this as X-bar, and it stands for the arithmetic mean of all the items in X, which can also be written in sigma notation as the sum of all the items divided by the number of items. In this example, X-bar evaluates to five. X-bar is used in other formulas like this one. Let's analyze it piece by piece. For each item in X, subtract the average value, square that result, and add them all up. If you want to do this calculation in Java, you'll have to go through the array twice. Once to find X-bar, the average, and again to subtract the average from the items individually and square them.