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What is Sigma? - Definition, Lesson & Quiz

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Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught Math at a public charter high school.

In addition to being the eighteenth letter of the Greek alphabet, sigma also means 'sum' and 'deviation' in the mathematics world. Learn what each symbol looks like and how each formula works.

What is Sigma?

Sigma is the eighteenth letter of the Greek alphabet and is equivalent to our letter S. In mathematics, the upper case sigma is used for the summation notation. The lower case sigma stands for standard deviation. Each has their own unique formula. And yes, both the upper case and lower case look vastly different from each other.

The Greek letter sigma.
sigma letters

Neither of them looks like our letter S, but they are both the Greek equivalent of it. If you notice, the two formulas that use these two symbols both start with the letter s.

Upper Case Sigma

The upper case sigma is used in the summation notation. This particular notation is also called sigma notation.

Summation notation formula.
sigma summation

This particular formula, as its name denotes, tells you to sum up the function evaluated at particular points determined by the little numbers on top and below the big sigma. It is used to add a series of numbers.

You will most likely see this used to sum up a function evaluated at certain points. In the real world, this can be used to figure out the interest you earn over a period of time if you have money saved in an interest-bearing account at a financial institution.

When performing math problems, you will most often see this in association with functions of various types. An example is the summation of f(n)=1/n evaluated at 1, 2, 3, and 4.

An example of the summation formula.
sigma summation example

The little numbers on top and below the big sigma determine the starting and ending evaluation values. You can see that I've plugged in the values 1, 2, 3, and 4 into the n in the formula to evaluate at each of the values and then I summed it all up.

The summation or sigma notation is fairly straightforward and easy to follow. Try it out with a few functions you are familiar with so you can get a better feel for how it works. Try different starting and ending values as well.

Lower Case Sigma

The lower case sigma is used for the standard deviation formula in statistics.

The standard deviation formula.
sigma std dev

If you haven't delved into statistics before, welcome! Don't let this huge formula scare you. The letter that looks like the letter u but with a longer line is the Greek letter 'mu', and in this formula it stands for the mean or average of a series of numbers. The x with the subscript i stands for each number in the series. Say, for example, my series of numbers goes like this: 5, 2, 8, and 1. The x sub 1 number is 5 and the x sub 3 number is 8. Do you see how it works? In this particular series, I only have four numbers and so my N is 4.

An example of the standard deviation formula at work.
sigma std dev ex

While the above example might look scary, it really isn't. If you break it down into its parts you will see that all I have done is plug and play. I've plugged my numbers into the equation where they belong and I solve it step by step, simplifying as I go.

Summary

The first and most common formula related to the sigma symbol is the summation notation using the upper case sigma. The less common formula is the one for standard deviation used in statistics. This second formula uses the lower case sigma.

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