Statistics Definitions > Summation Notation

## What is Summation Notation?

Summation (Σ) just means to “add up.” For example, let’s say you had 5 items in a data set: 1,2,5,7,9; you can think of these as x-values. If you were asked to add all of the items up in summation notation, you would see:

Σ(x) which equals 1 + 2 + 5 + 7 + 9 = 24.

When using summation notation, X_{1} means “the first x-value”, X_{2} means “the second x-value” and so on. For example, let’s say you had a list of weights: 100lb, 150lb, 153lb and 202lb. The weights and their corresponding x-values are:

X_{1}: 100lb

X_{2}: 150lb

X_{3}: 153lb

X_{4}: 202lb

The “i=1” at the base of Σ means “start at your first x-value”. This would be X_{1} (100lb in this example). The “n” at the top of Σ means “end at n”. In statistics, n is the number of items in the data set. So what this summation is asking you to do is “**add up all of your x-values from the first to the last**.” For this set of data, that would be:

100 lb + 150 lb + 153 lb + 202 lb = 605 lb.

**Note**: If you see a number above Σ, instead of n, it means to add up to a certain point. For example, a “3” above the Σ means to sum up the the third item (X_{3}) in the set.

**Why the difficult notation?** Why not just say “add up”? There *are* cases when you might want to start at a different point in the data set. Although you probably won’t come across these in an elementary statistics class, if you go onto more advanced stats (or calculus), you’ll come across many different variations. So introducing the Σ notation is getting you used to the format, much like x and y is introduced very early on in basic math.

Summation notation is also a shorthand that helps to avoid long equations. For example, take this lengthy expression, where a, b, and c are constants, and X And Y are random variables.

(aX_{1}+bY_{1}+c)+(aX_{2}+bY_{2}+c)+(aX_{3}+bY_{3}+c)+(aX_{4}+bY_{4}+c)+(aX_{5}+bY_{5}+c)+(aX_{5}+bY_{5}+c)

This can be written more succinctly in summation notation as:

## A More Complicated Example

One of the most challenging formulas you’ll come across in elementary statistics that involves summation notation is Pearson’s correlation coefficient:

There are multiple summations in the formula and although it’s time consuming to solve, it is fairly straightforward if you break it down into steps. Note that there are two summations of X in the formula:

ΣX

^{2}, which means to square the x-values and add them all up

and

(ΣX)

^{2}, which means to add up all of the x-values and then square.

## Calculus Example

In calculus, sigma (Σ) also represents adding many values together.The “k” in the summation is called the **index variable**, or counter; the function to the right of sigma (in this example, k^{2}) is the **summand** and the variable is the index of summation. The numbers to the top and bottom of the sigma sign are the **upper or lower limits** of summation. Sigma notation is used in calculus to evaluate sums of rectangular areas. You can think of the limits of summation here as where your rectangles start, and where they end.

## Using Sigma Notation: Example

**Sample problem:** Evaluate the sum of the rectangular areas in the figure below. Use Σ notation:

Step 1: **Multiply the lengths** of the base **by the height** of each rectangle.

1 * ^{1}⁄_{3} = ^{1}⁄_{3}

1 * ^{1}⁄_{4} = ^{1}⁄_{4}

1 * ^{1}⁄_{5} = ^{1}⁄_{5}

*Summation Notation (Sigma Notation): Definition and Use was last modified: June 14th, 2018 by *

Step 2: **Add up the numbers **you calculated in Step 1:

^{1}⁄_{3} + ^{1}⁄_{4} + ^{1}⁄_{5} = ^{47}⁄_{60}.

Step 3: **Write the summand** ^{1}⁄_{k} to the right of the sigma. The variables i, j, and k are usually used instead of x:

Step 4: **Write the place** where the summation ends at the top of Σ. This is a right-hand Riemann sum and so the measurement ends at the right of the last rectangle, at x=5.

Step 5: **Write the place where the summation starts **at the bottom of Σ, after the index of summation (in this case, the index of summation is k).

*That’s it!*

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