Use summation notation to express the sum of all numbers

Use summation notation to express the sum of a subset of numbers

Use summation notation to express the sum of squares

Many statistical formulas involve summing numbers.
Fortunately there is a convenient notation for expressing summation.
This section covers the basics of this summation
notation.

Let's say we have a variable X that represents the
weights (in grams) of 4 grapes. The data are shown in Table 1.

Table 1. Weights of 4 grapes.

Grape

X

1
2
3
4

4.6
5.1
4.9
4.4

We label Grape 1's weight X_{1},
Grape 2's weight X_{2}, etc. The following
formula means to sum up the weights of the four grapes:

The Greek letter capital sigma (Σ)
indicates summation. The "i = 1" at the bottom indicates
that the summation is to start with X_{1}
and the 4 at the top indicates that the summation will end with
X_{4}. The "X_{i}" indicates
that X is the variable to be summed as i goes from 1 to 4. Therefore,

because the expression on the left means to sum up all the values
of X and then square the sum (19² = 361),
whereas the expression on the right means to square the numbers
and then sum the squares (90.54, as shown).

Some formulas involve the sum of cross products.
Table 2 shows the data for variables X and Y. The cross products (XY) are shown in the third
column. The sum of the cross products is 3 + 4 + 21 = 28.

Table 2. Cross Products.

X

Y

XY

1
2
3

3
2
7

3
4
21

In summation notation, this is written as: ΣXY
= 28.