Summation Notation
Prerequisites
None
Learning Objectives
 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.
We label Grape 1's weight X1,
Grape 2's weight X2, etc. The following
formula means to sum up the weights of the four grapes:
The capital Greek letter sigma (Σ)
indicates summation. The "i = 1" at the bottom indicates
that the summation is to start with X1
and the 4 at the top indicates that the summation will end with
X4. The Xi indicates
that X is the variable to be summed as i goes from 1 to 4. Therefore,
= X1 + X2 +
X3 + X4 = 4.6
+ 5.1 + 4.9 + 4.4 = 19.0.
The symbol
indicates that only the first 3 scores are to
be summed. The index variable i goes from 1 to 3.
When all the scores of a variable (such as X)
are to be summed, it is often convenient to use the following
abbreviated notation:
Thus when no values of i are shown, it means to
sum all the values of X.
Many formulas involve squaring numbers before they
are summed. This is indicated as
ΣX²=
4.62 + 5.12
+ 4.92 + 4.42
= 21.16 + 26.01+ 24.01 + 19.36 = 90.54
Notice that:
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 totals are shown
in the bottom row and the cross products. are shown in the third
column. The sum of the cross products. is 28.
In summation notation, this is written as: ΣXY
= 28.
