Page:Online Statistics Education.pdf/53

 $$\sum_{i=1}^{4}X_{i}=X_{1}+X_{2}+X_{3}+X_{4}=4.6+5.1+4.9+4.4=19$$

The symbol

$$\sum_{i=1}^{3}X_{i}$$

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:

$$\sum{X}$$

Thus, 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

$$\sum{}X^{2}=4.6^{2}+5.1^{2}+4.9^{2}+4.4^{2}$$

$$=21.16+26.01+24.01+19.36=90.54.$$

Notice that:

$$\left( \sum{X}\right)^{2}\neq\sum{}X^{2}$$

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 28 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. 53