Whether doing calculations manually or using software, you need to ensure that you do your

formula calculations in the correct order (called the order of operation). If you evaluate the terms

and operations in the formula in the wrong order, you will get incorrect results. In a complicated

formula, the order in which you evaluate the terms and operations is governed by the interplay of

several rules arranged in a hierarchy. Most computer programs try to follow the customary

conventions that apply to typeset formulas, but you need to check software’s documentation to be

sure.

Here’s a typical set of operator hierarchy rules. Within each hierarchical level, operations are

carried out from left to right:

1. Evaluate any terms and operations within parentheses, brackets, curly braces, or absolute-

value bars first, including terms inside parentheses that follow the name of a function. Please note

that nested functions are evaluated inside out, so additional parentheses may be needed to prevent

any confusion.

2. Evaluate negation, factorials, powers, and roots.

3. Evaluate multiplication and division.

4. Evaluate addition and subtraction.

In a typeset fraction, evaluate terms and operations above the horizontal bar (the numerator)

first, then terms and operations below the bar (the denominator) next. After that, divide the

numerator by the denominator.

Equations

An equation has two expressions with an equal sign between them. Most equations appearing in this

book have a single variable name to the left of the equal sign and a formula to the right, like this:

. This style of equation defines the variable appearing on the left in terms of the

calculations specified on the right. In doing so, it also provides the “cookbook” instructions for

calculating the result, which in this case is the SEM for any values of SD and N.

The book also contains another type of equation that appears in algebra, asserting that the terms on the

left side of the equation are equal to the terms on the right. For example, the equation

asserts that x is a number that, when added to 2, produces a number that’s 3 times as large as the

original x. Algebra teaches you how to solve this expression for x, and it turns out that the answer is

.

Counting on Collections of Numbers