Basic Math Operations - Mathematics and Statistics for the Environment - page 48

■ EXAMPLE 2.20
Problem: How is the term (3/8) 2 written in expanded form?

Note: When parentheses are used, the exponent refers to the entire term within the parentheses.

Solution: In this example, (3/8) 2 means:

(3/8) 2 = (3/8 × 3/8)

Key Point: When a negative exponent is used with a number or term, a number can be rewritten
using a positive exponent:

6 -3 = 1/6 3

Another example is

11 -5 = 1/11 5

■ EXAMPLE 2.21
Problem: How is the term 8 -3 written in expanded form?

Solution:

1
8

1
888

− == ×× 3

8

3

Key Point: A number or letter written as, for example, 3 0 or X 0 does not equal 3 × 1 or X × 1, but
simply 1.

2.7 AVERAGES (ARITHMETIC MEAN)
Whether we speak of harmonic mean, geometric mean, or arithmetic mean, each represents the
“center,” or “middle,” of a set of numbers. They capture the intuitive notion of a “central tendency”
that may be present in the data. In statistical analysis, an “average of data” is a number that indicates
the middle of the distribution of data values.
An average is a way of representing several different measurements as a single number. Although
averages can be useful in that they tell us “about” how much or how many, they can also be mislead-
ing, as we demonstrate below. You will find two kinds of averages in environmental engineering
calculations: the arithmetic mean (or simply mean ) and the median.

■ EXAMPLE 2.22
Problem: The operator of a waterworks or wastewater treatment plant takes a chlorine residual
measurement every day; part of the operator's log is shown below. Find the mean.

DEFINITION

The mean (what we usually refer to as an average ) is the total of values of a set of observations
divided by the number of observations. We simply add up all of the individual measurements
and divide by the total number of measurements we took.