Statistics Review - Mathematics and Statistics for the Environment - page 110

Subscripts are used to represent individual observations within these sets of data. Thus, X i repre-
sents the income of the i th company, where i takes on the values 1, 2, 3, 4, and 5. Using this notation,
X 1 , X 2 , X 3 , X 4 , and X 5 stand for the incomes of the first company, the second company, and so on.
The data are arranged in some order, such as by size of income, the order in which the data were
gathered, or any other way suitable to the purposes or convenience of the investigator.
The subscript i is a variable used to index the individual data observations; therefore, X i , Y i , and
Z i represent the income, materials expenditures, and savings of the i ith company. For example, X 2
represents the income of the second company, Y 2 the materials expenditures of the second company,
and Z 5 the savings of the fifth company.
Suppose that we have data for two different samples: the net worths of 100 companies and the
test scores of 30 students. To refer to individual observations in these samples, we can let X i denote
the net worth of the i th company, where i assumes values from 1 to 100 (as indicated by the notation
i = 1, 2, 3, …, 100.) We can also let Y j denote the test score of the j th student, where j = 1, 2, 3, …,
20. The different subscript letters make it clear that different sample are involved. Letters such as
X, Y , and Z generally represent the different variables or types of measurements involved, whereas
subscripts such as i , j , k , and l designate individual observations (Hamburg, 1987).
Next, we turn our attention to the method of expressing summations of sets of data. Suppose we
want to add a set of four observations, denoted X 1 , X 2 , X 3 , and X 4 . A convenient way of designating
this addition is

4

∑ =+++
1

XXXXX

i 1 2 3 4

i =

where the symbol ∑ (Greek capital “sigma”) means the “sum of.” Thus, the following

4

∑ 1

X i

i =

is read “the sum of the X values going from 1 to 4.” For example, if X 1 = 5, X 2 = 1, X 3 = 8, and X 4 =
6, then

4

∑ =+++=
1

X i 5186 20

i =

In general, if there are n observations, we write

n

∑ =++++
1

XXXX … X

i 1 2 3 n

i =

Basic statistical terms include mean or average, median, mode, and range. The following is an
explanation of each of these terms.

7.3.1 m ean
Mean is one of the most familiar and commonly estimated population parameters. It is the total of
the values of a set of observations divided by the number of observations. Given a random sample,
the population mean is estimated by

n

∑ 1

X

i

= =

i

X

n