Descriptive statistical methods
are used to summarize all of the data in an existing database into fewer
numbers, making the data easier to visualize and understand. Faulkner and
Faulkner (2009) defi ne descriptive statistical methods as “ways of
organizing, describing, and presenting quantitative (numerical) data in a
manner that is concise, manageable, and understandable” (p. 155). Descriptive
statistics utilize univariate statistical methods to examine and
summarize data one variable at a time. We can calculate numeric values that
describe samples or populations. Numeric values that describe samples are
called statistics, whereas numeric values that describe populations are
called parameters. This chapter focuses on a review of the descriptive
statistical methods commonly used in social work research. Before we turn to
these individual methods, we will fi rst look at the steps involved in defining
the variables that will be used in a study, and in determining how and at what
level these variables will be measured.
DEFINING
VARIABLES
Variables are concepts or
characteristics that vary. Constants are those concepts in a research
study that do not vary. For example, suppose we are trying to determine
depression levels of a group of sixth grade girls. The concept in this study
that will vary (variable) is the depression level, and two concepts or
characteristics that do not vary (constants) are gender (girls) and grade in
school (sixth). The process of defining a variable is called conceptualization
. For example, in the previous example, we would first have to define what
we mean by depression. Some people define depression based on the presence of
negative emotions, while others define it as a series of behavioral symptoms.
Still others view it as a combination of both emotions and behaviors. Some may
prefer to ask the participants in the study to keep a log of how often they
feel depressed, the duration of each depressive episode, and at what level of
intensity the depression is experienced.
After a variable
has been conceptualized, the next step for the researcher is to determine how
the variable will be measured. This is called operationalization . Of
course, how the variable is conceptualized affects how it will be measured or
operationalized. There are standardized instruments, such as the Brief
Depression Rating Scale, that measure the presence of depressive emotions,
such as despair and anxiety, as well as behavioral symptoms that have been tied
to depression, such as sleep disturbances and suicidal behaviors (Kellner,
1986). The researcher may create a log that the participants can make an entry
in every time they experience depression, noting time, length, and intensity.
Researchers generally turn to the social science research literature for
assistance in conceptualizing and operationalizing variables. Many concepts of
interest to social workers have been defined and measured many times by researchers.
Often, these previously defined variables and measures can be adapted for use
in new research studies.
Values
and Value Categories
The way we
operationalize the variables of interest in our research study determines the
possible values our variable can take. For example, if we measure loneliness
using a selfreport scale from 0 (not lonely at all) to 10 (lonely most or all
of the time), then the variable “loneliness” can take the values 0 to 10. If we
defi ne it as the number of times the client reports feeling lonely during a
oneweek period, then the variable can be equal to 0 and greater. If we measure
it using the UCLA Loneliness Scale (Russell, 1996), then it can have any
value from 20 to 80 (the possible values of this scale).
Variables can be
classified as continuous or discrete depending on how they are operationalized
(i.e., the set of values that they can assume). A continuous variable is
a variable that can, in theory, take on an infinite value at any decimal point
in a given interval. Examples of continuous variables are temperature, height,
and length of time. Of course, while these variables can theoretically take on
an infinite number of values, they are actually measured discretely, on a fixed
number of decimal points. For example, while theoretically temperature is a
continuous variable, we may choose to measure it to the nearest degree. A discrete
variable can only take on a finite value, typically reflected as a whole number.
Examples of discrete variables are grades on a final exam, the number of
children in a family, and annual income. A discrete variable that assumes only
two values is called a dichotomous variable. A variable that designates
whether a person is in the experimental or control group is an example of a
dichotomous variable. This variable would have two values—assignment to the
experimental group or to the control group. Some discrete variables are also
referred to as categorical variables, because their values can be grouped into
mutually exclusive categories. In the case of categorical variables, such as
race, marital status, and religious orientation, the possible values of the
variable include all of the possible categories of the variable. These
categories are sometimes called attributes. For example, the attributes
of the variable “marital status” could be defined as 1) single, never married,
2) married, 3) divorced, 4) widowed, not remarried, 5) living with a signifi
cant other, not married, and 6) other. In cases like this, it is always useful
to include a category labeled “other” to include statuses that do not fi t into
the usual categories.
Measurement
It is important
to clearly conceptualize and operationalize each variable in a research study.
Variables must be defined in such a way that the researchers involved in the
study, as well as those who utilize the research after it is published,
understand the variables in the same way. Likewise, the measurement of the
variable must be defined in such a way that everyone involved in the study will
measure it in exactly the same way each time it is measured. In addition, if
another researcher wants to replicate your study at a later date, the
measurement strategies should be clear enough so that they can be accurately
replicated.
Measurement is a systematic
process that involves assigning labels (usually numbers) to characteristics of
people, objects, or events using explicit and consistent rules so that,
ideally, the labels accurately represent the characteristic measured.
Measurement is vulnerable to errors, both systematic and random. Random
measurement errors are errors that occur randomly. Systematic
measurement error is a pattern of error that occurs across many
participants. We will cover random and systematic measurement errors in more
detail in Chapter 3.
The goal of
developing clear and accurate measurement procedures is to develop instruments
with adequate reliability and validity. The reliability of a measurement
is the degree of consistency of the measure. It refl ects the amount of random
error in the measurement. If a measure is applied repeatedly to the same person
or situation, does it yield the same result each time? For example, suppose you
use a bathroom scale to measure a person’s weight. If it indicates the same
weight each time the person steps onto it, then the scale or measure is
reliable. It may not, however, be accurate. Suppose we go to the doctor’s office
and find out that the scale at home shows the person’s weight as 20 pounds
higher than his or her actual weight. The home scale is reliable, but not
accurate based on the assumption that the doctor’s scale is accurate.
The general definition
of measurement validity is the degree to which accumulated evidence and
theory support interpretations and uses of scores derived from a measure. The validity
of a measurement refers to the accuracy of a measure. A measure can be
reliable, as in the home bathroom scale above, but it may not be accurate or
valid. Validity reflects the amount of systematic error in our measurement
procedures. Suppose two observers of a student in a classroom are given a clear
list of behaviors to count, for the purpose of measuring behaviors that
correspond to symptoms of Attention Deficit Disorder (ADD). The researcher may
have defined the behaviors that suggest the hyperactive symptoms of ADD, but
failed to include the behaviors that suggest the inattentive symptoms of ADD.
Therefore, the two observers would be able to consistently or reliably count
the hyperactive behaviors of the study, but not the inattentive behaviors, and
thus would not accurately or validly be assessing the total symptoms of ADD.
Again, the key to creating measures that are both reliable and valid is to
clearly conceptualize and operationalize each variable of interest in one’s
research study.
Levels of Measurement
The way a
variable is operationalized will generally determine the level at which the
data will be collected. If the variable “age” is defined by categories “0–10
years old,” “11–20 years old,” “21–30 years old,” etc., then we would not know
the actual age of the participants, but only these approximations. If
participants are asked to enter their age, then we would know the actual number
of years of age of each participant. If participants are asked to enter their
birth date, then we would know their age to the day. Determining what level of
measurement we need for each variable is part of operationalizing a variable.
Variables can be defined at four basic levels of measurement: nominal, ordinal,
interval, and ratio. The level of measurement used for a variable determines
the extent to which the value of a variable can be quantified.
Nominal. The first level
of measurement is the nominal level. Nominallevel variables are
categorical variables that have qualitative attributes only. The attributes or
categories defined for a nominal variable must be exhaustive (meaning every
response fits into one of the categories) and mutually exclusive (meaning each
response fits into no more than one category). In other words, every possible
response will fit into one and only one category defined for a variable. Let us
return to the variable “marital status.” Suppose the categories were defined as
follows: 1 = single; 2 = married; 3 = divorced; and 4 = widowed. What if a
person is divorced and now living as a single person? In this case, two categories
could be selected, divorced and single; therefore, the categories are not
mutually exclusive. What if a couple has been living together for 10 years and
have 3 children together? Should they select “single?” In this case, there
really are no categories that fi t the couple’s situation, thus the categories
are not exhaustive. Other examples of nominallevel variables include race,
gender, sexual orientation, and college major. As mentioned previously, a dichotomous
variable is a variable measured at the nominal level that has only two
possible attributes. Responses for dichotomous
variables may include “yes/no,” “true/false,” “control group/ experimental
group,” “male/female,” and so on.
Ordinal
The second level
of measurement is the ordinal level. Like the nominallevel variables,
ordinallevel variables are also categorical, and the attributes must also be
exhaustive and mutually exclusive. In addition to these characteristics, the
attributes of an ordinallevel variable have an inherent order or ranking to
them. For example, a variable “education level” could be defined to include the
following attributes: 1 = less than high school education; 2 = graduated high
school; 3 = some college, no degree; 4 = 2year college degree; 5 = 4year
college degree; 6 = some graduate school, no degree; and 7 = graduate college
degree. Unlike the earlier example of marital status, there is an inherent
order or ranking to the attributes for this variable. If we listed them on a
measurement instrument, it would always be listed in this order (or possibly in
reverse order). In contrast, the attributes for the nominallevel variable, “marital
status,” could be listed in any order. Other examples of ordinallevel variables
include “client satisfaction” (1 = extremely dissatisfied; 2 = dissatisfyed; 3
= neutral; 4 = satisfied; 5 = extremely satisfied) and “level of agreement” (1
= completely disagree; 2 = somewhat disagree; 3 = neither disagree nor agree; 4
= somewhat agree; 5 = completely agree).
Interval. The third level
of measurement is the interval level. While the fi rst two levels are
considered categorical variables, the values of an intervallevel variable can
be validly measured with numbers. Building on the requirements of the preceding
levels, the attributes of intervallevel variables are also exhaustive,
mutually exclusive, and rankordered. In addition, the quantitative difference
or distance between each of the attributes is equal. Looking at the variable, “education
level,” in the preceding example, there is not an equal amount of “education”
between each of the categories. The difference between “graduated high school”
and “some college, no degree” is not the same as the difference between a “4year
college degree” and a “graduate college degree.” In an intervallevel variable,
there is equal distance between each attribute. For example, consider the
scores on an IQ test that range from 50 to 150. The difference between a score
of 50 and a score of 60 (10 points) is equal to the distance between a score of
110 and 120 (10 points).
Ratio. The fourth and
final level of measurement is the ratio level. The attributes of a
ratiolevel variable are exhaustive, mutually exclusive, rankordered, and have
equal distance between each attribute. One final requirement yields a
ratiolevel variable: the presence of an absolute zero point. A variable can be
measured at the ratio level only if there can be a complete absence of the
variable. Examples include “number of children,” “monthly mortgage payment,” or
“number of years served in prison.” Note how all of these could be given the
value of 0 to indicate an absence of the variable. In contrast, a temperature
of 0 degrees Fahrenheit does not indicate an absence of temperature; therefore,
temperature would be an intervallevel variable rather than a ratiolevel
variable. See Table 2.1 for an overview of the levels of measurement described.
It is important
to reiterate that a variable can often be defined at more than one level of
measurement, depending on how it is conceptualized and operationalized. The
researcher sometimes uses more than one variable at different levels of
measurement in order to capture a concept more fully. See Table 2.2 for an
example of how we can measure eating disordered behaviors using measurements at
all four levels.
Table 2.2
Level of Measurement

Characteristics

Examples

Nominal

Attributes are
exhaustive
Attributes are
mutually exclusive

Race
Gender
Sexual
orientation

Ordinal

Attributes are
exhaustive
Attributes are
mutually exclusive
Attributes
have an inherent order

Client
satisfaction
Highest
educational
Achievement
Level of
agreement

Interval

Attributes are
exhaustive
Attributes are
mutually exclusive
Attributes
have an inherent order
Differences
between attributes are equal

IQ Score
Temperature
SAT Scor

Ratio

Attributes are
exhaustive
Attributes are
mutually exclusive
Attributes
have an inherent order
Differences
between attributes are equal
Attributes
have an absolute 0point

Number of children
Monthly income
Number of
times
married
