## Social Items

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 self-report 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 one-week 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. Nominal-level 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 nominal-level 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 nominal-level variables, ordinal-level variables are also categorical, and the attributes must also be exhaustive and mutually exclusive. In addition to these characteristics, the attributes of an ordinal-level 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 = 2-year college degree; 5 = 4-year 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 nominal-level variable, “marital status,” could be listed in any order. Other examples of ordinal-level 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 interval-level variable can be validly measured with numbers. Building on the requirements of the preceding levels, the attributes of interval-level variables are also exhaustive, mutually exclusive, and rank-ordered. 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 “4-year college degree” and a “graduate college degree.” In an interval-level 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 ratio-level variable are exhaustive, mutually exclusive, rank-ordered, and have equal distance between each attribute. One final requirement yields a ratio-level 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 interval-level variable rather than a ratio-level 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 0-point Number of children Monthly income Number of times married

## Descriptive Statistical Methods

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 self-report 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 one-week 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. Nominal-level 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 nominal-level 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 nominal-level variables, ordinal-level variables are also categorical, and the attributes must also be exhaustive and mutually exclusive. In addition to these characteristics, the attributes of an ordinal-level 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 = 2-year college degree; 5 = 4-year 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 nominal-level variable, “marital status,” could be listed in any order. Other examples of ordinal-level 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 interval-level variable can be validly measured with numbers. Building on the requirements of the preceding levels, the attributes of interval-level variables are also exhaustive, mutually exclusive, and rank-ordered. 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 “4-year college degree” and a “graduate college degree.” In an interval-level 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 ratio-level variable are exhaustive, mutually exclusive, rank-ordered, and have equal distance between each attribute. One final requirement yields a ratio-level 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 interval-level variable rather than a ratio-level 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 0-point Number of children Monthly income Number of times married

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