Introduction to Statistics
Statistics and Behavioral Sciences
Statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information.
Population is the set of all the individuals of interest in a particular study.
Sample is a set of individuals selected from a population, usually intended to represent the population in a research study.
Random Sample everyone in the population has an equal chance of being selected.
Variable is a characteristic or condition that changes or has different values for different individuals.
Data (plural) are measurements or observations.
Data Set is a collection of measurements or observations.
Datum (singular) is a single measurement or observation and is commonly called a score or raw score.
Parameter is a value, usually a numerical value, that describes a population. A parameter is usually derived from measurements of the individuals in the population.
Statistic is a value, usually a numerical value, that describes a sample. A statistic is usually derived from measurements of the individuals in the sample.
Descriptive Statistics are statistical procedures used to summarize, organize, and simplify data.
Inferential Statistics consist of techniques that allow us to study samples and then make generalizations about the populations from which they were selected.
Sampling Error is the naturally occurring discrepancy, or error, that exists between a sample statistic and the corresponding population parameter.
Observations, Measurement, and Variables
Constructs are internal attributes or characteristics that cannot be directly observed but are useful for describing and explaining behavior.
Operational Definition identifies a measurement procedure (a set of operations) for measuring an external behavior and uses the resulting measurements as a definition and a measurement of a hypothetical construct. Note that an operational definition has two components. First, it describes a set of operations for measuring a construct. Second, it defines the construct in terms of the resulting measurements.
Discrete Variable consists of separate, indivisible categories. No values can exist between two neighboring categories.
Continuous Variable, there are an infinite number of possible values that fall between any two observed values. A continuous variable is divisible into an infinite number of fractional parts.
Nominal Scale consists of a set of categories that have different names. Measurements on a nominal scale label and categorize observations, but do not make any quantitative distinctions between observations.
Ordinal Scale consists of a set of categories that are organized in an ordered sequence. Measurements on an ordinal scale rank observations in terms of size or magnitude.
Interval Scale consists of ordered categories that are all intervals of exactly the same size. Equal differences between numbers on a scale reflect equal differences in magnitude. However, the zero point on an interval scale is arbitrary and does not indicate a zero amount of the variable being measured.
Ratio Scale is an interval scale with the additional feature of an absolute zero point. With a ratio scale, ratios of numbers do reflect ratios of magnitude.
Three Data Structures, Research Methods, and Statistics
Descriptive Research or the Descriptive Research Strategy involves measuring one or more separate variables for each individual with the intent of simply describing the individual variables.
Correlational Method, two different variables are observed to determine whether there is a relationship between them.
Experimental Method, one variable is manipulated while another variable is observed and measured. To establish a cause-and-effect relationship between the two variables, an experiment attempts to control all other variables to prevent them from influencing the results.
Manipulation The researcher manipulates one variable by changing its value from one level to another.
Control The researcher must exercise control over the research situation to ensure that other, extraneous variables do not influence the relationship being examined. Control usually involves matching different groups as closely as possible on those variables that we don’t want to manipulate.
Environmental Variables These are characteristics of the environment such as lighting, time of day, and weather conditions. A researcher must ensure that the individuals in treatment A are tested in the same environment as the individuals in treatment B.
Participant Variables These are characteristics such as age, gender, motivation, and personality that vary from one individual to another. Because no two people (or animals) are identical, the individuals who participate in research studies will be different on a wide variety of participant variables.
The individuals in a research study differ on a variety of participant variables such as age, weight, skills, motivation, and personality. The differences from one participant to another are known as Individual Differences.
Independent Variable is the variable that is manipulated by the researcher. In behavioral research, the independent variable usually consists of the two (or more) treatment conditions to which subjects are exposed. The independent variable is manipulated prior to observing the dependent variable.
Dependent Variable is the one that is observed to assess the effect of the treatment. The dependent variable is the variable that is measured in the experiment and its value changes in a way that depends on the status of the independent variable.
Individuals in a Control Condition do not receive the experimental treatment. Instead, they either receive no treatment or they receive a neutral, placebo treatment. The purpose of a control condition is to provide a baseline for comparison with the experimental condition.
Individuals in the Experimental Condition do receive the experimental treatment.
In a nonexperimental study, the “independent variable” that is used to create the different groups of scores is often called the Quasi-Independent Variable.
Statistical Notation
Order of Mathematical Operations
Any calculation contained within parentheses is done first.
Squaring (or raising to other exponents) is done second.
Multiplying and/or dividing is done third. A series of multiplication and/or division operations should be done in order from left to right.
Summation using the notation is done next.
Finally, any other addition and/or subtraction is done.