Chapter 1: Introduction C ap e oduc oto Statistics

St ti ti S i d Ob tiStatistics, Science, and Observations

• Definition: The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information.

• Statistics serve four general purposes:

– Statistics are used to organize and summarize the information so that the researcher can see what happened in the research study and can communicate the results to others.

– Statistics help the researcher to answer the general questions that initiated the research by determining exactly what conclusions are justified based on the

l h b i dresults that were obtained.

Statistics, Science, and Observations , ,cont.

– Statistical procedures help ensure that the information or observations are presented and interpreted in an accurate and informative way.

– In addition, statistics provide researchers with a set of standardized techniques that are recognized and understood h h h fthroughout the scientific community.

P l ti d S lPopulations and Samples

• Definition: A population is the set of all the individuals/units of interest in a particular study.

• Definition: A sample is a set of individuals selected from a population, usually intended to represent the population in a research study.

V i blVariables

• Definition: A variable is a characteristic or condition that can change or take on different values.

• Definition: Data (plural) are measurements or observations. – A data set is a collection of

measurements or observations. – A datum (singular) is a single

measurement or observation and is commonly called a score or raw score.

P t d St ti tiParameters and Statistics

• When describing data it is necessary to distinguish whether the data come from a population or a sample.

• A characteristic that describes a population for example, the population average, is called a parameter.

• On the other hand, a characteristic that describes a sample is called a statistic.

– Thus, the average score for a sample is an example of a statistic.

• Typically, the research process begins with a question about a population parameter.

• However, the actual data come from a sample pand are used to compute sample statistics.

Descriptive and Inferential Statistical pMethods• Definition: Descriptive statistics are statistical

procedures used to summarize, organize, and simplify data.

• Definition: Inferential statistics consist of techniques that allow us to study samples and then make generalizations about the populations from which they were selected.

• Definition: Sampling error is the discrepancy, or amount of error, that exists between a sample statistic and the corresponding

l ipopulation parameter.

S hStatistics in the Context of R hResearch (Example 1.1)

Data Structures, Research Methods, and , ,Statistics• Most research is intended to examine the

relationship between two or more variables.

• To establish the existence of a relationship, researchers must make observations-that is, measurements of the variables under study.

• The resulting measurements can be classified into two distinct data structures that also help to classify different research methods and different statistical techniques.

• In the following section (next couple of slides) we identify and discuss these two data structures.

Data Structures, Research Methods, and , ,Statistics cont.

– (1) Measuring two variables for each individual: The Correlational Method

• In the correlational method, two different variables are observed to determine whether there is a relationship between them.

• Occasionally, the correlational method produces scores that are not numerical values.

• This type of data is typically summarized in a table showing how many individuals are classified into each of the possible categories.

• Table 1.1 shows an example of this kind of summary table.

Data Structures, Research Methods, and , ,Statistics cont.

• The relationship between variables for non-numerical data, such as the data in Table 1.1, is evaluated using a statistical technique known as a chi-

(Chsquare test. (Chi-square tests are presented in Chapter 18. )

• The results from a correlational study d h fcan demonstrate the existence of a

relationship between two variables, but they do not provide an explanation for the relationshipexplanation for the relationship.

• In particular, a correlational study cannot demonstrate a cause-and-effect relationshipeffect relationship.

Data Structures, Research Methods, and , ,Statistics cont.

• To demonstrate a cause-and-effect relationship between two variables, researchers must use the experimental method, which is d d hdiscussed in the next section.

– (2) Comparing two (or more) groups of scores: The Experimental Method and

l h dnon-experimental methods

• The goal of an experimental study is to demonstrate a cause-and-effect

l i hi b i blrelationship between two variables.

Th E i t l M th dThe Experimental Method

• Specifically, an experiment attempts to show that changing the value of one variable will cause changes to occur in the second variable.

• To accomplish this goal, the experimental method has two characteristics that differentiate experiments from other types of

h dresearch studies:

– Manipulation: The researcher manipulates one variable by changing its value from

l l hone level to another.

– A second variable is observed (measured) to determine whether the manipulation

hcauses changes to occur.

Th E i t l M th d tThe Experimental Method cont.

– Control: The researcher must exercise control over the research situation to ensure that other, extraneous variables do not influence the relationship being

dexamined.

• There are two general categories of variables that researchers must consider:

– Participant Variables: These are characteristics such as age, gender, and intelligence that vary from one individual

hto another.

– Environmental Variables: These are characteristics of the environment such as li h i i f d d hlighting, time of day, and weather conditions.

Th E i t l M th d tThe Experimental Method cont.

• Researchers typically use one of the three basic techniques to control other variables:

– Random assignment

– Matchingg

– Holding them constant

• The Seven Factors Needed for a Classic Experimental DesignExperimental Design

– Independent variable

– Dependent variable

C l di i /– Control condition /group

– Experimental condition /group

– Random assignment

– Pre-test

– Post-test

Oth T f St diOther Types of Studies• Other types of research studies, known as

non-experimental or quasi-experimental, are similar to experiments because they also compare groups of scores.

• These studies do not use a manipulated ariable to differentiate the gro psvariable to differentiate the groups.

• Instead, the variable that differentiates the groups is usually a pre-existing participant variable (such as male/female) or a timevariable (such as male/female) or a time variable (such as before/after).

• Because these studies do not use the manipulation and control of true experiments, they cannot demonstrate cause and effect relationships.

• As a result, they are similar to correlational research because they simply demonstrate andresearch because they simply demonstrate and describe relationships.

V i bl d M tVariables and Measurement

• Constructs and Operational Definitions

– Constructs are internal attributes or characteristics that cannot be directly observed but are useful for describing and explaining behavior.

– An 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

f h h i lmeasurement of a hypothetical construct.

Di t d C ti V i blDiscrete and Continuous Variables• Definition: A discrete variable consists of

separate indivisible categoriesseparate, indivisible categories.

• Thus, no values can exist between two neighboring categories.

Di t i bl l– Discrete variables are commonly restricted to whole countable numbers.

• For example, the number of children i f il th b f t d tin a family or the number of students attending class.

– A discrete variable may also consist of b ti th t diff lit ti lobservations that differ qualitatively.

• For example, a psychologist observing patients may classify some

h i i di d thas having panic disorders, others as having dissociative disorders, and some as having psychotic disorders

Di t d C ti V i bl tDiscrete and Continuous Variables cont.

• Definition: For a continuous variable, there are an infinite number of possible values that fall between any two observed values.

• In other words, a continuous variable is divisible into an infinite number of fractional parts.

• Two other factors apply to continuous variables:

– When measuring a continuous variable, it should be very rare to obtain identical measurements for two different individuals. Because a continuous variable has an infinite number of possible values, i h ld b l i ibl fit should be almost impossible for two people to have exactly the same score.

Di t d C ti V i bl tDiscrete and Continuous Variables cont.

• When measuring a continuous variable, each measurement category is actually an interval that must be defined by boundaries called “real limits”. (upper real limit and lower real l )limit).

S l f M tScales of Measurement

• Nominal: A 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: An 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.

S l f M t tScales of Measurement cont.

• Interval: An interval scale consists of ordered categories that are all intervals of exactly the same size.

– Equal differences between numbers on 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: A 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.

O d f M th ti l O tiOrder 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 pleft to right.

• Summation using the ∑ notation is done next.

• Finally, any other addition and/or subtraction is done.