Introduction to Statistics
Chapter 1: Introduction to Statistics
Learning Outcomes
Know key statistical terms
Know key measurement terms
Know key research terms
Know the place of statistics in science
Understand summation notation
Math Skills Assessment
Statistics requires basic math skills.
Inadequate basic math skills puts students at risk in this course.
Appendix A, Math Skills Assessment:
Helps determine if students need a skills review.
Appendix A, Math Skills Review:
Provides a quick refresher course on necessary areas.
The final Math Skills Assessment identifies students' basic math skills competence.
1-1 Statistics and Behavioral Sciences
Definition of Statistics:
Statistics means "statistical procedures."
Uses of Statistics:
Organizes and summarizes information.
Determines what general conclusions are justified based on results obtained.
Goals of Statistical Procedures:
Accurate and meaningful interpretation.
Standardized evaluation procedures.
Populations and Samples
Population:
The set of all individuals of interest in a particular study.
Varies in size; often quite large.
Sample:
A set of individuals selected from a population.
Typically intended to represent the population in a research study.
Figure 1.1: The Relationship between a Population and a Sample
The Population:
All of the individuals of interest.
The Sample:
The individuals selected to participate in the research study.
Generalization:
Results from the sample are generalized to the population.
Variables and Data
Variable:
A characteristic or condition that changes or has different values for different individuals.
Data (plural):
Measurements or observations of a variable.
Data Set:
A collection of measurements or observations.
Datum (singular):
A single measurement or observation; commonly called a score or raw score.
Parameters and Statistics
Parameter:
A value, usually numerical, that describes a population.
Derived from measurements of the individuals in the population.
Statistic:
A value, usually numerical, that describes a sample.
Derived from measurements of the individuals in the sample.
Descriptive & Inferential Statistical Methods (1 of 2)
Descriptive Statistics:
Summarize, organize, and simplify data.
Familiar examples include tables, graphs, and averages.
Inferential Statistics:
Study samples to make generalizations about the populations from which they were selected.
Interpret experimental data.
Common Terminology:
"Sampling error" (also known as "margin of error").
"Statistically significant."
Descriptive & Inferential Statistical Methods (2 of 2)
Key Note:
A sample is never identical to the population.
Sampling Error:
The discrepancy, or amount of error, between a sample statistic and the corresponding population parameter.
Example of Margin of Error in Polls:
Example statement: "This poll was taken from a sample of registered voters and has a margin of error of plus or minus 4 percentage points."
Figure 1.3: The Role of Statistics in Research
Experimental Design Example:
Compare two studying methods.
Data:
Reading scores for samples A and B.
Sample A reading from computer screens and Sample B reading from printed pages.
Analysis Steps:
Step 2: Descriptive statistics to organize and simplify data.
Step 3: Inferential statistics to interpret the results.
Results Interpretation:
Samples show a 4-point average difference.
Interpretations may be:
No difference exists and the variation is due to chance (sampling error).
Actual difference exists and the data reflects this.
Learning Check 1 (1 of 2)
Scenario: A researcher investigates sleep effects on exam scores with a group of 75 boys. The boys are _.
Options: statistic, variable, parameter, or sample.
Learning Check 1 – Answer (1 of 2)
Answer: The boys are a sample.
Learning Check 1 (2 of 2)
True/False Statements:
Most research studies data from samples.
When samples differ from the population, there is a systematic difference between groups.
Learning Check 1– Answer (2 of 2)
True: Samples are used as measuring all individuals is often impractical.
False: Sampling error might produce unsystematic group differences.
1-2 Observations, Measurement, and Variables
Empirical Basis of Science:
Science is based on observation.
Data Collection:
Scores in research studies come from observing and measuring variables.
Measurement Process:
Involves applying clearly defined procedures for each variable.
Constructs & Operational Definitions
Constructs:
Internal attributes or characteristics that cannot be directly observed; useful for describing and explaining behavior.
Operational Definition:
Identifies procedures for measuring an external observable behavior, utilizing measurements as both definitions and indicators of hypothetical constructs.
Discrete and Continuous Variables
Discrete Variable:
Comprises separate, indivisible categories; no values exist between neighboring categories.
Continuous Variable:
Has an infinite number of possible values between any two observed values; divisible into endless fractional parts.
Figure 1.4: Example of Continuous Measurement
Continuity Illustration:
Scores like 148.5, 149, 150.3 illustrate continuous variables distinct from discrete.
Real Limits of Continuous Variables
Definition of Real Limits:
Boundaries of intervals for scores depicted on a continuous number line.
The real limit separating two adjacent scores is positioned precisely halfway between them.
Real Limits Specifics:
Each score has two real limits:
The upper real limit marks the top of the interval.
The lower real limit marks the bottom of the interval.
Scales of Measurement (1 of 2)
Measurement Purpose:
Assign individuals or events to categories, which can either be names or numerical values.
Scale Composition:
Categories for measuring variables make up the measurement scale.
The relationships among categories determine the nature of the scale.
Scales of Measurement (2 of 2)
Scale Characteristics and Examples:
Nominal Scale:
Characteristics: Label and categorize without quantitative distinctions.
Examples: Gender, diagnosis, experimental or control groups.
Ordinal Scale:
Characteristics: Categorizes observations and is organized by size or magnitude.
Examples: Rank in class, clothing sizes (S, M, L, XL), Olympic medals.
Interval Scale:
Characteristics: Ordered categories with equal size intervals and an arbitrary or absent zero.
Examples: Temperature, IQ, golf scores (above/below par).
Ratio Scale:
Characteristics: Ordered categories, equal intervals, and an absolute zero.
Examples: Number of correct answers, time to complete tasks, gain in height and/or weight since last year.
Learning Check 2 (1 of 2)
Scenario: A study looks at the optimal size for study groups. The variable size of the group is _.
Options: Discrete and interval, Continuous and ordinal, Discrete and ratio, Continuous and interval.
Learning Check 2 – Answer (1 of 2)
Answer: Discrete and ratio.
Learning Check 2 (2 of 2)
True/False Statements:
Variables that cannot be directly measured cannot be studied scientifically.
Research measurements follow specific procedures defining constructs.
Learning Check 2 – Answer (2 of 2)
False: Constructs can be operationally measured, despite being indirectly observed.
True: Operational definitions ensure consistent measurement and define constructs.
1-3 Three Data Structures, Research Methods, and Statistics
Data Structure 1: Descriptive Research (Individual Variables):
Measurement of one or more variables per individual.
Statistics describe the observed variable, utilizing both categorical and numerical variables without concern for relationships.
Relationships between Variables (1 of 3)
Defining Relationships:
Observing and measuring two or more variables aims to determine the nature of their relationship.
Data Structures for Relationships:
Measured variables can be classified into specific structures to analyze relationships.
Relationships between Variables (2 of 3)
Data Structure 2: Correlational Method:
One group of participants is studied.
Measurement of two variables for each participant aims to describe the type and magnitude of the relationship.
Patterns reveal relationships via a nonexperimental study method.
Limitations of the Correlational Method
Correlation Insights:
It can demonstrate relationships between variables but does not explain them.
It cannot establish cause-and-effect relationships between the variables.
Relationships between Variables (3 of 3)
Data Structure 3: Experimental and Nonexperimental Methods:
Involves comparing two or more groups of scores.
A specific variable defines the groups while the second variable's scores are recorded.
Research Method Types:
Both experimental and nonexperimental studies apply this structure.
Figure 1.6: Data Structure 3: Comparing Two (or More) Groups of Scores
Example of Comparison:
Experimental Group: Scores related to video game types and aggressive behavior.
Nonviolent group comparison vs. violent group analysis.
The Experimental Method
Goal:
Establish a cause-and-effect relationship between two variables.
Variable Manipulation:
The researcher determines the level of one variable, known as the independent variable.
Control Measures:
Control is applied to eliminate the influence of extraneous variables, including participant and environmental variables.
Terminology in the Experimental Method
Independent Variable:
The variable manipulated by the researcher, unaffected by other study variables.
Dependent Variable:
The variable observed to assess the treatment's effect, depending on the independent variable's level.
Control Conditions in an Experiment
Control Methods:
Random assignment of subjects, matching subjects, or maintaining constant influential variables.
Control Condition:
Individuals either receive no treatment or a neutral/placebo treatment for baseline comparison against the experimental condition.
Experimental Condition:
Participants receive the experimental treatment.
Nonexperimental Methods: Nonequivalent Groups and Pre-Post Studies
Nonequivalent Groups:
Comparing groups of scores without control over group assignment.
Pretest/Posttest:
Individuals measured at two points over time, implicating the passage of time as a variable.
Quasi-independent Variable:
The independent variable is not manipulated but used to categorize groups.
Figures of Nonexperimental Studies
Example (a):
Participant Location as a quasi-independent variable.
Measured verbal test scores serve as the dependent variable.
Example (b):
Time as a quasi-independent variable before and after therapy.
Depression scores measured across two different times.
Learning Check 3 (1 of 2)
Scenario: Researchers observed higher exam scores with increased sleep. This study is _.
Options: Descriptive, Experimental comparison of groups, Non-experimental group comparison, Correlational.
Learning Check 3 – Answer (1 of 2)
Answer: This study is correlational.
Learning Check 3 (2 of 2)
True/False Statements:
All research methods have an independent variable.
All research methods can show cause-and-effect relationships.
Learning Check 3 – Answer (2 of 2)
False: Correlational methods can operate without an independent variable.
False: Only experiments can effectively control participant and environmental influences.
1-4 Statistical Notation
Statistical Operations and Notations:
Familiar operations are applied alongside specific statistical notation.
Common symbols:
X (and Y) for scores,
N for the number of scores in a population,
n for the number of scores in a sample.
Summation Notation
Summation in Statistics:
Many statistical procedures involve summing scores using the summation sign Σ.
Followed by a symbol or equation indicating what needs summing.
Order of operations:
Perform operations in parentheses, squaring, and multiplication/division first;
Summation comes before addition or subtraction.
Learning Check 4 (1 of 2)
Expression: ∑X² + 47 instructs you to _.
Options: Square each score and add 47, add squared scores then add 47, or other variations.
Learning Check 4 – Answer (1 of 2)
Correct Process: Square each score, add up squared scores, then add 47 to that sum.
Learning Check 4 (2 of 2)
True/False Statements:
∑X² = (∑X)²
(∑X) • (∑X) = (∑X)²
Learning Check 4 – Answer (2 of 2)
False: Different operation orders yield different results.
True: Definition of (∑X)².
Clear Your Doubts, Ask Questions
Address any queries about concepts or equations to clarify understanding.