The Fundamentals of Psychological Statistics SY 2024-2025 PDF

THE FUNDAMENTALS OF PSYCHOLOGICAL STATISTICS KENNETH R. ALQUINO, RPm SY 2024-2025 1-C MWF (7:15 – 9:15PM) S (12:00-3:00PM) 1-D TTH (6:30 – 9:30PM) S (8:00-11:00AM) TOP 10 COUNTRIES WITH THE HIGHEST AVERAGE IQ - ULSTER INSTITUTE 2019: DEPARTMENT OF HEALTH (2024)  Do you pl...

THE FUNDAMENTALS OF PSYCHOLOGICAL STATISTICS KENNETH R. ALQUINO, RPm SY 2024-2025 1-C MWF (7:15 – 9:15PM) S (12:00-3:00PM) 1-D TTH (6:30 – 9:30PM) S (8:00-11:00AM) TOP 10 COUNTRIES WITH THE HIGHEST AVERAGE IQ - ULSTER INSTITUTE 2019: DEPARTMENT OF HEALTH (2024)  Do you plan to enroll next term? Yes No. I plan to take a gap year No. I plan to transfer to other schools STUDENT ATTRITION RATE 2024  PSYCHOMETRICIAN 2018-2023 80% 70% 60% 50% 40% 30% 20% 10% 0% 2017 2018 2019 2020 2021 2022 2023 2024 Passed Failed RPM 2018-2023 WHY LEARN STATISTICS 1. Understanding statistics to read psychological research 2. Understanding statistics to do your research 3. Understanding statistics to develop your analytical and critical thinking INTRODUCTION TO STATISTICS  DEFINE STATISTICS Statistics – a branch of mathematics that focuses on the organization, analysis, and interpretation of a group of numbers. Statistics is a method of pursuing the truth: Predicting the likelihood that your hunch is true. What will happen if we use this type of intervention? Is there a significant difference in students’ examination scores compared to different teaching methods?  A PIECE OF CAKE In behavioral science research, the primary goal is often to understand or explain behaviors, characteristics, or phenomena within a specific group or group of individuals. Population A population includes all individuals of interest in a study. It can be very broad (e.g., all women on Earth) or more specific (e.g., women who are registered voters in the U.S.). Populations aren't limited to people; they can include animals, organizations, products, etc.  A PIECE OF CAKE Sample A population includes all individuals of interest in a study. Due to the impracticality of studying entire populations, researchers select a sample, a smaller group from the population. A sample should be representative of the population to allow for generalizations. The size of samples can vary significantly depending on the study's design and objectives. TWO BRANCHES OF STATISTICS  A PIECE OF CAKE Generalization After analyzing the sample, researchers aim to generalize the findings to the entire population. This step is crucial for the research to provide meaningful insights about the broader group.  BASIC STATISTICAL CONCEPTS VARIABLES Typically, researchers are interested in specific characteristics of the individuals in the population (or in the sample), or they are interested in outside factors that may influence the individuals. A variable is a characteristic or condition that can have different values. It is something that can vary among individuals or over time. Examples: height, social class, test scores, stress levels, gender, educational levels, etc. VARIABLE Height Stress Level Blood Type  BASIC STATISTICAL CONCEPTS VALUES A value is a specific number or category assigned to a variable. It represents the numerical or classification of the variable for each individual. We assign value to a variable to quantify, measure, and analyze different attributes or characteristics. VARIABLE VALUE Height 160 cm, 175 cm, 180 cm Stress Level In a scale of 1-10 Blood Type A, B, O, AB  BASIC STATISTICAL CONCEPTS SCORES A score is the specific value assigned to an individual on a given variable. It indicates the individual's position or status regarding that variable. A datum (singular) is a single measurement or observation and is commonly called a score or raw score VARIABLE VALUE SCORE Height 160 cm, 175 cm, 180 cm 175 cm Stress Level In a scale of 1-10 7 on a scale Blood Type A, B, O, AB A Blood Type  BASIC STATISTICAL CONCEPTS 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 average score for the population—is called a parameter. A characteristic that describes a sample is called a statistic Every population parameter has a corresponding sample statistic, and most research studies involve using statistics from samples as the basis for answering questions about population parameters  TWO BRANCHES OF STATISTICS Descriptive Statistics – Descriptive statistics are used to organize, summarize, and simplify the data. They provide a way to present raw data in a more understandable and interpretable form. Tables and Graphs: Data can be displayed in tables or visualized through graphs such as histograms, bar charts, and scatter plots. This helps in seeing patterns, trends, and distributions at a glance. Measures of Central Tendency: These include the mean (average), median, and mode. These measures provide a central or typical value for the dataset. Measures of Variability: These include the range, variance, and standard deviation. They describe the spread or dispersion of the data points.  TWO BRANCHES OF STATISTICS Inferential Statistics – Inferential statistics are used to make generalizations or predictions about a population based on sample data. They allow researchers to conclude the immediate data available. Estimation: This includes point estimates and confidence intervals. For example, estimating the population mean from the sample mean. Hypothesis Testing: This involves testing assumptions about a population parameter. Common tests include t-tests, chi-square tests, and ANOVA. These tests help determine if observed patterns or differences in the data are statistically significant.  SAMPLING ERROR The concept of sampling error is fundamental to understanding how inferential statistics work. When researchers gather data from a sample instead of the entire population, they aim to make inferences about the population as a whole. However, because the sample is only a subset of the population, it is unlikely to perfectly represent the population's characteristics. This discrepancy between the sample statistic (e.g., sample mean, sample proportion) and the corresponding population parameter (e.g., population mean, population proportion) is known as sampling error. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Individual Variables: Descriptive Researches Some research studies are conducted simply to describe individual variables as they exist naturally. Collecting data without any manipulation or intervention Relationships Between Variables Most research, however, is intended to examine relationships between two or more variables. To understand and analyze these relationships, researchers rely on two primary data structures: correlational and experimental data. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS One Group with Two Variables Measured for Each Individual: The Correlational Method: One method for examining the relationship between variables is to observe the two variables as they exist naturally for a set of individuals. Aim: Examining the relationship between sleep quality and stress levels in adults. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Statistics for Correlational Methods When the data from a correlational study consists of numerical scores, the relationship between the two variables is usually measured and described using a statistic called a correlation. Correlation quantifies the strength and direction of the linear relationship between the two numerical variables. Interpretation: Positive Correlation, Negative Correlation, or No Correlation Non-numerical data: Chi-Square test – to evaluate the relationship between two categorical variables. Numerical/Dummy Code: Even though the original data are non-numerical, they can be coded with numbers for analysis purposes DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Limitations of the Correlational Method Correlation does not explain the relationship It cannot demonstrate a cause-and-effect relationship They do not provide information about whether one variable causes changes in another. There may be other variables, known as confounding variables, that are influencing both variables in the study. Example: There is a systematic relationship between wake-up time and academic performance among college students. Students who wake up late = poor academic performance. Possible explanations: Lifestyle, health, social, or other factors. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS To determine causality, researchers must use experimental methods. Comparing Two (or More) Groups of Scores: Experimental and Nonexperimental Methods This method involves comparing the scores of different groups to examine the relationship between variables. Statistics for Comparing Two (or More) Groups of Scores In research studies that compare groups of scores, the statistical procedures are designed to both describe the data and determine if the observed differences can be generalized to the entire population. Descriptive and Inferential Statistics: describe the data and compare them to determine whether the differences observed between groups are statistically significant. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Experimental and Nonexperimental Methods There are two distinct research methods that both produce groups of scores to be compared: the experimental and the nonexperimental strategies Experimental Research Method: The primary goal is to establish a cause-and- effect relationship between variables. Nonexperimental Research Method: The goal is to observe and describe relationships between variables without establishing causality. Researchers observe the variables as they naturally occur without manipulation or intervention. Example: A researcher might observe whether students who naturally wake up early tend to have higher academic performance compared to those who wake up late. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS The Experimental Method The primary objective is to demonstrate that changing the value of one variable causes changes in another variable. Characteristics of Experimental Method Manipulation: The researcher actively manipulates one variable by changing its value from one level to another. Control: To establish a clear cause-and-effect relationship, the researcher must control for extraneous variables that could influence the relationship being examined. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS The Experimental Method In experimental research, it's crucial to control participant and environmental variables to ensure that any observed effects are truly due to the manipulated independent variable. There are two general categories of variables that researchers must consider: Participant Variables Environmental Variables DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Participant Variables: Participant variables are characteristics such as age, gender, and intelligence that vary among individuals. Researchers must ensure that participant variables are balanced across experimental conditions to prevent them from confounding the results. Example: Investigating the impact of social media usage on self-esteem among adolescents. Recruiting both boys and girls aged 16 years old as participants in the study. Investigating the impact of violent games on aggression. Females and males were grouped separately. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Environmental Variables: In experimental research, controlling for environmental factors is crucial to ensure that any observed effects can be attributed to the variables being studied rather than extraneous influences Characteristics of the environment such as lighting, time of day, and weather conditions can potentially impact participants' behavior and responses. Example: Investigating the impact of music on students’ stress levels. Individuals in one treatment group (Group A) are tested in a different room compared to another treatment group (Group B). This would produce a confounded experiment because the researcher could not determine what causes the results. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Essential Techniques in Experimental Research for Controlling Variables 1. Random Assignment: This technique involves assigning participants to different treatment conditions randomly. The aim is to ensure that each participant has an equal chance of being assigned to any of the treatment groups. Researchers can distribute participant characteristics (such as intelligence, age, or other relevant factors) evenly across the groups. Random assignment can also be used to control environmental variables DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Essential Techniques in Experimental Research for Controlling Variables 2. Matching: Matching involves pairing participants in different treatment groups based on specific characteristics to ensure equivalence between groups. By matching participants on key variables, researchers can create equivalent groups, reducing the potential for confounding variables to influence the results. 3. Holding Variables Constant: This technique involves keeping certain variables constant across experimental conditions. By holding variables constant, researchers can isolate the effects of the manipulated variable without interference from other factors. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Terminology in the Experimental Method In experimental research, two specific types of variables are crucial: the independent variable and the dependent variable. Independent Variable: This is the variable that the researcher manipulates or controls to observe its effect on the dependent variable. Dependent Variable: This is the variable that is observed and measured to assess the effect of the independent variable. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Terminology in the Experimental Method Control Conditions in an Experiment: An experimental study evaluates the relationship between two variables by manipulating one variable (the independent variable) and measuring one variable (the dependent variable). Often an experiment will include a condition in which the participants do not receive any treatment. The scores from these individuals are then compared with scores from participants who do receive the treatment. The goal of this type of study is to demonstrate that the treatment has an effect by showing that the scores in the treatment condition are substantially different from the scores in the no- treatment condition. Control condition: The group that does not receive the experimental treatment Experimental condition: The group that does receive the experimental treatment Note: Independent variable always consists of at least two values to consider it a variable. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Nonexperimental Methods: Nonequivalent Groups and Pre-Post Studies Nonequivalent Groups Study: This type of study involves comparing two groups of scores, similar to an experiment. However, unlike in a true experiment where participants are randomly assigned to groups, in a nonequivalent group study, the researcher cannot control the assignment of participants to groups. Pre–Post Study: In a pre–post-study, participants are measured on the same variable (e.g., depression) before and after a treatment or intervention. While this design allows researchers to examine changes in the dependent variable over time, it lacks control over extraneous variables that may influence the results. DATA STRUCTURES, RESEARCH METHODS, & STATISTCS Terminology in Nonexperimental Research Independent Variable in Nonexperimental Research (Quasi-Independent Variable: In nonexperimental studies, the variable used to create groups is often referred to as the independent variable, just as in experimental research. However, it's crucial to recognize that in nonexperimental studies, the independent variable is not manipulated by the researcher. Dependent Variable: Similarly, the variable that is measured to obtain scores within each group is still referred to as the dependent variable in nonexperimental studies.  VARIABLES & MEASUREMENT Constructs and Operational Definitions In research, the measurement concept is fundamental, allowing us to quantify and analyze variables systematically. Variables can be broadly categorized into concrete variables and hypothetical constructs. Concrete Variables: Concrete variables are tangible and can be directly observed and measured. These variables are straightforward because they have physical manifestations that can be objectively quantified using standard measurement tools and methods. Height, Weight, Eye Colour, etc Hypothetical Constructs: Constructs are intangible and cannot be directly observed. They are used to explain and describe behaviors and internal characteristics. Intelligence, Anxiety, Stress, etc  VARIABLES & MEASUREMENT Operational Definitions Researchers use operational definitions to measure constructs, which specify the external behaviors or indicators that represent the construct. Identifying Behaviors: Determining which behaviors are representative of the construct. Defining Measurement Methods: Establishing how these behaviors will be quantified. For Example: Intelligence: Measured by performance on standardized tests like IQ tests. The test scores serve as an operational definition of intelligence. Anxiety: Measured by self-report questionnaires, physiological indicators (e.g., heart rate), or behavioral observations  VARIABLES & MEASUREMENT Another distinction that researchers sometimes make is between discrete variables and continuous variables. Discrete Variables: These are variables that can only take on specific, distinct values. They often represent counts of things or categorical distinctions. For example, the number of children in a family, the number of students in a class, or the type of car a person drives are all examples of discrete variables. You can't have fractional or in-between values for discrete variables. For instance, you can't have 2.5 children or 1.75 cars.  VARIABLES & MEASUREMENT Continuous Variables: These variables can take on an infinite number of values within a certain range. They are often measurements of things that can be broken down into smaller and smaller increments. Examples include height, weight, time, temperature, and age (when measured in years and fractions of a year). Unlike discrete variables, you can have any value within a given range for continuous variables. For example, you can be 5.3 feet tall or 98.6 degrees Fahrenheit.  VARIABLES & MEASUREMENT When working with continuous variables, it’s important to consider the nature of measurement and the concept of real limits. Real Limits in Continuous Measurement Each measurement is an interval rather than a specific point when measuring continuous variables. These intervals are defined by real limits, boundaries set exactly halfway between adjacent possible values. Example: If you measure someone's weight and record it as 150 pounds, the lower real limit is 149.5 pounds, and the upper real limit is 150.5 pounds. This means the actual weight is somewhere between the lower and upper limit. Importance of Real Limits Real limits help ensure precision in measurements and clarify that recorded values are not exact points but intervals within which the actual value falls. This is crucial in maintaining the accuracy and reliability of data in research involving continuous variables.  VARIABLES & MEASUREMENT In psychology and statistics, variables are used to measure and analyze different characteristics or attributes – Levels of Measurement Numerical Variables Equal-Interval Variables: Variables where the numbers represent equal amounts of what is being measured. GPA (Grade Point Average): The difference between a GPA of 2.5 and 2.8 is about the same as the difference between a GPA of 3.0 and 3.3. Stress Ratings: A difference between stress ratings of 4 and 6 is about the same as the difference between 7 and 9.  VARIABLES & MEASUREMENT Numerical Variables Ratio Variables: A special type of equal-interval variable that has an absolute zero point, indicating the complete absence of the variable. Number of Siblings: A zero value means having no siblings. Someone with four siblings has twice as many as someone with two. Other Examples: Age in years, earnings in a month, time, and distance.  VARIABLES & MEASUREMENT Numerical Variables Rank-Order Variables: Variables where the numbers represent the order or rank of values but do not indicate equal intervals between ranks. Class Standing: The rank in a graduating class, where the difference in GPA between being second and third might differ from the difference between being eighth and ninth. Other Examples: year and grade levels  VARIABLES & MEASUREMENT Categorical Variables Nominal Variables: Variables where the values are names or categories and do not have a numerical or ordered relationship. Gender: Values are categories such as female and male. Psychiatric Diagnosis: Categories include major depression, post- traumatic stress disorder, schizophrenia, and obsessive-compulsive disorder. Characteristics: Used for categorizing data into distinct groups or types without implying any order or quantity.  VARIABLES & MEASUREMENT Variables represent different levels of measurement, and the choice of measurement level affects the statistical methods used in analysis. Nominal, Ordinal Interval, and Ratio Importance of Levels of Measurement Measurement Selection: The choice of measurement level affects how data is collected, interpreted, and analyzed. Researchers select the level that best suits the research question and the nature of the variable. Statistical Analysis: Different levels of measurement allow for different types of statistical analysis.  VARIABLES & MEASUREMENT Properties of Scales: Three important properties make scales of measurement different from one another: magnitude, equal intervals, and an absolute 0. Magnitude – the property of moreness. Magnitude means that one value can be considered more or less than another value. Examples: On a scale from 0 to 10, a stress level of 8 indicates more stress than a stress level of 4. If one person is 6 feet tall and another person is 5 feet tall, the person who is 6 feet tall has more height.  VARIABLES & MEASUREMENT Equal Interval – consistent differences between points Equal interval means that the difference between two points is the same no matter where you are on the scale. Examples: If the difference between a stress level of 2 and 3 is the same as the difference between 7 and 8, the scale has equal intervals. Important Note: Psychological tests often don't have perfect equal intervals. It's not like a ruler where every inch is exactly the same.  VARIABLES & MEASUREMENT Absolute 0 – complete absence of the property Absolute 0 means that a score of zero represents a complete lack of whatever is being measured. Examples: A heart rate of 0 means no heartbeat at all. In the case of the Kelvin scale, absolute zero (0 K) represents the absence of thermal energy, meaning there is no molecular motion.  VARIABLES & MEASUREMENT Why psychological tests are typically interval? Lack of Absolute Zero: Most psychological constructs (intelligence, stress, personality traits) do not have a true zero point. A score of zero does not mean the absence of the trait. Measurement Nature: Psychological tests often measure constructs that are abstract and cannot be completely absent. These tests provide scores that allow for comparison but not for statements about the absolute absence of a trait.  VARIABLES & MEASUREMENT Statistics and Scales of Measurement Scales of measurement are important because they help determine the statistics that are used to evaluate the data. Certain statistical procedures are used with numerical scores from interval or ratio scales and other statistical procedures with nonnumerical scores from nominal or ordinal scales. (What is the total for three psychology majors, an English major, and two chemistry majors?) VARIABLES & MEASUREMENT  STATISTICAL NOTATION Scores Representation: A score represents a value obtained for a particular variable in a research study. These raw scores are the original, unchanged measurements collected during the study. Scores for a specific variable are commonly denoted by the letter X. Observations for Multiple Variables: In research studies, observations are often made for multiple variables for each individual. In such cases, each individual will have two scores—one for each variable. The data can be presented in two lists labeled X and Y for the two variables. Each pair of X and Y represents the observations made for a single participant. Number of Scores: The letter N is used to specify the total number of scores in a set. Uppercase N denotes the number of scores in a population, while lowercase n denotes the number of scores in a sample.  STATISTICAL NOTATION Summation Notation Many of the computations required in statistics involve adding a set of scores. Because this procedure is frequently used, a special notation refers to the sum of a set of scores. The Greek letter sigma, or Σ, stands for summation. The expression ΣX means to add all the scores for variable X: the sum of the scores.  STATISTICAL NOTATION Summation Notation To use summation notation correctly, keep in mind the following two points: 1) The summation sign, Σ, is always followed by a symbol or mathematical expression. ΣX = summation of scores Σ(X – 1) = Calculate all of the (x-1) values and then add the results Example: What is the value of Σ(X – 2) for the following scores: 6, 2, 4, 2?  STATISTICAL NOTATION Summation Notation 2). The summation process often includes several other mathematical operations, such as multiplication or squaring. The different operations must be done in the correct sequence to obtain the correct answer. Order of Mathematical Operations This order of mathematical operations aligns with the standard conventions of mathematical calculations. Parenthesis Exponents Multiplication Division Addition/Summation Subtraction  STATISTICAL NOTATION X (Scores) ΣX ΣX2 (ΣX)2 3 4 9 10 15 16 20  STATISTICAL NOTATION X (Scores) ΣX ΣX2 (ΣX)2 3 3 9 3 4 4 16 4 9 9 81 9 10 10 100 10 15 15 225 15 16 16 256 16 20 20 400 20 77 77 1087 5929  STATISTICAL NOTATION X (Scores) (X-1) (X-1)2 3 4 9 10 15 16 20  STATISTICAL NOTATION 2 X (Scores) (X-1) (X-1) 3 2 4 4 3 9 9 8 64 10 9 81 15 14 196 16 15 225 20 19 361