Comprehensive Psychological Statistics Notes

Overview of Psychological Statistics

• Instructor: Mary Antonette D. Malagueño, RPm
• Definition of Statistics: Statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information.
• Two Primary Purposes of Statistics:
  • To organize and summarize information effectively so that complex datasets become manageable and clear.
  • To assist researchers in answering the specific questions and hypotheses that initiated the research project.

Population, Samples, Parameters, and Statistics

• Population:

  • A population is defined as the set of all individuals, items, or elements of interest in a particular research study.
  • Research target groups are typically defined by common characteristics, and populations are often very large, making direct testing of all members unfeasible.

• Sample:

  • A sample is a set of individuals selected from a population.
  • A sample is intended to accurately represent the population in a research study, allowing observations from the sample to be generalized to the broader population.

• Parameter:

  • A parameter is a numerical value that describes a characteristic of an entire population.
  • Parameters are usually derived directly from measurements of all individuals within the population.

• Statistic:

  • A statistic is a numerical value that describes a characteristic of a sample.
  • Statistics are derived directly from measurements of the individuals included in the sample.

Descriptive and Inferential Statistical Methods

• Descriptive Statistics:

  • Descriptive statistics consist of statistical procedures used to summarize, organize, and simplify raw data.
  • Examples of descriptive statistical methods include:
  • Organizing raw scores into structured tables or visual graphs (e.g., annual sales trends, bar charts, pie charts, radar/spider charts, and bubble charts).
  • Computing summary measures such as averages, central tendencies, and measures of spread.

• Inferential Statistics:

  • Inferential statistics consist of methods that use sample data to make general statements, inferences, or conclusions about a population.
  • Because samples are subsets of populations, inferential methods evaluate the probability that sample results represent genuine population trends rather than random sampling error.

Variables, Data, and Constructs

• Variables:

  • A variable is any property or characteristic of an event, object, or person that changes or takes on different values under different conditions.
  • Variables can be broadly categorized into qualitative (categorical) and quantitative (numerical) types.

• Data / Raw Scores:

  • To observe and demonstrate changes in variables, measurements must be recorded.
  • The measurement obtained for each individual participant or object is called data, or more commonly, a score or raw score.

• Constructs (Hypothetical Constructs):

  • Constructs are internal attributes, traits, or characteristics that cannot be directly observed or measured directly.
  • Constructs are useful conceptual frameworks for describing, explaining, and predicting human behavior.
  • Examples of psychological constructs include intelligence, motivation, anxiety, and fear.

• Operational Definitions:
  • An operational definition identifies a specific measurement procedure (a set of concrete operations) for measuring an external, observable behavior.
  • It uses the resulting external behavioral measurements as both a functional definition and a quantitative metric for the underlying hypothetical construct.
  • Concrete examples of variables, operational definitions, and values:
  • Variable: Intelligence | Operational Definition: Score on the Verbal SAT test (a standardized test) | Value: $550$
  • Variable: Age | Operational Definition: Response to questionnaire (a self-report measure) | Value: $21$
  • Variable: Intelligence | Operational Definition: Speed of repairing engine (a behavioral definition) | Value: $2\text{ hrs}$
  • Variable: Intelligence | Operational Definition: Number of hairs on left thumb (an invalid or stupid definition) | Value: $7$

Classification of Variables

• Discrete Variables:

  • A discrete variable consists of separate, indivisible categories.
  • No numerical or categorical values can exist between two neighboring categories.
  • Examples include:
  • Gender classification: Male / Female
  • Binary conditions: True / False
  • Finite counts of people or objects
  • Age when recorded in distinct whole-year units

• Continuous Variables:

  • For a 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 or decimal parts.
  • Examples include:
  • Height
  • Weight
  • Distance
  • Time

Scales and Levels of Measurement

• Nominal Scale:
  • A nominal scale consists of a set of categories that have different names or labels.
  • Nominal measurements label and categorize observations, but make no quantitative distinctions between observations.
  • Examples: Eye color (Blue, Brown, Green), Smartphone brand (iPhone, Samsung, Moto), Transportation type (Bus, Train, Car).
  • Statistical Analysis Methods:
  • Descriptive statistics: Frequency distribution and mode.
  • Inferential statistics: Non-parametric statistical tests.

• Ordinal Scale:
  • An ordinal scale consists of a set of categories organized in an ordered sequence.
  • Ordinal measurements rank observations in terms of relative size, magnitude, or rank.
  • Examples: School grades (A, B, C), Education level (Bachelor's, Master's, PhD), Seniority level (Junior, Mid, Senior).
  • Statistical Analysis Methods:
  • Descriptive statistics: Frequency distribution, mode, median, and range.
  • Inferential statistics: Non-parametric statistical tests.

• Interval Scale:
  • An interval scale consists of ordered categories that are all intervals of exactly the same size.
  • Equal numerical differences between numbers on the scale reflect equal differences in physical magnitude.
  • The zero point on an interval scale is arbitrary and does not indicate a true zero amount or complete absence of the variable.
  • Examples: Temperature ($70^\circ$, $80^\circ$, $90^\circ$), IQ scores ($40$, $100$, $160$), Income ranges ($\$19\text{--}29\text{k}$, $\$30\text{--}39\text{k}$, $\$40\text{--}49\text{k}$).
  • Statistical Analysis Methods:
  • Descriptive statistics: Frequency distribution, mode, median, mean, range, standard deviation, and variance.
  • Inferential statistics: Parametric statistical tests (e.g., t-test, linear regression).

• Ratio Scale:
  • A ratio scale is an interval scale with the additional feature of an absolute, non-arbitrary zero point.
  • With a ratio scale, ratios of numbers reflect true ratios of magnitude.
  • Examples: Weight in kilograms ($50$, $70$, $90$), Number of staff members ($10$, $30$, $50$), Income in USD ($\$20\text{k}$, $\$40\text{k}$, $\$60\text{k}$).
  • Statistical Analysis Methods:
  • Descriptive statistics: Frequency distribution, mode, median, mean, range, standard deviation, variance, and coefficient of variation.
  • Inferential statistics: Parametric statistical tests (e.g., ANOVA, linear regression).

• Four Levels of Measurement Feature Matrix:
  • Categorizes and labels variables: Nominal (Yes), Ordinal (Yes), Interval (Yes), Ratio (Yes).
  • Ranks categories in order: Nominal (No), Ordinal (Yes), Interval (Yes), Ratio (Yes).
  • Has known, equal intervals: Nominal (No), Ordinal (No), Interval (Yes), Ratio (Yes).
  • Has a true or meaningful zero: Nominal (No), Ordinal (No), Interval (No), Ratio (Yes).

Statistical Notation and Summation Rules

• Summation Notation Basics:
  • The upper-case Greek letter sigma, $\Sigma$, is used to denote summation.
  • The expression $\Sigma X$ means to sum all the individual scores for variable $X$.
  • The summation sign $\Sigma$ reads as "the sum of", making $\Sigma X$ read as "the sum of scores".
  • Total number of scores in a sample is represented by $N$.
  • Introductory Example:
  • Given raw scores $X = \{10, 6, 7, 4\}$.
  • Sample size: $N = 4$.
  • Sum of scores calculation: $\Sigma X = 10 + 6 + 7 + 4 = 27$

Detailed Summation Computational Exercises

• Problem Set 1: Sum of Scores, Sum of Squares, and Square of Sums

  • Given raw dataset $X = \{3, 1, 7, 4\}$.
  • Step 1: Compute $X^2$ values for each score:
  • For $X = 3 \implies X^2 = 3^2 = 9$
  • For $X = 1 \implies X^2 = 1^2 = 1$
  • For $X = 7 \implies X^2 = 7^2 = 49$
  • For $X = 4 \implies X^2 = 4^2 = 16$
  • Step 2: Compute $\Sigma X$: $\Sigma X = 3 + 1 + 7 + 4 = 15$
  • Step 3: Compute $\Sigma X^2$: $\Sigma X^2 = 9 + 1 + 49 + 16 = 75$
  • Step 4: Compute $(\Sigma X)^2$: $(\Sigma X)^2 = (15)^2 = 225$

• Problem Set 2: Summation with Constant Subtractors

  • Given raw dataset $X = \{3, 1, 7, 4\}$.
  • Step 1: Compute $(X - 1)$ for each score:
  • For $X = 3 \implies (3 - 1) = 2$
  • For $X = 1 \implies (1 - 1) = 0$
  • For $X = 7 \implies (7 - 1) = 6$
  • For $X = 4 \implies (4 - 1) = 3$
  • Step 2: Compute $\Sigma(X - 1)$: $\Sigma(X - 1) = 2 + 0 + 6 + 3 = 11$
  • Step 3: Compute $(X - 1)^2$ for each score:
  • For $X = 3 \implies (2)^2 = 4$
  • For $X = 1 \implies (0)^2 = 0$
  • For $X = 7 \implies (6)^2 = 36$
  • For $X = 4 \implies (3)^2 = 9$
  • Step 4: Compute $\Sigma(X - 1)^2$: $\Sigma(X - 1)^2 = 4 + 0 + 36 + 9 = 49$

• Problem Set 3: Two-Variable Summation and Cross-Products

  • Given dataset for Person A, B, C, D across variables $X$ and $Y$:
  • Person A: $X = 3$, $Y = 5$
  • Person B: $X = 1$, $Y = 3$
  • Person C: $X = 7$, $Y = 4$
  • Person D: $X = 4$, $Y = 2$
  • Step 1: Compute product $XY$ for each individual:
  • Person A: $XY = 3 \times 5 = 15$
  • Person B: $XY = 1 \times 3 = 3$
  • Person C: $XY = 7 \times 4 = 28$
  • Person D: $XY = 4 \times 2 = 8$
  • Step 2: Compute $\Sigma X$: $\Sigma X = 3 + 1 + 7 + 4 = 15$
  • Step 3: Compute $\Sigma Y$: $\Sigma Y = 5 + 3 + 4 + 2 = 14$
  • Step 4: Compute $\Sigma XY$: $\Sigma XY = 15 + 3 + 28 + 8 = 54$