Statistics for Psychologists Study Notes

Course Overview

• Course Title: PSYC331 - Statistics for Psychologists
• Instructor: John E. K. Dotse, PhD
  • Office Location: Main Building, Psychology
  • Office Hours: Wednesdays, 10:30 am - 12:30 pm
  • Email: jekdotse@ug.edu.gh

Lecture 1: Introduction to Statistics

Course Instructions

• Mode of Teaching: Face-to-face
• Calculator Requirement: All students are required to procure calculators.
• Teaching Assistants: Available for tutorial assistance.
• Attendance: Students are advised against skipping lectures due to the technical nature of the course.
• Practice Method: Most computations will be conducted through self-practice.

Assessment and Grading

• Assessment Format: Assessments will include both online and paper-based (in-person) formats.
• Tests:
  • Test 1: Interim Assessment (online - MCQ) - 25%
  • Test 2: Interim Assessment (online - MCQ) - 25%
  • Test 3: Final Exam (paper-based) - 50%
• Communication: Dates and times for assessments, along with any changes, will be communicated via the SAKAI platform and during lectures.

Reading List

• Opoku, J. Y. (2007). Tutorials in Inferential Social Statistics. (2nd Ed.). Accra: Ghana Universities Press. Pages 3 - 22.

Introduction to Inferential Statistics

Understanding Statistics

• Statistics is fundamental for research within Social Sciences, providing tools for data analysis.
  • Technical Nature: Despite its technicality, statistics is not inherently difficult.
  • Comparison with Mathematics: Statistics fundamentally differs from pure mathematics, involving straightforward calculations.
  • Importance: Essential for conducting research projects focusing on data analytics.

Computations in Statistics

• Independent t-test: An example of a computational formula relevant in the course.

Course Syllabus Outline

• Key topics include:
  • Descriptive Statistics
  • Inferential Statistics
  • Properties/Levels of Measurement Scales
  • Summation Notations
  • Computation of Mean and Standard Deviation

What is Statistics?

• Definition: A branch of mathematics concentrated on the organization, analysis, and interpretation of numerical data (Aron, Coups, & Aron, 2014).
• Branches of Statistical Methods:
  • Descriptive Statistics: Summarizes and describes quantitative information or data.
  • Inferential Statistics: Draws conclusions about a population based on sampled data.

Descriptive Statistics

Purpose

• Summarizes and provides an overall picture of a sample and its variables.

Types of Measures

1. Measures of Central Tendency:
   • Mean: Sum of all scores divided by the number of observations.
   • Median: Middle score when all scores are arranged in order.
   • Mode: Most frequently occurring score.
   • Quartiles: Values that divide a data set into four equal parts.
2. Measures of Dispersion/Variation:
   • Standard Deviation: Represents average variability of scores around their mean.
   • Variance: Average squared deviation from the mean.
   • Range: Difference between the highest and lowest scores.

Implications

• Descriptive statistics allow for educated guesses about a population, but they do not reveal precise population attributes.

Inferential Statistics

Overview

• Techniques that enable researchers to draw conclusions about a population based on a sample.
  • Due to challenges of studying entire populations, samples are used as substitutes.
  • Assumes carefully selected samples represent the population.
• Concepts:
  • Sample Statistic: Estimate computed from a sample.
  • Population Parameter: Estimate computed from a whole population.
  • Uses Greek letters (e.g., μ for mean, σ for standard deviation) for population parameters.
  • Uses English letters (e.g., , s) for sample statistics.

Applications

• Generalizes sample findings to the broader population.

Basic Concepts in Statistics

Variables

• Definition: Characteristics that can assume various values.
  • Types:
  1. Continuous Variables: Numeric with infinite values (e.g., time).
  2. Discrete Variables: Countable numeric values (e.g., number of students).
  3. Categorical Variables: Allocates categories (e.g., gender).

Examples of Variable Types

Properties of Measurement Scales

• Measurement scales have four potential properties:
  1. Identity: Numbers assigned for identification (e.g., male = 1, female = 2).
  2. Magnitude: Numbers represent ordered relationships.
  3. Equal Intervals: Measurement unit intervals are consistent (e.g., temperature).
  4. Absolute Zero Point: Theoretical point at which the attribute is absent (e.g., distance).

Types of Measurement Scales

1. Ratio Scale: All properties; highest measure (e.g., height, weight).
2. Interval Scale: Lacks an absolute zero but maintains equal intervals (e.g., temperature).
3. Ordinal Scale: Natural order without magnitude representation (e.g., satisfaction levels).
4. Nominal Scale: Distinct categories without inherent ordering (e.g., gender).

Levels of Measurement Summary

Exercises on Measurement Scales

• Tasks to identify the appropriate measurement scale in various scenarios involving standardized tests, clinical assessments, and categorization of variables.

Statistical Tests and Measurement Scales

• The choice of statistical tests is influenced by the measurement scale used to collect data.
  • Parametric Tests: Applicable for ratio or interval data (e.g., t-test, F test).
  • Non-Parametric Tests: Used for ordinal or nominal data (e.g., Mann-Whitney U test, Chi-Square test).

Summation Notations in Statistics

Key Concepts

• Summation Symbol (Σ): Represents the total of a set of numbers.
• Example Case: Calculating sum for dataset: 2, 3, 5, 7, 11 as x1, x2, x3, x4, x5.

Various Summation Examples

1. Sum of Squares:
   • $ ext{∑X}^2$ denotes summation of squares of scores.
2. Square of Sums:
   • $( ext{∑X})^2$ represents the sum of scores squared.
3. Sum of Products:
   • $ ext{∑XY}$ for multiplying corresponding pairs of two variables.

The Mean and Standard Deviation

The Mean (Arithmetic Mean)

• Calculated as the total of data divided by the count of observations.
  • Example Calculation: $rac{(2 + 4 + 6 + 8 + 10 + 12)}{6} = rac{42}{6} = 7$.

Variability Measures

1. Standard Deviation: Indicates dispersion about the mean.

• Calculation Importance: Summing differences directly leads to zero variances, necessitating squaring differences.
• Variance calculated as $rac{ ext{Total Squared Differences}}{N}$.
  • Example Calculation: For scores, total variability $= 70$ leading to variance indicator.

Variance & Standard Deviation Formulas

• Population Variance: ext{σ}^2
• Sample Variance: ext{s}^2
• Standard Deviation formulas derived from their respective variances.

Calculation Example

• Given scores: $ ext{∑X}^2$ derived as: 2, 4, 6, 8, 10, 12 leading to ext{∑X}^2 = 364 and $( ext{∑X})^2 = 1764$.