Quantitative Research Methods Lecture 4

Quantitative Research Methods in Political Science

• Lecture 4: The Normal Curve and Z Scores
• Course Instructor: Michael E. Campbell
• Course Number: PSCI 2702 (A)
• Date: 09/26/2024

Quick Recap (Lecture 1)

• Role of statistics in social sciences
• Use of systematic processes
• Difference between facts and values
• Characteristics of variables (discrete vs. continuous)
• Levels of measurement

Quick Recap Cont'd (Lecture 2)

• Causality (causal relationships)
• Independent and dependent variables
• Conceptualization and operationalization
• Instruments and instrumentation
• Systematic and random measurement error (reliability and validity)

Quick Recap Cont'd (Lecture 3)

• Descriptive and univariate statistics (proportions, percentages, rates, ratios)
• Measures of central tendency (mode, median, mean)
• Measures of dispersion (IQR, range, variance, standard deviation)
• Frequency distribution tables
• Graphs and charts (pie, bar, histograms)
• Foundation for the Normal Curve

The Normal Curve

• Theoretical model used in statistics
• Can precisely describe empirical distributions
• Perfectly smooth frequency polygon
• Unimodal (single mode/peak)
• Symmetrical (unskewed)
• Mean, median, and mode are equal

The Normal Curve Cont'd

• Resembles unskewed distributions from previous lectures (Chapter 3)
• Bell-shaped (Gaussian curve)
• Tails extend infinitely
• Does not exist perfectly in nature
• Several variables approximate a normal distribution

The Normal Curve Cont'd

• Tool for descriptive statements about empirical distributions
• Used in inferential statistics to generalize from samples to populations
• Distances along the horizontal axis encompass the same proportion of area under the curve when measured in standard deviations
• Distances between any point and the mean cut off the same proportion of total area when measured in standard deviations

The Normal Curve Example

• Data represent IQ scores for children and adults
• Each distribution is symmetrical (unskewed)
• Each sample size is 1000
• Children's mean IQ = 100, standard deviation = 20
• Adults' mean IQ = 100, standard deviation = 10
• Larger spread for children's data due to higher standard deviation

The Normal Curve Example Cont'd

• Larger spread of data for children's IQ scores due to higher standard deviation
• Two scales: IQ units and standard deviations
• No difference between these scales conceptually
• One standard deviation above/below the mean for children (e.g., 120/80)

The Normal Curve Example Cont'd

• Same logic applies for adult IQ
• One standard deviation above/below the mean for adults (e.g., 110/90)
• Standard deviation affects the spread of the data

Area Under the Normal Curve

• When measured in standard deviations, distances along the horizontal axis on any normal curve always encompass the same proportion of area under the curve
• Between +/- 1 standard deviation: 68.26% of the area
• Between +/- 2 standard deviations: 95.44% of the area
• Between +/- 3 standard deviations: 99.72% of the area

Area Under the Normal Curve Cont'd

• Shows areas corresponding to standard deviations

Z Scores (Standard Scores)

• Express scores after standardization to the theoretical normal curve
• Convert original units (e.g., weight, time, IQ) to Z scores
• Mean = 0, standard deviation = 1

Computing Z Scores

• Formula for Z scores: Z = (X - μ) / σ
• Where:
  • X = individual score
  • μ = sample mean
  • σ = sample standard deviation

Computing Z Scores Cont'd

• Calculate mean and standard deviation for a dataset
• Apply formula to compute Z scores for each data point

Computing Z Scores Cont'd

• Sample Z scores for given data
• Example values for Z scores

Positive and Negative Z Scores

• Positive Z scores are to the right of the mean
• Negative Z scores are to the left of the mean

The Standard Normal Curve Table

• Table provides areas under the normal curve for various Z scores
• Aids in determining probabilities
• Offers an abridged version

The Standard Normal Curve Table Cont'd

• Z score columns
• Area between mean and Z column
• Area beyond Z column
• Examples showing how to extract Z score values

Positive Z Score Example

• Illustrates how to find area between the mean and a positive Z score
• Convert raw score into Z score
• Use Z score table

Positive Z Score Example Cont'd

• Example from the Z score table to solve calculations
• Shows values to demonstrate area under the curve for provided examples

Negative Z Score Example

• Handles negative Z scores with the same method as positive Z scores

Negative Z Score Example Cont'd

• Example of how to derive Z scores to demonstrate examples to find area under the curve
• Shows how to determine the area calculations

Finding the Total Area Below a Score

• Finding area below a positive Z score
• Finding area below a negative Z score
• Z-score calculation methods

Finding a Z Score Above a Positive Score

• Calculating the area above a positive score in similar manner to positive Z score example
• Example of how to extract Z score values from Z table

Finding a Z Score Above a Positive Score Cont'd

• Example demonstrating methods for calculating area above a positive Z score calculation
• Illustrates how to use the Z score table for calculation

Finding the Area Above/Below Z Summarized

• Finding areas above positive/negative Z scores
• Summarized table of the techniques

Finding Raw Scores

• Finding raw scores when the percentile is known
• Illustrates the process

Finding Raw Score Example

• Demonstrating finding raw scores with example of adult IQ scores
• Calculating proportion
• Applying Z score equation

Finding Raw Score Example Cont'd

• Example continuing the process demonstrates the use of raw score Z table lookup

Finding the Area Between Two Scores on Opposite Sides of the Mean

• Finding area between two scores on opposite sides of the mean by adding the areas
• Using children's IQ sample as example

Finding the Area Between Two Scores on Opposite Sides of the Mean Cont'd

• Examples of how to calculate area under the curve examples for examples
• Demonstrates how to determine the area between the two scores

Finding the Area Between Scores on Same Side of Mean

• Finding the area between scores on the same side of the mean by subtracting
• Examples of how to derive the examples

Finding the Area Between Scores on Same Side of Mean Cont'd

• Demonstration of calculating area between scores with example

Using the Normal Curve to Estimate Probabilities

• Utilizing the normal curve for calculating probability
• Theoretical normal curve as a distribution of probabilities
• Estimating probabilities of events

Probabilities

• Formula for probabilities
• Definition of "events"

Probabilities Example

• Demonstrates probabilities with example of selecting a king of hearts from a deck of cards

Probabilities Example

• Explanations of how to calculate probability of selections

Using the Normal Curve to Estimate Probabilities Cont'd

• Probability ranges (0.00 to 1.00)
• Higher probability values indicate higher likelihood
• Example demonstrating probability calculation

Using the Normal Curve to Estimate Probabilities Cont'd

• Calculating probability for instances where success and total events are known (e.g., rolling a die)
• Probability examples

Probability Distributions

• Listing probabilities of each event gives a probability distribution
• Example using a single die

Discrete and Continuous Probability Distributions - Overview

• Differentiating between discrete and continuous variables
• Discrete variables take whole numbers
• Continuous variables can take decimals

Discrete and Continuous Probability Distributions Cont'd

• Discrete Probability distribution
• Continuous Probability distribution (normal curve)
• Calculating probabilities using infinite nature of continuous variables

Probabilities for Continuous Variables

• Combining probability and normal curve knowledge to estimate probabilities for continuous variables

Probabilities for Continuous Variables Example

• Example using the normal distribution of children's IQ scores to find probability of score between 95 and 100

Probabilities for Continuous Variables Cont'd

• Probability calculations using standard normal curve table
• Finding probabilities given the Z scores

Probabilities for Continuous Variables Example #2

• Calculating probability that a randomly selected child has an IQ less than 123

Probabilities for Continuous Variables Example #2

• Example calculations for probabilities that demonstrate examples of Z score calculations

Probabilities at a Glance

• Probability of selecting a case close to the mean is high for normal distributions.
• Probability decreases with distance from the mean
• The majority of cases cluster around the mean
• Probability of randomly selecting a case that is within 1 standard deviation of the mean is 0.6826

Probabilities at a Glance Cont'd

• Probability is low for cases beyond 3 standard deviations from the mean
• Area under the curve for cases beyond 3 standard deviations is small (0.0013 to 0.0026)

This quiz covers Lecture 4 of the Quantitative Research Methods in Political Science course, focusing on the Normal Curve and Z Scores. It extends the foundational knowledge gained in previous lectures about the role of statistics, causality, and descriptive statistics. Prepare to test your understanding of these key concepts in political science research methods.