Statistics Chapter 5 Quiz

Introduction

• This lecture covers foundational concepts from the course, based on student questions.
• It does not include all final exam material.
• Consult the study sheet, additional PowerPoints, and assigned readings for complete information.
• Review course assignments for relevant material.

Introduction Cont'd

• Research is a systematic process for gathering information to answer questions, examine ideas, or test theories. (Healey, Donoghue, and Prus 2023, 10)
• Quantitative research differs from qualitative research through its use of statistics.
• Statistics are mathematical techniques to organize and manipulate data.
• These tools analyze data, identify trends, examine relationships, generalize findings, and revise theories. (Healey, Donoghue, and Prus 2023, 16)

Natural vs. Social Science

• Quantitative methods apply the scientific method, an organized series of steps for studying empirical reality.
• The scientific method was initially developed to study the natural world.
• The social world is complex, with various factors affecting results, making precise control impossible.
• Researchers can consider factors through control variables but cannot control all possible influences.

The Role of Statistics in Social Science Research

• Quantitative research follows a process:
  • A theory
  • Development of hypotheses
  • Observation of phenomena
  • Empirical generalizations

Variables

• Variables are traits that change in value across cases.
• Variables have three characteristics:
  • Mutual Exclusivity: Response categories are clearly defined and do not overlap.
  • Exhaustiveness: Response categories encompass all possible outcomes.
  • Homogeneity: Response categories consistently measure the same characteristic.

Mutual Exclusivity

• Response categories must not overlap.
• A case cannot fall into two categories.

Exhaustiveness

• Response categories must cover all possible outcomes.
• No other outcomes can exist beyond listed responses.

Homogeneity

• Response categories must measure the same characteristic.
• All answers within a variable must measure the same attribute.

Discrete and Continuous Variables

• Discrete variables cannot be subdivided (e.g., whole numbers).
• Continuous variables can be infinitely subdivided (e.g., time).

Levels of Measurement

• There are three levels of measurement:
  • Nominal
  • Ordinal
  • Interval/Ratio

Nominal Level Variables

• The lowest level of measurement.
• Response categories are simply labels (even if numerical).
• Cannot be ranked or ordered.

Ordinal Level Variables

• Ranked variables, more precise than nominal variables.
• Categories can be ordered as "more or less" or "higher or lower."
• Exact distances between categories cannot be measured.

Interval/Ratio Level Variables

• The highest level of measurement.
• Exact distances between scores can be measured.
• A true zero point exists that means the absence of something (e.g., zero income).

Conceptualization and Operationalization

• Before working with variables, understand their meaning/concepts.
• A concept is an idea or mental construct to organize, map, and understand phenomena.
• For example, what is a political party?

Conceptual Definition

• A precise description of a concept.
• Explains the concept's measurable properties and units of analysis (e.g., people, nations).
• Defines the meaning of the concept within the context of the research.

Operational Definition

• Specifies how to measure a concept.
• Explains the instrument to be used in data collection and specifically defines how the observed data will represent the concept.

Process of Conceptualization and Operationalization

• Clarify the concept: identify the concept's key characteristics (observable and vary).
• Develop a conceptual definition: clearly describes the concept and its variations, outlining relevant units and applying criteria.
• Operationalize: specify how this concept will be measured empirically. (e.g., use surveys to measure voter turnout).
• Collect Data or relevant Variables: gather or identify data suitable to support the identified concept.

Descriptive vs. Inferential Statistics

• Descriptive statistics summarize and describe data distribution for single or multiple variables.
• Inferential statistics generalize data from a sample to a larger population.

Measures of Central Tendency

• Mode: Most frequent score.
• Median: Middle score.
• Mean: Average score.

Measures of Dispersion

• Measures of dispersion describe the variability or heterogeneity within a data distribution.
• Common measures include range, variance, and standard deviation.

The Normal Curve

• A theoretical model in statistics.
• Paired with mean and standard deviation to accurately describe data distributions.
• It represents a symmetrical, unimodal (single peak) frequency distribution (i.e., mean, median, and mode are all identical).
• Data points cluster near the mean.
• Standard deviations measure the distance along the horizontal axis, each encompasses an equal proportion of the area under the curve.

Z Scores

• Convert raw scores into standardized scores in terms of standard deviations from the mean.
• Indicates the location of a score relative to the mean.
• Reveals the proportion of area falling between the score and the mean as well as the area beyond the z score.

Samples and Sampling

• Inferential statistics use samples instead of entire populations.
• Samples must be representative for meaningful generalizations.
• Equal Probability of Selection Method (EPSEM) is the appropriate method to collect probability samples. Even representative probability samples may not be fully representative due to sampling error.

Sampling Distribution

• A theoretical, probabilistic distribution of a statistic across all possible samples of a given size from a population.
• Used in inferential statistics to link a sample to the population from which it is drawn.
• Theoretical, not empirical – it's based on probability, not direct observation of all possible samples.

Theorems Underpinning the Sampling Distribution

• The central limit theorem is crucial because it allows for inferences about population traits from samples even if the population distribution is not normal in its shape. In instances where sample size is large enough (namely, when n>100) the inherent theoretical properties of central limit theorem dictate that the sample distribution will be approximately normal in its shape.

Estimation Procedures

• Point Estimates: a single value that estimates a population parameter (e.g., sample mean to estimate population mean).
• Confidence Intervals: a range of values within which the population parameter is likely to fall, giving a clearer sense of the possible spread of potential values in a population.

Unbiased Estimator

• Mean of the sampling distribution matched to the observed value's mean in a population. Larger samples yield results with higher confidence.

Efficient Estimator

• Estimators are clustered around the population mean with smaller standard deviations, leading to higher accuracy of estimating the population mean from a sample measure. Large sample sizes are more efficient to yield more accurate estimation from the sample.

Confidence Intervals

• Defined alpha levels (typically 0.05), which represent the probability of error.
• Determine Z or t critical values for the specific level.
• Calculating confidence interval involves sample mean, estimated standard error of the mean, and z/t critical value.

Hypothesis Testing

• A statistical method to determine if the relationship between variables or if differences between groups, exists within a population.
• Involves a null hypothesis (no difference or relationship) and a research hypothesis (difference or relationship). Rejects the null hypothesis if evidence from the sample supports a relationship.

Chi Square, p-value, and Measures of Association

• Chi-square tests for relationships between nominal or ordinal variables (e.g., is voter turnout affected by age group).
• p-value indicates statistical significance of the findings. p-value less than pre-determined alpha level suggests a relationship exists in the population.
• Measures of Association (e.g., phi, Cramer's V, and others) measure the strength of relationship between variables, but only the direction.