Untitled Quiz

Inferential Statistics Course Notes

• Course Objective 1: To equip students with skills to summarize and interpret data using descriptive statistics and visualization techniques.
• Course Objective 2: To develop a foundational understanding of probability and its applications in data science.
• Course Objective 3: To enable students to perform hypothesis testing and construct confidence intervals for statistical inference.
• Course Objective 4: To teach students how to build and assess linear and logistic regression models for predictive analysis.
• Course Objective 5: To provide hands-on experience with statistical software for data manipulation, analysis, and visualization.

Course Outcomes

• CO1: Summarize and describe dataset features (mean, median, mode, variance, standard deviation), using graphs (histograms, box plots, scatter plots).
• CO2: Understand probability theory (random variables, probability distributions, law of large numbers) to model uncertainty in data.
• CO3: Apply statistical inference (hypothesis testing, confidence intervals, p-value computation) to draw conclusions from sample data about larger populations.
• CO4: Apply linear and logistic regression for identifying relationships, making predictions, and evaluating model performance.
• CO5: Use statistical software for data analysis (cleaning, transformation, visualization, various statistical methods).

Unit-3: Inferential Statistics Syllabus

• Inferential Statistics & Hypothesis Testing: Statistical Inference Terminology, Hypothesis Testing, Parametric Tests, Non-parametric Tests
• Industry Application: Hypothesis Testing using Excel, Industry Practices & Applications of Statistics

Suggestive Readings

• Text Books: Hastie, Trevor, et al., The elements of statistical learning, Montgomery, Douglas C., and George C. Runger, Applied statistics and probability for engineers, Probability and Statistics, Jeffrey S. Rosenthal.
• Reference Books: Practical Statistics for Data Scientists (Peter Bruce et al.), An Introduction to Statistical Learning (Gareth James et al.), Think Stats.

What is a Statistic?

• Population: The whole group of individuals of interest.
• Parameter: A value that describes a population.
• Sample: A part of a population.
• Statistic: A value that describes a sample.
• Note: Psychology (PSYCH) often uses samples.

Descriptive & Inferential Statistics

• Descriptive Statistics: Organizing, summarizing, and presenting data.
• Inferential Statistics: Generalizing from samples to populations, hypothesis testing, and relationships among variables.

Descriptive Statistics Types

• Frequency Distributions: Number of subjects that fall into categories.
• Graphical Representations: Graphs and Tables (histograms, box plots, scatter plots).
• Summary Statistics: Single numbers (mean, median, mode) to describe data.

Frequency Distributions Examples

• Cross-tabulation: Categorizing data based on multiple variables (e.g., Democrats/Republicans, male/female).
• Calculating percentages and proportions based on totals from frequency distributions.

Central Limit Theorem

• The larger the sample size, the closer a distribution will approximate a normal distribution.

Normal Distribution

• Half the scores are above the mean, and half are below (symmetrical).

Summary Statistics

• Measures of central tendency: Mean for typical average score
• Measures of variability: Range, Variance, Standard Deviation, and Standard Error of the Mean

Measures of Central Tendency

• Quantitative Data: Mode (most frequent), Median (middle value), Mean (average).
• Qualitative Data: Mode only.

Mean (Notation)

• Sample mean = X
• Population mean = μ
• Summation sign = ∑
• Sample size = n
• Population size = N

Special Property of the Mean

• The sum of all deviations from the mean equals zero.

Inferential Statistics

• Using sample data to evaluate a hypothesis about a population.

Null Hypothesis

• Ho: The claim we're evaluating (no difference between means).

Alternative Hypothesis

• Ha: The claim we're trying to find evidence for (there is a difference).

Hypothesis Testing Decision Possibilities

• Null hypothesis is true ⇒ Do not reject Ho
• Null hypothesis is false ⇒ Reject Ho

Possible Outcomes in Hypothesis Testing (Decision)

• Null is True: Correct decision / Error (Type I Error)
• Null is False: Error (Type II Error) / Correct decision

Alpha (α)

• Probability of making a Type I error (rejecting a true null hypothesis)

Beta (β)

• Probability of making a Type II error (failing to reject a false null hypothesis)

Power

• Ability to reduce type II error

Inferential Statistics Tests for Mean Differences

• T-test (Independent/Correlated/Within-Subjects): For comparisons between 2 groups, or repeated measurements in one group.
• Analysis of Variance (ANOVA): For comparing more than 2 groups.

Meta-Analysis

• Statistical averaging of results from independent studies about the same phenomenon.

Other Important Distributions

• Poisson: Distribution of counts occurring at a rate, e.g., web visits.
• Exponential: Intervals between events.
• Binomial/Multinomial: Categorical outcomes, e.g., die tosses.

Statistical Significance Testing

• Practical significance vs. statistical significance.

Method 1: Ablation

• Train a model with all the features, and then calculate performance Qo.
• Remove a feature, retrain the model, and calculate the Performance Q1
• If Q1 is significantly worse than Q0, the feature is useful (keep it). Otherwise, discard.
• Note: Check significantly worse using a statistical test, e.g., t-test or bootstrap sampling.

Method 2: Mutual Information

• MI measures the relationship one feature has to another. Higher MI means potential to be important.

Method 3: Chi-Squared

• Used for comparing contingency table counts to determine feature dependence.

Measurement

• Basic properties (min, max, mean, standard deviation).
• Relationships (scatter plots, regression).
• Model accuracy and performance.

Variables

• Characteristics or conditions that change.
• Values can vary.
• Categorical (discrete or ordinal classes).
• Numerical (can be continuous).

Population

• Total group of all possible values.

Sample

• Portion of the population.

Statistical Notation

• Uppercase (X) represents a random variable.
• Lowercase (x) represents specific samples from a population.

Normal Distributions

• Complete characteristic are based on mean and variance.
• Standard Deviation = square root of variance

Central Limit Theorem

• The theoretical foundation behind generalizing sample results to populations.

Bootstrap Sampling

• A method to estimate the variability of a statistic based on resampling.

Types of Data

• Numerical/Quantitative: Numerical quantities.
  • Continuous: Can take on any value within a range. (Height, Weight).
  • Discrete: Can take on only certain values. (Number of students in a class, number of equipment in a project).
• Categorical/Qualitative: Placed into groups.
  • Nominal: No inherent order (Gender, Hair Color).
  • Ordinal: Natural order between categories. (Customer satisfaction surveys, student grades).

Validation

• A third set of data used for parameter tuning and validating generalized results based on training.

Train/Test Split

• Splitting data into two subsets for training and testing.
• Prevents overfitting in the model.