Module 1 - Statistics

Data Distributions

Introduction to Statistical Methods

Course Overview

• Class Title: STAT 2331
• Institution: Southern Methodist University (SMU)

What is Statistics?

• Real-World Examples:
  • Points scored by each player on a basketball team
  • Unemployment rate
  • Percentage of students who indulge in binge drinking
• Definition of Statistics:
  • Statistics are numbers calculated from data. They involve:
  • Designing studies.
  • Making data-informed decisions.
• Additional Aspects of Statistics:
  • Statistics is not just about numbers; it is fundamentally about:
  • Collecting data.
  • Analyzing data.
  • Interpreting data to influence decisions.

Why Should You Care?

• Growth of the Field:
  • The field of statistics has grown exponentially over the past few decades.
• Related Fields:
  • Data science and machine learning are recent outgrowths of statistics.
• Job Security:
  • Statisticians enjoy excellent job security with opportunities in multiple sectors:
  • Academia
  • Industry
  • Consulting
  • Government
• Everyday Relevance:
  • Statistics permeate our daily lives, affecting decision-making and understanding of the world.

Questions Statistics Can Answer

• Questions that statistics can address include:
  • Are large doses of vitamin C beneficial?
  • Which team is most likely to win a tournament?
  • How can you analyze whether a diet really works?
  • What is the chance your tax return will be audited?
  • Is there bias against women in appointing managers?
  • Are rates of anxiety and depression increasing over time?

Module Objectives

• Key Learning Outcomes:
  • Distinguish between statistics and parameters.
  • Explain the impact of variability on sampling and inference.
  • Distinguish between categorical and quantitative data.
  • Describe and display categorical data:
  • Graphically (e.g., charts).
  • Numerically (e.g., through totals or percentages).
  • Describe and display quantitative data:
  • Graphically (e.g., histograms).
  • Numerically (e.g., means or medians).

Four Main Phases of Statistical Analysis

• Research Question:
  • Choose a question you are interested in answering.
• Design:
  • Outline how to obtain data to answer the question of interest.
• Description:
  • Summarize the data obtained.
• Inference:
  • Make decisions in the face of uncertainty based on the data.

Probability versus Statistics

• Scenario 1:

  • Known probability: 21% of the people in a population of 1000 aged 60 or older have diabetes.
  • Calculate the probability that 3 out of 10 sampled people will have diabetes.
  • This scenario involves: Probability.

• Scenario 2:

  • Unknown population percentage: sample indicates 2 out of 10 people have diabetes.
  • Use data to predict the population percentage.
  • This scenario involves: Statistics.
  • Focus in this class: Scenario 2.

The Language of Statistics

• Subjects:
  • The subjects we observe or measure (often people).
• Target Population:
  • The specific subjects in which we are interested.
• Sample Frame:
  • The list from which the sample is selected.
• Sample:
  • A subset of the population for which we plan to have data.
• Parameter:
  • A numerical summary of the population.
• Statistic:
  • A numerical summary of a sample.

Examples in Statistical Analysis

Example 1: Dieting Obese Women Study

• Research Question:
  • Do dieting obese women following a new diet plan lose more weight over 6 weeks than those who don’t?
• Design:
  • Randomly sample 30 women following the plan and 30 not following it.
  • Weigh everyone at the start and again at 6 weeks.
• Description:
  • Total weight loss recorded; mean weight loss: 18 pounds for the new plan, 12 pounds for those not on it.
• Inference:
  • Conclusion: Women on the new diet plan lose more weight than those not on it.
  • Variable Identification:
  • Subjects
  • Target Population
  • Sample Frame
  • Sample
  • Statistic(s)
  • Parameter(s)
• Importance of Random Sampling:
  • Ensures results are generalizable and minimizes bias.
• Impact of Larger Sample Sizes:
  • Generally leads to more reliable results.

Inference: The Goal of Statistical Analysis

• Research Questions:
  • Focused on specific populations.
• Design Considerations:
  • Create a list (sample frame) and gather random data.
• Description of Data:
  • Summarizes sample findings with statistics.
• Inference:
  • Leads to hypotheses or predictions about population parameters.

Statistical Analysis Starts with a Data Set

Example Data Set Overview

• Key Variables:
  • Categorical Variables: Labels for the types of products.
  • Quantitative Variables: Sensory score, number of cookies, calories, fat content.

Types of Variables

Categorical Variables

• Definition:
  • A variable that fits observations into categories.
• Sub-types:
  • Nominal: Cases are classified into groups without any order.
  • Examples: Gender, color.
  • Ordinal: Groups maintain a natural order.
  • Examples: Educational levels, satisfaction ratings.

Quantitative Variables

• Definition:
  • Observations take numerical values.
• Sub-types:
  • Discrete: Finite list of values with no intermediate values.
  • Examples: Number of students.
  • Continuous: Infinite values possible between any two fixed points.
  • Examples: Weight, height.

Exploratory Data Analysis

Components of Exploratory Data Analysis

• Steps Involved:
  • Look at each variable individually.
  • Analyze relationships between variables.
  • Display data visually (charts, graphs).
  • Summarize data numerically.
• Data Example:
  • From 1900 to 2016, 1303 unprovoked shark attacks reported in the USA:
  • California: 187
  • Florida: 828
  • Hawaii: 230
  • Texas: 58
  • Related questions for investigations:
  • Identify variable of interest.
  • Establish type and sub-type.
  • Define values for the variable.

Exploring a Single Categorical Variable

Methods of Analysis

• Frequency Tables:
  • List possible values for a variable along with their counts.
• Pie Charts:
  • Visualize the percentage of each category within the whole dataset.
• Bar Charts:
  • Use bars to represent counts or proportions for each category.

Example Data for Categorical Analysis

Categorical Data Summary

• Characteristics:
  • Observations fit within groups (e.g., named categories, ordered groups).
• Key Features:
  • Count, percentage, or proportion calculated.
    
• Graphical Displays:
  • Include frequency tables and various chart types.

Exploring a Single Quantitative Variable

Features Analyzed

• Shape:
  • Distribution of the sample values.
• Center:
  • Measured using mean, median, or mode.
• Spread:
  • Measured using standard deviation, variance, range, or interquartile range (IQR).
• Graphical Displays:
  • Histograms and boxplots help visualize data patterns.

Histograms

Basic Features

• Definition:
  • A graphical representation that divides quantitative data into intervals (bins).
• Count Observations in Each Bin:
  • Created a frequency table reflecting the grouped data.
• Graphical Display:
  • Bins on the horizontal axis, frequency on the vertical axis.

Example Histogram Data

Describing the Shape of Quantitative Data

Key Shapes to Identify

• Number of Modes:
  • Unimodal, bimodal, uniform distributions.
• Skewness:
  • Symmetric, skewed right (positive skew), skewed left (negative skew).

Comparing Distributions

Hypothetical Examples

• Common Quantitative Variables: Include those typically having specific skewness patterns.
• Examples:
  • IQ (generally normally distributed).
  • Life expectancy for humans (slightly skewed right).
  • Annual incomes of adults (typically right skewed).

Measuring the Center of Quantitative Data

Measures of Central Tendency

• Mode:
  • The most frequently occurring value.
• Mean:
  • Average calculated as the sum of observations divided by the sample size.
  • Mathematical Expression: \text{Mean} = \frac{\sum x_i}{n}
• Median:
  • Middle value when observations are ordered.
  • Example Series: [2, 4, 6, 8, 10] – Median = 6.

Comparing Mean and Median

Impact of Outliers

• Outlier Definition:
  • A data point that is far outside the general distribution of the dataset.
• Comparative Behavior:
  • Median: Resistant to outliers; remains stable with extreme values.
  • Mean: Sensitive to outliers; can skew dramatically.

Measuring the Spread of Quantitative Data

Key Metrics

• Range:
  • Calculated as the difference between maximum and minimum values.
    \text{Range} = \text{Maximum} - \text{Minimum}
• Variance:
  • Reflects the average squared deviation from the mean:
    s^2 = \frac{1}{n-1}\sum{i=1}^{n}(xi - \bar{x})^2
• Standard Deviation (SD):
  • Square root of the variance:
    s = \sqrt{\frac{1}{n-1}\sum{i=1}^{n}(xi - \bar{x})^2}
• Interquartile Range (IQR):
  • Difference between the third quartile (Q3) and the first quartile (Q1):
    \text{IQR} = Q3 - Q1

Identifying Outliers

Criteria for Outlier Detection

• Mild Outliers:
  • Values that are less than Q1 - 1.5 \times IQR or greater than Q3 + 1.5 \times IQR.
• Extreme Outliers:
  • Values less than Q1 - 3 \times IQR or greater than Q3 + 3 \times IQR.

Boxplots

Features of Boxplots

• Summary Visualization:
  • Represents the five-number summary: minimum, Q1, median, Q3, and maximum.
• Display characteristics of the data:
  • Spread and potential outliers.
• Whiskers:
  • Extend from the quartiles to the farthest non-outlier points.

Example Boxplot Analysis

Example Data:

• Example Data:
  • Set: [1, 2, 7, 11, 12, 18, 23, 25, 98, 197]
• Five Number Summary:
  • Minimum = 1
  • Q1 = 7
  • Median = 12
  • Q3 = 25
  • Maximum = 197

Comparing Histograms and Corresponding Boxplots

Insight on Distribution:

• Histograms:
• Boxplots:
  • Both graphically represent spread and outliers while offering an intuitive understanding of the central tendency.

Conclusion: Understanding Quantitative Data

Summary of Key Concepts

• Characteristics:
  • Numerical values used in operation.
• Sub-types:
  • Discrete vs. Continuous Variables: Indicates the type of numerical data.
• Graphical Displays:
  • Histogram and boxplot for analyzing spread and identifying outliers.
• Numerical Summaries:
  • Essential features: shape, center (mean/median), spread (variance, standard deviation, IQR), and outliers.
  • Use resistant measures for skewed data (median and IQR).