Ch1: Statistics Introduction

Observational Study vs. Experiment

• Observational Study: No attempt is made to control or influence the variables of interest. Data is simply observed and recorded.

• Experiment: One or more variables are manipulated to determine how they influence the variable of interest.

Population Parameter vs. Sample Statistic

• Population Parameter: A numerical characteristic of a population (e.g., population mean μ, population proportion p).

• Sample Statistic: A numerical summary calculated from a sample (e.g., sample mean x̄, sample proportion p̂).

Types of Data

• Categorical Data: Values that represent categories or groups (e.g., gender, color).

• Quantitative Data: Values that represent measurable quantities (e.g., height, weight).

Variable Types

• Nominal Data: Categories without a natural order (e.g., eye color).

• Ordinal Data: Categories with a natural order (e.g., class rank).

• Discrete Quantitative Data: Countable values (e.g., number of siblings).

• Continuous Quantitative Data: Any value within a range (e.g., height).

Ch2: Organizing and Summarizing Data

Graphical Summaries

• Bar Charts: Used for single categorical variables.

• Pie Charts: Show proportions of categories.

• Histograms: Used for quantitative data to show frequency distributions.

• Scatterplots: Show the relationship between two quantitative variables.

Ch3: Numerically Summarizing Data

Measures of Location (Central Tendency)

• Mean (x̄): The arithmetic average. Sensitive to outliers.

• Median: The middle value when data is ordered. Resistant to outliers.

• Mode: The most frequently occurring value.

Measures of Variability (Dispersion)

• Range: Difference between the maximum and minimum values.

• Interquartile Range (IQR): (middle 50% of data).

• Variance: Average squared deviation from the mean.

  • Population variance:

  • Sample variance:

• Standard Deviation: Square root of variance.

  • Population:

  • Sample:

Boxplots and Outliers

• Boxplot: Visual summary using quartiles and median.

• Outlier Detection: Values more than 1.5 × IQR above Q3 or below Q1 are considered outliers.

Ch4: Describing the Relation Between Two Variables

Scatterplots and Correlation

• Scatterplot: Graphs two quantitative variables to identify relationships.

• Correlation Coefficient (r): Measures strength and direction of linear relationship.

Regression

• Regression Line: Best-fit line through data points, used for prediction.

Ch5: Probability

Probability Basics

• Probability of an Event (A):

• Sum of Probabilities: All possible outcomes sum to 1.

• Relative Frequency:

Conditional Probability

Probability Rules

• Addition Rule (for mutually exclusive events):

• Multiplication Rule (for independent events):

Ch6: Discrete Probability Distributions

Random Variables

• Discrete Random Variable: Takes on countable values (e.g., number of heads in coin tosses).

• Probability Distribution: Lists all possible values and their probabilities.

Expected Value and Variance

• Expected Value:

• Variance:

Binomial Distribution

• Models the number of successes in a fixed number of independent trials.

• Binomial Probability Formula:

  • n = number of trials

  • p = probability of success

  • k = number of successes

• Mean:

• Standard Deviation:

Ch7: The Normal Probability Distribution

Normal Distribution

• Continuous, symmetric, bell-shaped distribution.

• Characterized by mean and standard deviation .

• Notation:

Standard Normal Distribution

• Special case with and .

• Z-score:

• Used to find probabilities using standard normal tables.

Empirical Rule

• 68% of data within 1 standard deviation of mean

• 95% within 2 standard deviations

• 99.7% within 3 standard deviations

Ch8: Sampling Distributions

Sampling Distribution of the Sample Mean

• Distribution of sample means from all possible samples of a given size from a population.

• Central Limit Theorem: For large n, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

• Mean:

• Standard Error:

Ch9: Estimating the Value of a Parameter

Confidence Intervals

• Definition: Range of values likely to contain the true population parameter.

• General Form:

• Interpretation: "We are 95% confident that the true mean lies between [Lower Bound] and [Upper Bound]."

Factors Affecting Interval Width

Ch10: Hypothesis Tests Regarding a Parameter

Hypothesis Testing Core Concept

• Statistical method to decide whether sample data provides enough evidence to support a specific claim about a population parameter.

• Null Hypothesis (H0): No effect or difference (status quo).

• Alternative Hypothesis (Ha): There is an effect or difference.

• Significance Level (α): Probability of Type I error (rejecting H0 when true), commonly 0.05.

• P-value: Probability of observing data as extreme as the sample, assuming H0 is true.

Types of Errors

• Type I Error (α): Rejecting a true H0.

• Type II Error (β): Failing to reject a false H0.

Summary Table

Choosing the Right Test

• Use Z-test if population standard deviation is known.

• Use T-test if population standard deviation is unknown and estimated from the sample.