📈Preparatory Statistics Unit 1 – Intro to Statistics & Data Analysis

Key Concepts and Terminology

• Statistics involves collecting, analyzing, interpreting, and presenting data to make informed decisions
• Population refers to the entire group of individuals, objects, or events of interest
• Sample is a subset of the population used to draw conclusions about the whole population
• Parameter is a numerical summary measure that describes a characteristic of a population
• Statistic is a numerical summary measure computed from sample data used to estimate a population parameter
• Descriptive statistics involves methods for organizing, summarizing, and presenting data
• Inferential statistics involves methods for using sample data to make estimates, decisions, predictions, or other generalizations about a population

Types of Data and Variables

• Variables can be classified as either quantitative (numerical) or qualitative (categorical)
• Quantitative variables take on numerical values and can be further classified as discrete (countable values) or continuous (measurable values)
  • Examples of discrete variables include the number of siblings or the number of cars owned
  • Examples of continuous variables include height, weight, or temperature
• Qualitative variables take on non-numerical values that can be classified into categories or groups
  • Nominal variables have categories with no natural ordering (eye color, gender, race)
  • Ordinal variables have categories with a natural ordering (education level, income bracket, survey ratings)

Descriptive Statistics

• Measures of central tendency describe the center or typical value of a dataset
  • Mean is the arithmetic average of a set of values
  • Median is the middle value when the data is arranged in order
  • Mode is the most frequently occurring value
• Measures of variability describe the spread or dispersion of a dataset
  • Range is the difference between the maximum and minimum values
  • Variance measures the average squared deviation from the mean
  • Standard deviation is the square root of the variance
• Measures of position describe the relative standing of a value within a dataset
  • Percentiles indicate the percentage of values that fall below a given value
  • Quartiles divide the data into four equal parts (Q1, Q2 or median, Q3)
  • Z-scores measure the number of standard deviations a value is from the mean

Probability Basics

• Probability is a numerical measure of the likelihood that an event will occur
• Classical probability is used when outcomes are equally likely and is calculated as the number of favorable outcomes divided by the total number of possible outcomes
• Empirical probability is based on historical data or observations and is calculated as the relative frequency of an event
• The complement of an event A, denoted as A', is the event "not A" or everything in the sample space that is not included in A
• The addition rule for mutually exclusive events states that $P(A \text{ or } B) = P(A) + P(B)$
• The multiplication rule for independent events states that $P(A \text{ and } B) = P(A) \cdot P(B)$
• Conditional probability is the probability of an event A occurring given that another event B has already occurred, denoted as $P(A|B)$

Sampling and Data Collection

• Simple random sampling selects a sample such that every possible sample of the same size has an equal chance of being selected
• Stratified sampling divides the population into homogeneous subgroups (strata) and then takes a simple random sample from each stratum
• Cluster sampling divides the population into clusters, randomly selects some of the clusters, and then samples all individuals within the chosen clusters
• Systematic sampling selects individuals from a list by starting at a random point and then selecting every kth element thereafter
• Convenience sampling selects individuals who are easily accessible or available, but may not be representative of the population
• Observational studies observe individuals and measure variables of interest without attempting to influence the responses
• Experiments deliberately impose some treatment on individuals to observe their responses
  • The independent variable (explanatory variable) is the variable that is manipulated or controlled in an experiment
  • The dependent variable (response variable) is the variable that is measured or observed in an experiment

Visualizing Data

• Bar charts display the distribution of a categorical variable using vertical or horizontal bars
• Pie charts display the relative frequencies of categories as slices of a circle
• Histograms divide the range of a quantitative variable into intervals (bins) and display the frequency or relative frequency of observations in each interval using vertical bars
• Stem-and-leaf plots display the individual values of a quantitative variable by splitting each value into a "stem" (leading digit(s)) and a "leaf" (trailing digit)
• Scatterplots display the relationship between two quantitative variables by plotting ordered pairs (x, y) as points in the coordinate plane
• Time-series plots display the values of a variable over time, with time on the horizontal axis and the variable of interest on the vertical axis
• Box plots (box-and-whisker plots) display the distribution of a quantitative variable using five summary statistics (minimum, Q1, median, Q3, maximum)

Statistical Inference

• Point estimation uses sample statistics to estimate population parameters
  • A point estimator is a formula or method that produces a single value as an estimate of a population parameter
  • An unbiased estimator has an expected value equal to the true value of the parameter being estimated
• Interval estimation uses sample data to construct an interval of plausible values for a population parameter
  • A confidence interval is an range of values that is likely to contain the true value of a population parameter with a certain level of confidence (e.g., 95%)
  • The margin of error is the maximum expected difference between the point estimate and the true value of the parameter
• Hypothesis testing is a procedure for using sample data to test a claim or conjecture about a population parameter
  • The null hypothesis ($H_0$) is a statement of "no effect" or "no difference" that is assumed to be true unless there is strong evidence against it
  • The alternative hypothesis ($H_a$) is a statement that contradicts the null hypothesis and is accepted if there is sufficient evidence against $H_0$
  • The significance level ($\alpha$) is the probability of rejecting $H_0$ when it is actually true (Type I error)
  • The p-value is the probability of obtaining a sample statistic as extreme as the one observed, assuming that $H_0$ is true
  • If the p-value is less than $\alpha$, we reject $H_0$ and conclude that there is sufficient evidence to support $H_a$

Real-World Applications

• Quality control uses statistical methods to monitor and maintain the quality of products or services
  • Control charts are used to detect unusual variations in a process over time
  • Acceptance sampling involves taking a random sample from a batch of items and deciding whether to accept or reject the entire batch based on the number of defective items in the sample
• Market research uses surveys, focus groups, and other methods to gather data about consumer preferences, opinions, and behaviors
• Medical research uses statistical methods to design and analyze clinical trials, epidemiological studies, and other types of health-related research
  • Randomized controlled trials randomly assign participants to treatment and control groups to assess the effectiveness of a drug, therapy, or other intervention
  • Observational studies, such as cohort studies and case-control studies, investigate the relationship between risk factors and health outcomes
• Social sciences, such as psychology, sociology, and political science, use statistical methods to study human behavior, attitudes, and interactions
  • Surveys and polls are used to gather data from a representative sample of a population
  • Regression analysis is used to model the relationship between variables and make predictions
• Business analytics uses statistical methods to analyze data and inform decision-making in areas such as finance, marketing, and operations management
  • Time series analysis is used to model and forecast future values of a variable based on its past values
  • A/B testing compares two or more versions of a website, app, or marketing campaign to determine which performs better