Statistics is a powerful tool for understanding data and making informed decisions. This intro to statistics covers key concepts like descriptive vs. inferential methods, populations and samples, and types of variables. These fundamentals are essential for analyzing data effectively.

Understanding these basic statistical concepts lays the groundwork for more advanced analysis. By grasping terms like parameters, statistics, and probability, you'll be better equipped to interpret data, draw meaningful conclusions, and apply statistical thinking to real-world problems.

Introduction to Statistics

Descriptive vs inferential statistics

• Descriptive statistics summarize and describe the main features of a data set without drawing conclusions beyond the data at hand
  • Involve measures such as mean, median, mode, standard deviation, and graphs to characterize the data
  • Focus on organizing, summarizing, and presenting data in a meaningful way (tables, charts)
• Inferential statistics use sample data to make inferences, predictions, or generalizations about a larger population
  • Involve hypothesis testing, confidence intervals, and regression analysis to draw conclusions that extend beyond the immediate data
  • Allow researchers to make data-driven decisions and predictions based on a representative sample (political polls, medical trials)

Key terms in statistical studies

• Population refers to the entire group of individuals, objects, or events of interest in a study
  • Often too large to study in its entirety (all college students, all smartphones)
• Sample is a subset of the population selected for study that is ideally representative of the population
  • Allows for more efficient data collection and analysis (100 randomly selected college students)
• Parameter is a numerical summary measure that describes a characteristic of a population
  • Usually unknown and estimated using sample data
  • Examples: population mean ($\mu$), population standard deviation ($\sigma$)
• Statistic is a numerical summary measure computed from sample data used to estimate the corresponding population parameter
  • Examples: sample mean ($\bar{x}$), sample standard deviation ($s$)
• Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population

Numerical and categorical variables

• Numerical variables are quantitative variables that take on numeric values and can be discrete or continuous
  • Discrete variables have countable values, often integers (number of siblings, number of cars owned)
  • Continuous variables have measurable values that can take on any value within a range (height, weight, temperature)
  • Examples: age, test scores, income
• Categorical variables are qualitative variables that take on values in distinct categories or groups and can be nominal or ordinal
  • Nominal variables have categories with no inherent order (gender, race, blood type)
  • Ordinal variables have categories with a natural order (education level: high school, bachelor's, master's, doctorate; income brackets)
  • Examples: eye color, marital status, political affiliation

Probability and Statistical Inference

• Probability is the measure of the likelihood that an event will occur
• Distribution refers to the pattern of values in a dataset or population
• Variance measures the spread of data points around the mean
• Correlation measures the strength and direction of the relationship between two variables
• Hypothesis testing involves:
  • Null hypothesis: a statement of no effect or no difference
  • Alternative hypothesis: a statement of an effect or a difference