🎲Data, Inference, and Decisions Unit 1 – Data, Inference, and Decisions: Introduction

Key Concepts and Terminology

• Data refers to raw facts, observations, or measurements collected through various methods
• Information is data that has been processed, organized, and given context to provide meaning and value
• Variables are characteristics or attributes of interest that can take on different values across observations
  • Quantitative variables are numerical and can be measured or counted (age, height, income)
  • Qualitative variables are categorical and describe qualities or characteristics (gender, color, occupation)
• Descriptive statistics summarize and describe the main features of a dataset (mean, median, mode, standard deviation)
• Inferential statistics use sample data to make generalizations or predictions about a larger population
• Probability is the likelihood of an event occurring, expressed as a number between 0 and 1
• Hypothesis testing is a statistical method used to determine whether there is enough evidence to support a claim about a population parameter

Types of Data and Their Characteristics

• Nominal data consists of categories with no inherent order or numerical value (blood type, country of origin)
• Ordinal data has categories with a natural order but no consistent scale between values (education level, customer satisfaction ratings)
• Interval data has a consistent scale between values but no true zero point (temperature in Celsius or Fahrenheit)
• Ratio data has a consistent scale and a true zero point, allowing for meaningful ratios between values (height, weight, income)
• Discrete data can only take on specific, countable values (number of children in a family, number of defective products)
• Continuous data can take on any value within a range and is typically measured (time taken to complete a task, weight of an object)
• Cross-sectional data is collected at a single point in time (a survey of consumer preferences)
• Time series data is collected over a period of time at regular intervals (daily stock prices, monthly sales figures)

Data Collection Methods

• Surveys involve asking participants a series of questions to gather information about their opinions, behaviors, or characteristics
  • Surveys can be administered online, by phone, or in person
  • Questions should be clear, unbiased, and relevant to the research objectives
• Interviews are one-on-one conversations with participants to gather detailed, qualitative data
  • Interviews can be structured (following a set of predetermined questions) or unstructured (allowing for more open-ended exploration of topics)
• Observations involve watching and recording the behavior of individuals or groups in a natural or controlled setting
• Experiments manipulate one or more variables to determine their effect on an outcome variable
  • Participants are typically divided into treatment and control groups
  • Randomization helps ensure that any differences between groups are due to the manipulation rather than pre-existing differences
• Secondary data is data that has been previously collected by someone else for another purpose (government statistics, academic research, company reports)

Introduction to Probability and Statistics

• Probability is the foundation of inferential statistics and helps quantify uncertainty
• The probability of an event (A) is denoted as P(A) and ranges from 0 (impossible) to 1 (certain)
• Independent events are not influenced by the occurrence of other events (rolling a die multiple times)
• Dependent events are influenced by the occurrence of other events (drawing cards from a deck without replacement)
• Conditional probability is the probability of an event (A) occurring given that another event (B) has already occurred, denoted as P(A|B)
• The law of large numbers states that as the number of trials increases, the average of the results will converge to the expected value
• The central limit theorem states that the distribution of sample means will approximate a normal distribution, regardless of the shape of the population distribution, as the sample size increases

Basic Inferential Techniques

• Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the entire population
  • Simple random sampling ensures that each individual has an equal chance of being selected
  • Stratified sampling divides the population into subgroups (strata) and then randomly samples from each subgroup
• Confidence intervals provide a range of values that are likely to contain the true population parameter with a certain level of confidence (95% confidence interval)
• Hypothesis testing involves comparing a sample statistic to a hypothesized population parameter to determine whether there is enough evidence to support or reject the claim
  • The null hypothesis (H0) represents the status quo or no effect
  • The alternative hypothesis (Ha) represents the research claim or expected effect
• T-tests compare the means of two groups to determine whether they are significantly different from each other
• ANOVA (analysis of variance) tests compare the means of three or more groups to determine whether they are significantly different from each other

Decision-Making Frameworks

• Decision trees visually represent the possible outcomes of a series of decisions, along with their associated probabilities and values
• Expected value is the average outcome of a decision, calculated by multiplying each possible outcome by its probability and summing the results
• Sensitivity analysis examines how changes in the input variables affect the outcome of a decision
• Cost-benefit analysis compares the expected costs and benefits of a decision to determine whether it is worthwhile
• Multi-criteria decision analysis (MCDA) evaluates alternatives based on multiple, often conflicting, criteria
  • Criteria are assigned weights based on their relative importance
  • Alternatives are scored on each criterion and then combined using the weights to determine an overall score

Data Visualization and Interpretation

• Data visualization helps communicate complex data in a clear and accessible format
• Bar charts compare the values of different categories using horizontal or vertical bars
• Line graphs show trends or changes over time by connecting data points with lines
• Scatter plots display the relationship between two continuous variables using points on a coordinate plane
• Pie charts show the proportions of different categories within a whole using slices of a circle
• Histograms display the distribution of a continuous variable using bars that represent the frequency of values within each bin
• Box plots summarize the distribution of a continuous variable using quartiles and outliers
• Heat maps use color intensity to represent the magnitude of values in a matrix or grid

Ethical Considerations in Data Analysis

• Privacy concerns arise when collecting, storing, and analyzing personal or sensitive data
  • Data should be anonymized or aggregated to protect individual identities
  • Informed consent should be obtained from participants before collecting their data
• Bias can enter the data analysis process at various stages, from data collection to interpretation
  • Sampling bias occurs when the sample is not representative of the population
  • Measurement bias occurs when the data collection instruments or methods are flawed
• Transparency involves being open and clear about the data sources, methods, and limitations of the analysis
• Reproducibility ensures that the analysis can be replicated by others using the same data and methods
• Responsible use of data and results involves considering the potential consequences and ensuring that they are not misused or misinterpreted
• Fairness and non-discrimination require that data analysis does not perpetuate or amplify existing biases or disparities
  • Algorithms and models should be regularly audited for fairness and adjusted as needed