📉Statistical Methods for Data Science Unit 1 – Statistical Thinking in Data Science

Key Concepts and Foundations

• Statistical thinking involves understanding the role of variability, uncertainty, and randomness in data analysis and decision-making
• Emphasizes the importance of data collection, exploration, and interpretation in the context of real-world problems
• Focuses on drawing meaningful insights and making data-driven decisions while acknowledging the limitations and potential biases in the data
• Incorporates domain knowledge and subject matter expertise to formulate relevant questions and hypotheses
• Utilizes probability theory and statistical inference to quantify uncertainty and make predictions
• Emphasizes the iterative nature of data analysis, involving data cleaning, preprocessing, modeling, and evaluation
• Stresses the importance of effective communication and visualization of results to stakeholders and decision-makers

Probability Theory Essentials

• Probability is a measure of the likelihood of an event occurring, expressed as a value between 0 and 1
• Joint probability is the probability of two or more events occurring simultaneously, calculated as the product of their individual probabilities
• Conditional probability is the probability of an event occurring given that another event has already occurred, denoted as $P(A|B)$
• Bayes' theorem relates conditional probabilities and can be used to update probabilities based on new evidence: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
• Independence of events occurs when the occurrence of one event does not affect the probability of another event
• Random variables are variables whose values are determined by the outcome of a random experiment (discrete or continuous)
• Probability distributions describe the likelihood of different values of a random variable (uniform, binomial, normal)
• Expected value is the average value of a random variable over many trials, calculated as the sum of each value multiplied by its probability

Types of Data and Distributions

• Categorical data consists of discrete categories or groups with no inherent order (gender, color)
• Ordinal data has categories with a natural order or ranking, but the differences between categories may not be equal (survey responses: strongly agree to strongly disagree)
• Numerical data is quantitative and can be further classified as discrete (integer values) or continuous (any value within a range)
• Normal distribution is a symmetric, bell-shaped curve characterized by its mean and standard deviation
  • 68% of data falls within one standard deviation of the mean
  • 95% of data falls within two standard deviations of the mean
• Skewed distributions are asymmetric, with a longer tail on one side (right-skewed or left-skewed)
• Bimodal distributions have two distinct peaks or modes, indicating two dominant values or groups in the data
• Uniform distribution has equal probability for all values within a given range

Descriptive Statistics and Exploratory Data Analysis

• Measures of central tendency summarize the typical or central value of a dataset (mean, median, mode)
  • Mean is the arithmetic average of all values
  • Median is the middle value when the data is sorted
  • Mode is the most frequently occurring value
• Measures of dispersion quantify the spread or variability of a dataset (range, variance, standard deviation)
  • Range is the difference between the maximum and minimum values
  • Variance is the average squared deviation from the mean
  • Standard deviation is the square root of the variance
• Exploratory data analysis (EDA) is the process of investigating and summarizing the main characteristics of a dataset
• EDA techniques include visualizing data through histograms, box plots, scatter plots, and heatmaps
• Identifying outliers, missing values, and potential relationships between variables is a key aspect of EDA
• Data preprocessing steps such as data cleaning, transformation, and normalization are often performed during EDA

Inferential Statistics and Hypothesis Testing

• Inferential statistics involves drawing conclusions about a population based on a sample of data
• Hypothesis testing is a formal procedure for determining whether there is sufficient evidence to reject a null hypothesis in favor of an alternative hypothesis
• Null hypothesis ($H_0$) represents the default or status quo, often stating no difference or no effect
• Alternative hypothesis ($H_a$ or $H_1$) represents the claim or research question being tested
• Type I error (false positive) occurs when rejecting a true null hypothesis, with the probability denoted as $\alpha$ (significance level)
• Type II error (false negative) occurs when failing to reject a false null hypothesis, with the probability denoted as $\beta$
• p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true
• Confidence intervals provide a range of plausible values for a population parameter based on the sample data and a specified level of confidence (90%, 95%, 99%)

Regression Analysis Basics

• Regression analysis is a statistical method for modeling the relationship between a dependent variable and one or more independent variables
• Simple linear regression models the relationship between two continuous variables using a straight line: $y = \beta_0 + \beta_1x + \epsilon$
  • $\beta_0$ is the y-intercept
  • $\beta_1$ is the slope or coefficient
  • $\epsilon$ is the error term
• Multiple linear regression extends simple linear regression to include multiple independent variables: $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_kx_k + \epsilon$
• Least squares estimation is a method for estimating the regression coefficients by minimizing the sum of squared residuals
• R-squared ($R^2$) is a measure of the proportion of variance in the dependent variable explained by the independent variable(s)
• Assumptions of linear regression include linearity, independence, homoscedasticity, and normality of residuals
• Logistic regression is used when the dependent variable is binary or categorical, modeling the probability of an event occurring

Data Visualization Techniques

• Data visualization is the graphical representation of data to convey insights and patterns effectively
• Scatter plots display the relationship between two continuous variables, with each point representing an observation
• Line plots connect data points in a sequence, often used for time series data or to show trends
• Bar plots compare categorical variables using rectangular bars, with the height or length representing the value
• Histograms display the distribution of a continuous variable by dividing the data into bins and showing the frequency or density of observations in each bin
• Box plots (box-and-whisker plots) summarize the distribution of a variable by displaying the median, quartiles, and outliers
• Heatmaps use color-coding to represent values in a matrix, often used for correlation matrices or confusion matrices
• Pie charts show the proportion or percentage of categories in a dataset, with each slice representing a category

Practical Applications in Data Science

• Predictive modeling involves building models to make predictions or estimates based on historical data (customer churn, sales forecasting)
• Anomaly detection identifies unusual or rare events, observations, or patterns that deviate significantly from the norm (fraud detection, network intrusion)
• Recommender systems suggest items, products, or services to users based on their preferences, behavior, or similarity to other users (movie recommendations, product suggestions)
• Customer segmentation divides a customer base into distinct groups based on shared characteristics, behaviors, or preferences for targeted marketing and personalization
• Sentiment analysis determines the sentiment, opinion, or emotion expressed in text data (social media posts, product reviews)
• Time series analysis examines data collected over time to identify trends, seasonality, and make forecasts (stock prices, weather patterns)
• A/B testing compares two or more versions of a product, website, or app to determine which performs better based on a specific metric (click-through rate, conversion rate)
• Optimization techniques are used to find the best solution or decision given a set of constraints and objectives (resource allocation, supply chain management)