📉Statistical Methods for Data Science Unit 3 – Descriptive Stats & Exploratory Analysis

Key Concepts and Terminology

• Descriptive statistics involves methods for organizing, summarizing, and presenting data in a meaningful way
• Measures of central tendency (mean, median, mode) provide information about the typical or central value in a dataset
• Variability refers to the spread or dispersion of data points around the central tendency
• Exploratory data analysis (EDA) is an approach to analyzing datasets to summarize their main characteristics, often with visual methods
• Univariate analysis examines one variable at a time, while bivariate analysis explores relationships between two variables
• Outliers are data points that significantly deviate from the rest of the dataset and can heavily influence statistical measures
• Skewness measures the asymmetry of a distribution, indicating if data is skewed left (negative) or right (positive)
  • Negative skew has a longer left tail, while positive skew has a longer right tail

Types of Data and Variables

• Categorical (qualitative) variables have values that can be grouped into categories or labels (gender, color)
  • Nominal variables have categories with no inherent order (blood type)
  • Ordinal variables have categories with a natural order or ranking (education level)
• Numerical (quantitative) variables have values that represent quantities and can be measured or counted
  • Discrete variables can only take on certain values, often integers (number of siblings)
  • Continuous variables can take on any value within a range (height, temperature)
• Understanding the type of variable is crucial for selecting appropriate summary statistics and visualization methods
• Variables can also be classified as independent (predictor) or dependent (response) based on their role in the analysis

Measures of Central Tendency

• Mean is the arithmetic average of a dataset, calculated by summing all values and dividing by the number of observations
  • Sensitive to extreme values or outliers, which can pull the mean in their direction
• Median is the middle value when the dataset is ordered from lowest to highest
  • Robust to outliers and preferred for skewed distributions
  • If the dataset has an even number of observations, the median is the average of the two middle values
• Mode is the most frequently occurring value in the dataset
  • Can be used for categorical or numerical data
  • A dataset can have no mode (no repeating values), one mode (unimodal), or multiple modes (bimodal, trimodal, etc.)
• Choosing the appropriate measure of central tendency depends on the data type, distribution, and presence of outliers

Measures of Variability

• Range is the difference between the maximum and minimum values in a dataset
  • Provides a rough measure of spread but is sensitive to outliers
• Variance measures the average squared deviation from the mean, giving more weight to values far from the center
  • Calculated as the sum of squared deviations divided by the number of observations (or n-1 for sample variance)
  • Units are squared, making interpretation difficult
• Standard deviation is the square root of the variance, expressing variability in the same units as the original data
  • Roughly 68% of data falls within one standard deviation of the mean in a normal distribution
• Interquartile range (IQR) is the difference between the 75th and 25th percentiles, covering the middle 50% of the data
  • Robust to outliers and useful for comparing the spread of different datasets

Data Visualization Techniques

• Histograms display the distribution of a single numerical variable using bins or intervals
  • Useful for identifying the shape, center, and spread of the data
• Box plots (box-and-whisker plots) summarize the five-number summary (minimum, Q1, median, Q3, maximum) of a numerical variable
  • Helps detect outliers and compare distributions across categories
• Scatter plots display the relationship between two numerical variables using points in a coordinate plane
  • Can reveal patterns, trends, or clusters in the data
• Bar charts compare categorical variables by representing the frequency or proportion of each category with rectangular bars
  • Stacked or grouped bar charts can display multiple categories or subgroups
• Line plots connect data points in order, typically over time or another continuous variable
  • Useful for showing trends, patterns, or changes in the data

Exploratory Data Analysis (EDA) Steps

• Data cleaning involves identifying and handling missing values, outliers, or inconsistencies in the dataset
  • Techniques include removal, imputation, or transformation of problematic observations
• Data transformation may be necessary to improve the interpretability or meet assumptions of statistical methods
  • Common transformations include log, square root, or Box-Cox for skewed data, and standardization (z-scores) for comparing variables on different scales
• Univariate analysis examines the distribution, central tendency, and variability of each variable separately
  • Helps understand the overall characteristics and potential issues in the data
• Bivariate analysis explores relationships between pairs of variables, often using correlation measures or visualization techniques
  • Pearson correlation coefficient measures the linear relationship between two numerical variables
  • Chi-square test assesses the association between two categorical variables
• Multivariate analysis considers multiple variables simultaneously to identify complex relationships or patterns
  • Techniques include multiple regression, principal component analysis (PCA), or clustering algorithms

Statistical Software and Tools

• R is a popular open-source programming language and environment for statistical computing and graphics
  • Offers a wide range of packages for data manipulation, analysis, and visualization
• Python is a versatile programming language with libraries like NumPy, Pandas, and Matplotlib for data analysis and visualization
  • Scikit-learn provides tools for machine learning and more advanced statistical modeling
• Tableau is a powerful data visualization and business intelligence platform
  • Allows users to create interactive dashboards and explore data through a drag-and-drop interface
• Excel is a spreadsheet application that offers basic data analysis and visualization capabilities
  • Pivot tables and charts can be used for quick exploration and summary of small to medium-sized datasets
• JMP is a statistical software package developed by SAS, providing a graphical user interface for data analysis and visualization
  • Offers features like dynamic linking, data mining, and design of experiments

Real-world Applications and Examples

• Market research: Descriptive statistics help summarize customer preferences, purchasing behavior, or demographic information
  • Example: A company analyzes survey data to identify the most popular product features among different age groups
• Quality control: Measures of central tendency and variability are used to monitor and maintain product consistency
  • Example: A manufacturing plant tracks the mean and standard deviation of product dimensions to ensure they meet specifications
• Healthcare: EDA techniques can uncover patterns or risk factors associated with diseases or treatment outcomes
  • Example: Researchers explore the relationship between patient characteristics (age, gender, lifestyle) and the effectiveness of a new drug
• Finance: Data visualization helps communicate complex financial information to stakeholders or identify trends in market data
  • Example: An investment firm creates interactive dashboards to monitor portfolio performance and risk exposure
• Social sciences: Statistical methods are applied to analyze survey results, test hypotheses, or examine social phenomena
  • Example: A psychologist uses box plots to compare the distribution of anxiety scores between treatment and control groups