6: Statistical Foundations for Analytics
- Page ID
- 48067
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Explain the role of statistics in data analytics and evidence-based decision-making.
- Distinguish between descriptive statistics, probability, inferential statistics, and predictive modeling.
- Calculate and interpret measures of central tendency, including mean, median, and mode.
- Calculate and interpret measures of dispersion, including range, variance, standard deviation, percentiles, and interquartile range (IQR).
- Describe how distribution shape, skewness, and modality affect statistical interpretation.
- Explain foundational probability concepts such as random experiments, sample spaces, events, and probability distributions.
- Identify and interpret major probability distributions, including normal, binomial, Poisson, and uniform distributions.
- Explain the importance of the normal distribution and the Central Limit Theorem in statistics.
- Distinguish between populations, samples, parameters, and statistics.
- Explain the purpose of random sampling and recognize the effects of sampling error and sampling bias.
- Construct and interpret point estimates and confidence intervals for population parameters.
- Explain the meaning of confidence level, standard error, and margin of error.
- Formulate null and alternative hypotheses for statistical testing.
- Interpret p-values and statistical significance in the context of hypothesis testing.
- Distinguish between Type I and Type II errors and explain the role of significance level and statistical power.
- Apply and interpret one-sample statistical procedures such as t-tests and confidence intervals for means.
- Explain and interpret correlation as a measure of association between variables.
- Describe the purpose of regression analysis and interpret slope, intercept, and
R^2in simple models. - Distinguish correlation, regression, and causation, and explain why correlation does not imply causation.
- Recognize confounding variables, lurking variables, and Simpson’s paradox as threats to valid interpretation.
- Describe how statistical methods are applied in business, healthcare, and social science settings.
- Identify common pitfalls in statistical reasoning, including sampling bias, p-hacking, overgeneralization, multiple testing, overfitting, and misinterpreting non-significant results.
- Use statistical thinking to make more reliable, cautious, and meaningful analytic conclusions.
- 6.1: Introduction - The Role of Statistics in Data Analytics
- This chapter introduces statistics as the foundational science of learning from data, explaining its pivotal role in data analytics for making evidence-based decisions amid uncertainty. It outlines the key statistical methods used in a typical workflow, including descriptive statistics, probability theory, inferential statistics, and correlation/regression techniques, which enable analysts to turn raw data into reliable knowledge.
- 6.2: Descriptive Statistics - Summarizing Data
- This chapter introduces descriptive statistics as methods to summarize the essential features of a dataset. It explains how to calculate and interpret measures of central tendency (mean, median, mode) to understand the "typical" value, and measures of dispersion (range, variance, standard deviation) to understand the data's spread or variability.
- 6.3: Probability Theory and Key Distributions
- This chapter introduces statistics as the foundational science of learning from data, explaining its pivotal role in data analytics for making evidence-based decisions amid uncertainty. It outlines the key statistical methods used in a typical workflow, including descriptive statistics, probability theory, inferential statistics, and correlation/regression techniques, which enable analysts to turn raw data into reliable knowledge.
- 6.4: Inferential Statistics - From Samples to Populations
- This chapter explains inferential statistics, which uses data from a sample to draw conclusions about a larger population. It covers two primary methods for doing this: creating confidence intervals to estimate population parameters and using hypothesis testing with p-values to assess specific claims.
- 6.5: Real-World Applications of Statistical Analysis
- This chapter illustrates how statistical analysis is the backbone of decision-making across diverse domains like business, healthcare, and social sciences. It provides real-world examples, such as A/B testing, clinical trials, and public opinion polls, to demonstrate the practical application of statistical concepts to draw conclusions and inform actions.
- 6.6: Common Pitfalls in Statistical Reasoning
- This chapter outlines common pitfalls in statistical reasoning, such as confusing correlation with causation, sampling bias, and misinterpreting p-values. It stresses that awareness of these issues and applying critical thinking are essential for drawing reliable and actionable conclusions from data.