Statistics Probability
The course teaches core concepts of descriptive statistics, probability theory, distributions, inferential methods, and regression/correlation, focusing on intuition and real‑world application using calculator‑level calculations for data‑driven decision making.
Who Should Take This
It is ideal for undergraduate students, recent graduates, and early‑career professionals in business, engineering, health sciences, or social research who need to interpret data, design experiments, or build predictive models, but lack advanced math backgrounds. Learners seek practical statistical intuition to support evidence‑based decisions.
What's Included in AccelaStudy® AI
Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
20 Activity Formats
Course Outline
66 learning goals
1
Descriptive Statistics
4 topics
Measures of Central Tendency
- Define mean, median, and mode and describe the conditions under which each measure best represents the center of a dataset
- Apply mean, median, and mode calculations to real-world datasets and explain why the median is preferred for skewed income or housing price distributions
- Analyze the relationship between mean, median, and mode in symmetric vs skewed distributions and use their relative positions to infer distribution shape
- Apply weighted mean calculations to real-world scenarios such as grade point averages and portfolio returns where observations have unequal importance
Measures of Dispersion
- Define range, interquartile range (IQR), variance, and standard deviation and describe what each reveals about data spread
- Apply the empirical rule (68-95-99.7) to normal distributions to estimate the proportion of data within specified standard deviations of the mean
- Compare population vs sample variance formulas and explain why Bessel's correction (n-1 denominator) produces an unbiased estimate of population variance
Data Visualization for Statistics
- Describe common statistical plots including histograms, box plots, stem-and-leaf plots, and scatter plots and explain what distributional features each reveals
- Apply box plot interpretation to identify median, quartiles, whiskers, and outliers and use side-by-side box plots to compare group distributions
- Evaluate the effectiveness of different plot types for communicating specific statistical insights and identify misleading visualization practices
Distribution Shape & Outliers
- Describe skewness and kurtosis as measures of distribution shape and explain how they complement measures of center and spread
- Apply the IQR method and z-score method to detect outliers and determine whether extreme values should be retained, transformed, or removed
- Analyze how outliers affect descriptive statistics and regression models and evaluate robust alternatives such as trimmed means and median-based measures
2
Probability Theory
3 topics
Basic Probability Concepts
- Define sample spaces, events, and probability axioms and describe the classical, frequentist, and subjective interpretations of probability
- Apply the addition rule and multiplication rule to calculate probabilities of compound events including mutually exclusive and independent events
- Apply the complement rule and counting techniques (permutations, combinations) to solve probability problems involving selections and arrangements
- Analyze common probability misconceptions including the gambler's fallacy and confusion between independence and mutual exclusivity
Conditional Probability
- Define conditional probability and explain the difference between P(A|B) and P(B|A) using real-world examples such as medical screening
- Apply the law of total probability to calculate event probabilities by partitioning the sample space into exhaustive, mutually exclusive scenarios
- Apply Bayes' theorem to update probabilities given new evidence and solve diagnostic test problems involving sensitivity, specificity, and prevalence
- Analyze the base rate fallacy and explain why ignoring prior probabilities leads to systematic errors in interpreting conditional probabilities
Independence & Random Variables
- Define statistical independence and explain how to test whether two events are independent using the multiplication rule
- Describe discrete and continuous random variables, probability mass functions, probability density functions, and cumulative distribution functions
- Apply expected value and variance calculations for discrete random variables and interpret their meaning in real-world decision-making contexts
3
Probability Distributions
3 topics
Discrete Distributions
- Describe the binomial distribution including its parameters (n, p), assumptions (fixed trials, independence, constant probability), and applications to counting successes
- Apply the binomial probability formula and cumulative probabilities to solve problems involving fixed numbers of independent trials
- Describe the Poisson distribution including its rate parameter lambda and applications to modeling rare events in fixed intervals of time or space
- Analyze when to use binomial vs Poisson vs geometric distributions based on the structure of the random process being modeled
Continuous Distributions
- Describe the normal (Gaussian) distribution including its parameters (mean, standard deviation), bell-curve shape, and symmetry properties
- Apply z-score standardization to convert normal distribution values to the standard normal and use z-tables to find probabilities and percentiles
- Describe the uniform and exponential distributions and apply them to model equally-likely outcomes and waiting times respectively
- Apply normal probability calculations to real-world problems including quality control limits, test score percentiles, and measurement tolerances
Sampling Distributions & CLT
- Describe the concept of a sampling distribution and explain how repeated sampling produces a distribution of sample statistics
- Apply the Central Limit Theorem to determine when sample means are approximately normally distributed regardless of population shape
- Analyze how sample size affects the standard error of the mean and explain the practical implications for study design and precision of estimates
4
Inferential Statistics
4 topics
Confidence Intervals
- Describe the concept of a confidence interval including the point estimate, margin of error, and the correct interpretation of confidence level
- Construct confidence intervals for population means using z-intervals and t-intervals and determine which is appropriate based on sample size and known variance
- Construct confidence intervals for population proportions and determine the sample size required to achieve a desired margin of error
- Analyze common misinterpretations of confidence intervals and explain why a 95% CI does not mean there is a 95% probability the parameter lies within the interval
Hypothesis Testing
- Describe the hypothesis testing framework including null and alternative hypotheses, test statistics, p-values, and significance levels (alpha)
- Apply one-sample and two-sample t-tests to compare means and interpret results in the context of the research question
- Apply chi-squared tests for independence and goodness-of-fit to evaluate relationships between categorical variables
- Describe Type I and Type II errors, statistical power, and the relationship between sample size, effect size, and the probability of detecting a true effect
- Analyze the distinction between statistical significance and practical significance and evaluate when a statistically significant result may not be meaningful
- Apply paired t-tests to before-and-after experimental designs and explain how pairing reduces variability compared to independent samples
ANOVA Basics
- Describe one-way ANOVA including its purpose, assumptions (normality, equal variances, independence), and the F-statistic as a ratio of between-group to within-group variance
- Apply one-way ANOVA to compare means across three or more groups and interpret the F-test result to determine whether group means differ significantly
- Analyze the need for post-hoc tests (Tukey HSD, Bonferroni correction) after a significant ANOVA result and explain the multiple comparisons problem
Non-Parametric Basics
- Describe when non-parametric tests are preferred over parametric tests including violations of normality assumptions and ordinal data
- Apply the Mann-Whitney U test and Wilcoxon signed-rank test as non-parametric alternatives to t-tests for comparing group distributions
- Evaluate the trade-offs between parametric and non-parametric approaches in terms of statistical power, assumptions, and interpretability
5
Regression & Correlation
3 topics
Correlation Analysis
- Describe the Pearson correlation coefficient including its range (-1 to +1), interpretation of strength and direction, and linearity assumption
- Apply Spearman rank correlation to assess monotonic relationships and explain when it is preferred over Pearson correlation
- Analyze the critical distinction between correlation and causation and identify confounding variables, spurious correlations, and reverse causation in examples
Simple Linear Regression
- Describe the simple linear regression model including slope, intercept, residuals, and the least squares method for finding the best-fit line
- Apply simple linear regression to make predictions and interpret the slope as the expected change in the response per unit change in the predictor
- Interpret R-squared as the proportion of variance explained by the regression and explain its limitations as a sole measure of model quality
- Analyze residual plots to check regression assumptions including linearity, constant variance (homoscedasticity), and normality of residuals
- Apply hypothesis tests for regression coefficients to determine whether the slope is significantly different from zero and interpret the results
Regression Pitfalls
- Identify common regression pitfalls including extrapolation beyond observed data range, influential outliers, and Simpson's paradox
- Analyze the impact of outliers and leverage points on regression coefficients and describe strategies for robust regression analysis
6
Bayesian Thinking
2 topics
Bayesian Framework
- Describe the Bayesian framework including prior probability, likelihood, and posterior probability and explain how beliefs are updated with evidence
- Apply Bayesian updating to simple problems such as iteratively revising disease probability as additional test results become available
- Analyze how the choice of prior affects posterior estimates and describe informative vs non-informative priors in practical applications
Bayesian vs Frequentist
- Compare frequentist and Bayesian interpretations of probability, hypothesis testing, and parameter estimation using concrete examples
- Evaluate the advantages and limitations of Bayesian approaches including natural uncertainty quantification, prior sensitivity, and computational demands
Scope
Included Topics
- Descriptive statistics (measures of central tendency, dispersion, shape), probability theory (axioms, conditional probability, Bayes' theorem, independence), probability distributions (discrete: binomial, Poisson, geometric; continuous: normal, exponential, uniform), inferential statistics (confidence intervals, hypothesis testing, t-tests, chi-squared tests, ANOVA basics), regression and correlation (simple linear regression, Pearson and Spearman correlation, coefficient of determination), Bayesian thinking (prior/posterior/likelihood, Bayesian updating, comparison with frequentist approach)
Not Covered
- Measure theory and formal probability axiomatization
- Advanced multivariate methods (MANOVA, factor analysis, structural equation modeling)
- Survival analysis and time-to-event methods
- Non-parametric statistics beyond basic rank tests
- Stochastic processes and Markov chains
- Advanced Bayesian computation (MCMC, variational inference)
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