Title: Statistics Essentials — LearnCoachAssist

Q: What is a population in statistics?
A: The entire set of individuals, items, or measurements that share a common property and are the subject of a statistical study.

Q: What is a sample?
A: A subset of a population selected to represent it for analysis.

Q: Why do we use samples instead of populations?
A: Because measuring the entire population is often impractical, costly, or impossible.

Q: What is a parameter?
A: A numerical value that describes a characteristic of a population (e.g., μ for mean, σ for standard deviation).

Q: What is a statistic (in this context)?
A: A numerical value that describes a characteristic of a sample (e.g., x̄, s).

Q: What is descriptive statistics?
A: Methods for summarizing and describing the main features of a dataset.

Q: What is inferential statistics?
A: Methods for drawing conclusions about a population from sample data.

Q: Define the mean.
A: The arithmetic average: the sum of values divided by the number of values.

Q: Define the median.
A: The middle value of an ordered dataset; if even count, the average of the two middle values.

Q: Define the mode.
A: The value or values that occur most frequently in a dataset.

Q: When is the median preferred over the mean?
A: When the data are skewed or contain outliers, because the median is more robust.

Q: What is the range?
A: The difference between the maximum and minimum values in a dataset.

Q: What is variance?
A: The average of squared deviations from the mean; measures spread.

Q: What is standard deviation?
A: The square root of the variance; spread expressed in the original units.

Q: Why do we square the deviations when computing variance?
A: To prevent positive and negative deviations from cancelling out and to weight larger deviations more heavily.

Q: What is the interquartile range (IQR)?
A: Q3 − Q1; the spread of the middle 50% of the data.

Q: Define a quartile.
A: A value that divides ordered data into four equal parts (Q1, Q2 = median, Q3).

Q: What is a percentile?
A: A value below which a given percentage of observations fall.

Q: What is the z-score?
A: (x − μ) / σ; how many standard deviations a value is from the mean.

Q: What is a positively skewed distribution?
A: A distribution with a long right tail; mean > median.

Q: What is a negatively skewed distribution?
A: A distribution with a long left tail; mean < median.

Q: What does kurtosis measure?
A: The "tailedness" or peakedness of a distribution relative to the normal.

Q: What is a categorical variable?
A: A variable whose values are categories (e.g., color, gender).

Q: What is a numerical variable?
A: A variable measured on a numeric scale (e.g., height, age).

Q: What distinguishes discrete and continuous numerical variables?
A: Discrete take countable values (e.g., number of children); continuous take any value in a range (e.g., weight).

Q: What is a frequency distribution?
A: A table showing how often each value or class of values occurs.

Q: What is a histogram?
A: A bar chart of a frequency distribution for a continuous variable, with bars touching.

Q: What is a boxplot?
A: A graphical summary using minimum, Q1, median, Q3, and maximum, often showing outliers.

Q: What is a scatterplot?
A: A plot of pairs of numerical values to visualize the relationship between two variables.

Q: What is correlation?
A: A measure of the strength and direction of linear association between two variables.

Q: What is the range of Pearson's r?
A: −1 to +1.

Q: What does r = 0 mean?
A: No linear association (variables may still have a non-linear relationship).

Q: Does correlation imply causation?
A: No; correlation can arise from coincidence, confounding, or reverse causation.

Q: What is a confounding variable?
A: A third variable that influences both the independent and dependent variables, creating a spurious association.

Q: What is simple linear regression?
A: A model predicting one dependent variable from one independent variable using a straight line.

Q: What is the equation of a regression line?
A: ŷ = b0 + b1·x, where b0 is the intercept and b1 is the slope.

Q: What is the least squares method?
A: A way of fitting a regression line by minimizing the sum of squared residuals.

Q: What is a residual?
A: The difference between observed and predicted values (y − ŷ).

Q: What is R-squared?
A: The proportion of variance in the dependent variable explained by the model.

Q: What does R² = 0.8 mean?
A: 80% of the variability in the response is explained by the model.

Q: What is multicollinearity?
A: High correlation between independent variables in a regression, which destabilizes coefficient estimates.

Q: What is heteroscedasticity?
A: Non-constant variance of residuals across the range of fitted values.

Q: What is probability?
A: A measure between 0 and 1 of how likely an event is to occur.

Q: What is the law of large numbers?
A: As sample size grows, the sample mean converges to the population mean.

Q: What is the central limit theorem?
A: The sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the population's shape.

Q: Why is the central limit theorem important?
A: It justifies using normal-based inference even when the underlying data are non-normal, given a large enough sample.

Q: What is a random variable?
A: A variable whose value is a numerical outcome of a random phenomenon.

Q: Discrete vs continuous random variable?
A: Discrete takes countable values; continuous takes values in a continuous range.

Q: What is a probability mass function?
A: A function that gives the probability of each possible value of a discrete random variable.

Q: What is a probability density function?
A: A function whose area over an interval gives the probability of a continuous random variable falling in that interval.

Q: What is the cumulative distribution function?
A: F(x) = P(X ≤ x); the probability that the variable is at most x.

Q: What is expected value?
A: E[X] = Σ x · P(X = x) for discrete variables; the long-run average.

Q: What is variance of a random variable?
A: E[(X − E[X])²]; the expected squared deviation from the mean.

Q: What is the Bernoulli distribution?
A: A distribution with two outcomes (success/failure) with probabilities p and 1−p.

Q: What is the binomial distribution?
A: The distribution of the number of successes in n independent Bernoulli trials with probability p.

Q: Mean and variance of Binomial(n, p)?
A: Mean = np; variance = np(1−p).

Q: What is the Poisson distribution?
A: A model for the count of events occurring in a fixed interval at a constant rate λ.

Q: Mean and variance of Poisson(λ)?
A: Both equal to λ.

Q: What is the geometric distribution?
A: The distribution of trials needed to get the first success in repeated independent Bernoulli trials.

Q: What is the normal distribution?
A: A symmetric, bell-shaped continuous distribution defined by mean μ and standard deviation σ.

Q: What is the standard normal distribution?
A: A normal distribution with mean 0 and standard deviation 1.

Q: What is the 68-95-99.7 rule?
A: About 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations of the mean in a normal distribution.

Q: What is the t-distribution?
A: A symmetric distribution similar to the normal but with heavier tails; used when sample size is small or σ is unknown.

Q: What are degrees of freedom?
A: The number of values free to vary when estimating a statistic; often n − 1 for variance estimates.

Q: What is the chi-square distribution?
A: The distribution of the sum of squared standard normal variables; used in tests of variance and categorical data.

Q: What is the F-distribution?
A: The distribution of the ratio of two scaled chi-square variables; used in ANOVA and regression.

Q: What is sampling without replacement?
A: Selecting items without putting them back; changes probabilities across selections.

Q: What is simple random sampling?
A: A sample where every member of the population has an equal chance of being chosen.

Q: What is stratified sampling?
A: Splitting the population into strata and randomly sampling from each; improves precision when strata differ.

Q: What is cluster sampling?
A: Randomly selecting whole groups (clusters) and surveying everyone in them; reduces cost.

Q: What is systematic sampling?
A: Selecting every kth element after a random start.

Q: What is sampling bias?
A: A systematic error from non-random selection that distorts estimates.

Q: What is selection bias?
A: A bias arising from how units are included in the study, making the sample unrepresentative.

Q: What is response bias?
A: A bias from inaccurate or untruthful responses, e.g., social desirability.

Q: What is non-response bias?
A: Bias from differences between people who respond and those who don't.

Q: What is a sampling distribution?
A: The probability distribution of a statistic computed from many samples of the same size.

Q: What is standard error?
A: The standard deviation of a statistic's sampling distribution; e.g., σ/√n for the mean.

Q: What is a confidence interval?
A: A range of plausible values for a parameter, with an associated confidence level.

Q: Interpret a 95% confidence interval correctly.
A: If we repeated sampling many times, about 95% of such intervals would contain the true parameter.

Q: What does the confidence level affect?
A: The width of the interval: higher confidence → wider interval.

Q: What is hypothesis testing?
A: A procedure for using data to evaluate whether a claim about a population is plausible.

Q: What is the null hypothesis?
A: The default assumption, typically of no effect or no difference, denoted H₀.

Q: What is the alternative hypothesis?
A: The claim we are testing for evidence, denoted H₁ or Ha.

Q: What is a p-value?
A: The probability of observing data at least as extreme as the sample, assuming H₀ is true.

Q: What is the significance level α?
A: A pre-chosen threshold (often 0.05) for rejecting H₀.

Q: What does "reject H₀" mean?
A: The evidence is strong enough (p < α) to favor H₁; not proof, but strong evidence.

Q: What is a Type I error?
A: Rejecting H₀ when it is true (false positive).

Q: What is a Type II error?
A: Failing to reject H₀ when it is false (false negative).

Q: What is statistical power?
A: The probability of correctly rejecting a false null (1 − β).

Q: Factors that increase power?
A: Larger sample size, larger effect, lower variance, higher α.

Q: What is a one-tailed test?
A: A test with a directional alternative (e.g., μ > 5).

Q: What is a two-tailed test?
A: A test for any difference (e.g., μ ≠ 5).

Q: When do we use a z-test?
A: When population σ is known or n is large.

Q: When do we use a t-test?
A: When σ is unknown and we estimate it from the sample.

Q: What is a paired t-test?
A: A t-test comparing two related measurements on the same subjects (e.g., before/after).

Q: What is an independent samples t-test?
A: A t-test comparing means of two unrelated groups.

Q: What is ANOVA?
A: Analysis of Variance: tests whether means of three or more groups differ.

Q: What is the F-statistic in ANOVA?
A: The ratio of between-group variance to within-group variance.

Q: What does a significant ANOVA tell you?
A: At least one group mean differs; not which one — that requires post-hoc tests.

Q: What are post-hoc tests?
A: Pairwise comparisons (e.g., Tukey's HSD) used after a significant ANOVA.

Q: What is a chi-square test of independence?
A: A test of whether two categorical variables are independent.

Q: What is a chi-square goodness-of-fit test?
A: A test of whether observed frequencies match expected proportions.

Q: What is the Mann-Whitney U test?
A: A non-parametric test for whether two independent samples come from the same distribution.

Q: What is the Wilcoxon signed-rank test?
A: A non-parametric paired-sample test based on ranks.

Q: What is the Kruskal-Wallis test?
A: A non-parametric alternative to one-way ANOVA.

Q: What is a non-parametric test?
A: A test that does not assume a specific distribution (e.g., normality) for the data.

Q: What is the law of total probability?
A: P(A) = Σ P(A|B_i)·P(B_i) over a partition {B_i}.

Q: What is Bayes' theorem?
A: P(A|B) = P(B|A)·P(A) / P(B).

Q: What is a prior probability?
A: The probability of a hypothesis before seeing the data.

Q: What is a posterior probability?
A: The updated probability after observing data, via Bayes' rule.

Q: What is the likelihood?
A: P(data | hypothesis); how plausible the data are under a hypothesis.

Q: What is conditional probability?
A: The probability of A given B: P(A|B) = P(A and B) / P(B).

Q: What does independence of A and B mean?
A: P(A and B) = P(A)·P(B); knowing B tells you nothing about A.

Q: What are mutually exclusive events?
A: Events that cannot occur together; P(A and B) = 0.

Q: What is the complement rule?
A: P(not A) = 1 − P(A).

Q: What is the addition rule?
A: P(A or B) = P(A) + P(B) − P(A and B).

Q: What is the multiplication rule?
A: P(A and B) = P(A)·P(B|A).

Q: What is permutation?
A: An arrangement of items where order matters: nPr = n!/(n−r)!.

Q: What is a combination?
A: A selection of items where order doesn't matter: nCr = n!/(r!(n−r)!).

Q: What is the difference between sampling distribution and population distribution?
A: The sampling distribution is the distribution of a statistic over many samples; the population distribution describes the underlying data.

Q: What is bias of an estimator?
A: The expected difference between the estimator and the true parameter.

Q: What is an unbiased estimator?
A: An estimator whose expected value equals the parameter.

Q: Is the sample mean unbiased for the population mean?
A: Yes; E[x̄] = μ.

Q: Why divide by n − 1 in sample variance?
A: Bessel's correction; it makes the sample variance an unbiased estimator of population variance.

Q: What is the maximum likelihood estimator (MLE)?
A: The parameter value that maximizes the likelihood of the observed data.

Q: What is a confidence interval for a proportion?
A: p̂ ± z·√(p̂(1−p̂)/n) for large samples.

Q: What is the formula for the standard error of a proportion?
A: √(p(1−p)/n).

Q: What is statistical significance?
A: When evidence is strong enough that an observed effect is unlikely to be due to chance under H₀.

Q: What is practical significance?
A: Whether an effect is large enough to matter in the real world, regardless of p-value.

Q: What is effect size?
A: A standardized measure of the magnitude of a phenomenon (e.g., Cohen's d).

Q: What is Cohen's d?
A: (mean1 − mean2) / pooled SD; an effect size for the difference of means.

Q: What is multiple testing?
A: Conducting many tests, which inflates the chance of a Type I error.

Q: What is the Bonferroni correction?
A: Dividing α by the number of tests to control family-wise error.

Q: What is FDR (False Discovery Rate)?
A: The expected proportion of false positives among rejected hypotheses; controlled by methods like Benjamini-Hochberg.

Q: What is a Q-Q plot?
A: A plot comparing quantiles of data to quantiles of a reference distribution; checks distributional assumptions.

Q: What is the Shapiro-Wilk test?
A: A test of whether a sample comes from a normal distribution.

Q: What is a residual plot?
A: A plot of residuals vs fitted values; used to check assumptions of regression.

Q: What does a funnel-shaped residual plot indicate?
A: Heteroscedasticity (non-constant variance).

Q: What is an outlier?
A: An observation far from the bulk of the data.

Q: What is a leverage point?
A: An observation with an extreme x-value in regression.

Q: What is an influential point?
A: An observation whose removal substantially changes the model fit.

Q: What is Cook's distance?
A: A measure of the influence of a point on regression coefficients.

Q: What is multiple regression?
A: Regression with two or more independent variables.

Q: What is the interpretation of a regression coefficient?
A: The expected change in y for a one-unit change in that x, holding other variables constant.

Q: What is logistic regression?
A: Regression for a binary outcome using the logistic function; outputs probabilities.

Q: What is the logit?
A: log(p / (1 − p)); the link function in logistic regression.

Q: What is an odds ratio?
A: exp(β); the multiplicative change in odds for a one-unit increase in the predictor.

Q: What is overfitting?
A: A model that captures noise in training data and generalizes poorly.

Q: What is underfitting?
A: A model too simple to capture the underlying structure.

Q: What is cross-validation?
A: A method of estimating model performance by splitting data into training and validation folds.

Q: What is k-fold cross-validation?
A: Splitting data into k subsets; training on k−1 and testing on the remaining, rotating k times.

Q: What is the bias-variance tradeoff?
A: A balance between error from overly simple models (bias) and overly complex models (variance).

Q: What is regularization?
A: Adding a penalty for complexity to a model to discourage overfitting (e.g., L1, L2).

Q: What is L1 (Lasso) regularization?
A: Adds |β| penalty; can shrink coefficients to zero, doing feature selection.

Q: What is L2 (Ridge) regularization?
A: Adds β² penalty; shrinks coefficients toward zero but rarely to zero.

Q: What is a time series?
A: A sequence of observations recorded over time.

Q: What is autocorrelation?
A: Correlation of a time series with a lagged version of itself.

Q: What is stationarity?
A: A time series whose statistical properties (mean, variance) don't change over time.

Q: What is a moving average?
A: A smoothed series computed by averaging values within a window.

Q: What is ARIMA?
A: AutoRegressive Integrated Moving Average; a class of time-series models combining AR and MA components with differencing.

Q: What is a survey?
A: A data collection method using questions posed to respondents.

Q: What is an observational study?
A: A study where the researcher observes without intervening.

Q: What is an experimental study?
A: A study where the researcher manipulates one or more variables to observe effects.

Q: Why is random assignment important in experiments?
A: It balances unmeasured confounders across groups, supporting causal inference.

Q: What is a placebo?
A: A sham treatment used to control for psychological effects in trials.

Q: What is double-blind?
A: Both participants and researchers don't know who is in which group.

Q: What is a control group?
A: A group that does not receive the experimental treatment, used as a baseline.

Q: What is a treatment group?
A: A group that receives the experimental intervention.

Q: What is a within-subjects design?
A: An experimental design where each subject experiences all conditions.

Q: What is a between-subjects design?
A: A design where different subjects are assigned to different conditions.

Q: What is regression to the mean?
A: The tendency for extreme observations to be closer to the mean on subsequent measurements.

Q: What is Simpson's paradox?
A: A trend in aggregated data that reverses when subgroups are considered.

Q: What is the gambler's fallacy?
A: The mistaken belief that past random events affect future independent random events.

Q: What is the hot-hand fallacy?
A: The mistaken belief that streaks of success predict more success in independent trials.

Q: What is the base rate fallacy?
A: Ignoring base-rate information in favor of specific-case information.

Q: What is survivorship bias?
A: A bias from focusing on entities that "survived" some process while ignoring those that didn't.

Q: What is publication bias?
A: The tendency for journals to publish only statistically significant or positive findings.

Q: What is p-hacking?
A: Manipulating data analysis until non-significant results become significant.

Q: What is HARKing?
A: Hypothesizing After Results are Known; presenting post-hoc findings as a priori.

Q: What is a meta-analysis?
A: A statistical synthesis of results from multiple studies on the same question.

Q: What is a forest plot?
A: A graphical display of effect estimates and confidence intervals from multiple studies in a meta-analysis.

Q: What is a funnel plot?
A: A plot used to detect publication bias in meta-analyses.

Q: What is a contingency table?
A: A table cross-tabulating frequencies of two categorical variables.

Q: What is a marginal distribution?
A: The distribution of a single variable obtained by summing across the other in a contingency table.

Q: What is a conditional distribution?
A: The distribution of one variable given a fixed level of another.

Q: What is a joint probability?
A: P(A and B); the probability of two events occurring together.

Q: What is a frequentist interpretation of probability?
A: Probability as the long-run frequency of an event in repeated trials.

Q: What is a Bayesian interpretation of probability?
A: Probability as a degree of belief, updated with evidence.

Q: What is a credible interval?
A: The Bayesian analog of a confidence interval: a range containing the parameter with stated probability.

Q: What is conjugate prior?
A: A prior whose posterior comes from the same family, making Bayesian updating tractable.

Q: What is bootstrapping?
A: A resampling technique for estimating sampling distributions by drawing with replacement from the data.

Q: What is a permutation test?
A: A non-parametric test that builds the null distribution by shuffling labels.

Q: What is Bessel's correction explained again?
A: Dividing by n − 1 corrects for the bias introduced by estimating the mean from the same sample.

Q: What is Pearson correlation vs Spearman correlation?
A: Pearson measures linear association; Spearman measures monotonic association via ranks.

Q: What is point estimate?
A: A single value used as an estimate of a parameter (e.g., sample mean).

Q: What is interval estimate?
A: A range used to estimate a parameter, e.g., a confidence interval.

Q: What does "consistent estimator" mean?
A: An estimator that converges to the true parameter as sample size grows.

Q: What is an efficient estimator?
A: An unbiased estimator with the smallest possible variance among all unbiased estimators.

Q: What is the Cramér-Rao lower bound?
A: A theoretical lower bound on the variance of any unbiased estimator.

Q: What is a likelihood ratio test?
A: A hypothesis test comparing likelihoods under two nested models.

Q: What is Akaike Information Criterion (AIC)?
A: A model-selection metric: 2k − 2 ln(L); lower is better, penalizing complexity.

Q: What is BIC (Bayesian Information Criterion)?
A: A model-selection metric: k·ln(n) − 2 ln(L); penalizes complexity more heavily than AIC.

Q: What is a categorical/dummy variable in regression?
A: A binary indicator (0/1) used to encode categories.

Q: What is the reference category?
A: The omitted category against which other dummy levels are compared.

Q: What is an interaction term in regression?
A: A product term (x1·x2) capturing how one predictor's effect depends on another.

Q: What is a Bayesian network?
A: A graphical model encoding conditional dependencies among variables.

Q: What is a Markov chain?
A: A stochastic process where the next state depends only on the current state.

Q: What is an absorbing state?
A: A state in a Markov chain that, once entered, cannot be left.

Q: What is the law of small numbers?
A: A cognitive bias: assuming small samples reflect population properties.

Q: What is statistical learning?
A: A field combining statistics and machine learning for prediction and inference.

Q: What is supervised learning?
A: Learning a function from labeled input-output pairs.

Q: What is unsupervised learning?
A: Finding structure in unlabeled data (e.g., clustering, PCA).

Q: What is principal component analysis (PCA)?
A: A dimensionality-reduction technique that finds orthogonal directions of maximum variance.

Q: What is k-means clustering?
A: An algorithm partitioning data into k clusters by minimizing within-cluster variance.