DSST Principles of Statistics key terms

The default claim that a tested effect equals a specified value, usually zero or no difference. Data are assessed against it, and it is rejected only when sample evidence is sufficiently unlikely under that assumption.

The competing claim a researcher hopes to support, stating that a parameter differs from, exceeds, or falls below the null value. It determines whether the test is two-sided or one-sided.

The probability, computed assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. It is not the probability that the null is true.

The threshold, conventionally 0.05, chosen before analysis such that the null is rejected when the p-value falls below it. It fixes the maximum tolerable Type I error rate.

Rejecting a true null hypothesis, a false positive. Its probability equals the significance level when assumptions hold, and it becomes more likely as many tests are run without adjustment.

Failing to reject a null hypothesis that is actually false, a missed detection. Its probability is denoted beta, and one minus beta is the power of the test.

The probability of correctly rejecting a false null hypothesis. It rises with larger effect sizes, larger samples, lower variability, and more lenient significance levels.

A standardized measure of how large a difference or association is, independent of sample size. Examples include Cohen's d and the correlation coefficient, and it drives practical interpretation.

A value computed from sample data, such as z, t, chi-square, or F, that summarizes how far the estimate lies from the null value in standard-error units and is compared to a reference distribution.

The number of independent pieces of information available after estimating parameters from data. It indexes the reference distribution used for t, chi-square, and F tests.

The result that sums and means of many independent observations tend toward a normal distribution as sample size grows, regardless of the population shape, which justifies large-sample normal-based inference.

The probability distribution of a statistic across repeated samples of the same size from the same population. Its spread is measured by the standard error and underlies all inferential procedures.

The standard deviation of a statistic's sampling distribution. For a sample mean it equals the population standard deviation divided by the square root of the sample size.

The inflation of false-positive risk when many hypothesis tests are conducted on one dataset. Procedures such as Bonferroni or Tukey adjustments control the familywise error rate.

A procedure that partitions total variability into between-group and within-group components and compares their ratio with an F test to judge whether several population means are equal.

A symmetric, bell-shaped continuous distribution defined by mean and standard deviation, with about 68 percent of values within one standard deviation and 95 percent within two.

The normal distribution with mean zero and standard deviation one. Any normal value converts to it through standardization, allowing use of a single table of probabilities.

The number of standard deviations a value lies from its mean, computed as the value minus the mean divided by the standard deviation. It permits comparison across different scales.

The distribution of the count of successes in a fixed number of independent trials, each with the same success probability. Its mean is n times p and variance is n p (1 minus p).

A single random experiment with exactly two possible outcomes, success or failure, and a fixed success probability. A sequence of such trials generates a binomial distribution.

A discrete distribution for counts of rare events occurring independently in a fixed interval, defined by a single rate parameter that equals both its mean and variance.

A continuous distribution describing waiting time between events in a Poisson process. It is right-skewed, and its mean is the reciprocal of the event rate.

A distribution in which every value in an interval, or every outcome in a finite set, has equal probability density or probability. It has a rectangular shape in the continuous case.

A function for a continuous random variable whose value gives relative likelihood, with total area under the curve equal to one. Probability comes from areas under the curve, not point heights.

A function giving the probability that a random variable is less than or equal to a given value. It increases from zero to one and works for both discrete and continuous variables.

A symmetric, heavier-tailed distribution used when the population standard deviation is unknown and estimated from the sample. Its exact shape depends on degrees of freedom.

A right-skewed distribution of a sum of squared standard normal variables, used for tests of independence, goodness of fit, and variances. It is indexed by degrees of freedom.

A right-skewed distribution of a ratio of two independent chi-square variables each divided by its degrees of freedom. It underlies analysis of variance and tests comparing variances.

Systematic distortion arising when the selected sample is not representative of the population. It does not shrink with larger samples and invalidates standard inference.

Selecting units so that every member of the population has a known, nonzero chance of inclusion, which supports unbiased estimation and valid standard errors.

A model predicting a quantitative response from one predictor using a straight line, estimated by least squares. The slope measures the expected change in the response per unit change in the predictor.

An estimation principle choosing parameter values that minimize the sum of squared vertical distances between observed and fitted values. For simple regression it yields the line through the means.

The difference between an observed value and the value predicted by the fitted model. Residual plots reveal curvature, unequal variance, and outliers that violate model assumptions.

The proportion of total variation in the response explained by the regression model, ranging from zero to one. In simple regression it equals the square of the correlation coefficient.

A number between minus one and one measuring the strength and direction of a linear association between two quantitative variables. It is unaffected by which variable is treated as response.

A model relating a response to two or more predictors simultaneously. Each slope estimates the effect of its predictor holding the others constant, which can differ sharply from a simple regression slope.

Strong linear relationships among predictors in a regression model. It inflates standard errors and destabilizes individual slope estimates even when the overall model fits well.

A diagnostic for each predictor measuring how much its slope variance is inflated by association with other predictors. Values near one indicate little collinearity; large values signal trouble.

A range of plausible values for a population parameter, computed so that a stated percentage of such intervals, across repeated samples, would contain the true value.

The long-run proportion of intervals that capture the true parameter if sampling were repeated many times, commonly 90, 95, or 99 percent. It describes the procedure, not any single interval.

The half-width of a confidence interval, equal to a critical value times the standard error. It shrinks with larger samples and grows with higher confidence levels.

Converting values to z-scores by subtracting the mean and dividing by the standard deviation, placing different variables on a common scale for comparison or model fitting.

The value below which a given percentage of observations in a distribution falls. Percentiles read directly from a cumulative distribution function and describe relative standing.