Statisticsdrill

Statistics formulas, one page per concept

Every statistics formula people search for, from standard deviation to t tests and regression, with the direct answer, the formula, a worked example with exact numbers, the Excel function, the usual mistakes, and the Statisticsdrill problems that practice it.

40 concepts, each with the direct answer, syntax, worked examples, the mistakes people make and the drills that practice it. Updated November 2026.

Descriptive statistics

  • Mean, median and modeHow to calculate the mean, median and mode of a data set by hand and in Excel, when each average is the right one, and the mistakes that give a wrong answer.
  • Standard deviationThe standard deviation formula, sample versus population versions, a worked example with exact numbers and the Excel functions STDEV.S and STDEV.P.
  • VarianceHow to calculate variance step by step, why the sample version divides by n - 1, how it relates to standard deviation, and Excel VAR.S and VAR.P.
  • z-scoreThe z-score formula z = (x - mean) / sd, worked examples, how to read a z-score as a percentile with the normal table, and the Excel function STANDARDIZE.
  • PercentileHow to find a percentile of a data set and the percentile rank of one value, with exact worked examples, Excel PERCENTILE.INC and common mix-ups.
  • Coefficient of variationThe coefficient of variation (CV) is the standard deviation divided by the mean, as a percent. Formula, worked examples, Excel and when it misleads.
  • Weighted averageHow to calculate a weighted average: multiply each value by its weight and divide by the total weight. Grade and price examples plus Excel SUMPRODUCT.
  • Range and interquartile rangeThe range is max minus min; the interquartile range is Q3 minus Q1. Worked examples with exact numbers, Excel QUARTILE.INC and why methods differ.
  • Skewness and kurtosisSkewness measures asymmetry and kurtosis measures tail weight. How to read the sign, a quick formula from mean and median, and Excel SKEW and KURT.
  • OutliersHow to detect outliers with the 1.5 IQR fence rule and the z-score rule, worked examples with exact fences, and what to do with outliers once found.

Probability

  • Probability rulesThe basic probability rules with dice and coin examples: the addition rule for A or B, the multiplication rule for independent events and the complement rule.
  • Conditional probabilityHow to calculate conditional probability P(A|B) = P(A and B) / P(B), read it from a two-way table, check independence and avoid confusing P(A|B) with P(B|A).
  • Bayes' theoremBayes theorem updates a probability with new evidence. The formula, a medical screening example with exact numbers, the natural frequency shortcut and errors.
  • Expected valueExpected value is the probability-weighted average of all outcomes. Formula, worked examples with dice, raffles and bets, and the Excel SUMPRODUCT formula.
  • Permutations and combinationsWhen order matters use permutations nPr = n! / (n - r)!; otherwise combinations nCr = n! / (r! (n - r)!). Examples plus Excel PERMUT and COMBIN.

Distributions

  • Binomial distributionThe binomial formula P(X = k) = nCk p^k (1 - p)^(n - k), its mean np and standard deviation sqrt(np(1 - p)), exact worked examples and Excel BINOM.DIST.
  • Poisson distributionThe Poisson formula P(X = k) = e^(-lambda) lambda^k / k! for counts of events per interval, worked examples with exact results and Excel POISSON.DIST.
  • Normal distributionHow to find normal distribution probabilities with z-scores, read the standard normal table, find a percentile cutoff, and use NORM.DIST and NORM.INV.
  • Empirical rule (68-95-99.7)The empirical rule: 68% of normal data lies within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. Worked examples for intervals and tails.
  • Central limit theoremThe central limit theorem makes sample means approximately normal with sd sigma / sqrt(n). Explanation, worked probability examples and the n = 30 rule.
  • Standard errorThe standard error of the mean is s / sqrt(n); for a proportion it is sqrt(p(1 - p) / n). Worked examples, SE versus standard deviation and the Excel formula.

Confidence intervals and tests

  • Confidence interval for a meanHow to calculate a confidence interval for a mean with x-bar plus or minus t times s over root n, a full 95% example, when to use z, and Excel CONFIDENCE.T.
  • Confidence interval for a proportionHow to calculate a confidence interval for a proportion with p-hat plus or minus z times sqrt(p(1 - p) / n), worked survey examples and the conditions.
  • Margin of errorMargin of error is the critical value times the standard error: z times sqrt(p(1 - p) / n) for a proportion, t times s over root n for a mean. Poll examples.
  • Sample sizeSample size formulas n = (z sigma / E)^2 for a mean and n = z^2 p(1 - p) / E^2 for a proportion, worked examples giving 97 and 385, and rounding rules.
  • One-sample t-testHow to run a one-sample t-test: compute t = (x-bar - mu0) / (s / sqrt(n)), find the p-value with n - 1 degrees of freedom and decide. Example and Excel.
  • Two-sample t-testHow to compare two independent means with a two-sample t-test: the Welch formula, degrees of freedom, a worked example and the Excel T.TEST function.
  • Paired t-testHow to run a paired t-test on before and after data: take the differences, compute t = d-bar / (s_d / sqrt(n)) with n - 1 degrees of freedom. Example, Excel.
  • p-valueA p-value is the probability of a result at least as extreme as the observed one if the null is true. How to compute it from z and t, and common misreadings.
  • Hypothesis testing stepsThe five steps of a hypothesis test: state H0 and H1, choose alpha, compute the test statistic, find the p-value or critical value, decide. Worked example.
  • Type I and Type II errorsA Type I error rejects a true null (false positive, probability alpha); a Type II error misses a false null (false negative, probability beta). Examples.
  • Chi-square testHow to run a chi-square test of independence or goodness of fit: expected counts, sum of (O - E)^2 / E, degrees of freedom and p-value. Examples and Excel.
  • Two-proportion z-testHow to test whether two proportions differ with the two-proportion z-test: pooled proportion, standard error, z statistic and p-value, with a worked example.
  • A/B test significanceHow to tell whether an A/B test result is statistically significant: compute relative lift, run the two-proportion z-test, read the p-value and avoid peeking.
  • Statistical powerStatistical power is the probability a test detects a real effect, 1 - beta. What drives it, the 80% convention and sample size formulas with examples.

Regression and correlation

  • Correlation coefficient (Pearson r)How to calculate the Pearson correlation coefficient r by hand and with Excel CORREL, a worked example with exact numbers and what values of r mean.
  • Linear regressionHow to fit a regression line y = a + bx: slope b = Sxy / Sxx, intercept a = y-bar - b x-bar, a worked example, prediction and Excel SLOPE and INTERCEPT.
  • R-squaredR-squared is the share of variation in y explained by the regression, 1 - SSE / SST. Worked example with exact numbers, link to r, Excel RSQ and its limits.
  • ResidualsA residual is observed y minus predicted y. How to compute residuals from a fitted line, a worked example, and residual plot patterns that signal a bad model.
  • CovarianceHow to calculate sample and population covariance, a worked example with exact numbers, how covariance becomes correlation, and Excel COVARIANCE.S and .P.

Reading is not practicing

Every concept here links to drills that check your answer on live data. All of them are free to try.

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