# Statistics Flashcards and Spaced Review: [Your course or exam] > A question-and-answer bank of 72 statistics flashcards grouped by topic, with Leitner box instructions, a spaced > review calendar (1, 3, 7, 14 and 30 days), rules for writing good cards, an export note for Anki, and a tracking > table. It is for students preparing for exams, thesis writers who need the core ideas at their fingertips, and > analysts refreshing their basics. It speeds up learning by combining retrieval practice (answering from memory) with > spacing (reviewing just before you would forget), the two techniques with the strongest evidence for long-term > retention. The worked examples use fictional data; replace them with your own. > Source basis: the Leitner box method (Leitner 1972) and spacing and retrieval research (Cepeda et al. 2006; > Karpicke and Roediger 2008; Kornell 2009), written in our own words. ## Document control | Field | Value | |---|---| | Course or project | [STAT 201, Lakemoor University (fictional)] | | Learner or analyst | [Your name] | | Deck started | YYYY-MM-DD | | Software and versions | Calendar code: R 4.5.1 and Python 3.14 with pandas 3.1.0; optional Anki for review | | Date | YYYY-MM-DD | | Version | [0.1] | ## How spaced review works Memory fades quickly after you learn something, and each successful recall tends to make the next fade slower. Spaced reviews beat the same amount of massed review (cramming) for long-term retention, and a review that takes some effort to recall usually helps more than an easy one. So the gaps between reviews grow: one day, three days, a week, two weeks, a month. These exact gaps are a common practical choice, not a law: research supports expanding or spaced gaps in general, and the best gap depends on how long you need to remember (Cepeda et al. 2006). Cards you know well come up rarely; cards you miss come back tomorrow. Answering from memory (retrieval) matters as much as the timing: reading a card's answer without first trying to recall it does little. The **Leitner system** puts the schedule into five boxes (or five sections of a list). Every card starts in box 1. When you answer a card correctly, it moves up one box; when you miss it, it goes back to box 1. Each box has its own review interval. Leitner's original scheme used boxes of growing size rather than fixed day counts; the intervals below are a widely used variant, so adjust them to your exam date and workload. | Box | Review every | Meaning | |---|---|---| | 1 | 1 day | new or recently missed | | 2 | 3 days | recalled once | | 3 | 7 days | recalled twice | | 4 | 14 days | recalled three times | | 5 | 30 days | well learned; a miss still sends it back to box 1 | ```mermaid flowchart TD N["New card"] --> B1["Box 1: review daily"] B1 -- "Correct" --> B2["Box 2: every 3 days"] B2 -- "Correct" --> B3["Box 3: every 7 days"] B3 -- "Correct" --> B4["Box 4: every 14 days"] B4 -- "Correct" --> B5["Box 5: every 30 days"] B2 -- "Wrong" --> B1 B3 -- "Wrong" --> B1 B4 -- "Wrong" --> B1 B5 -- "Wrong" --> B1 ``` ## Spaced review calendar For each topic, plan five reviews after the day you first learn it. The code below prints the dates for an example start date; change it to yours. ```r start <- as.Date("2026-11-02") gaps <- c(1, 3, 7, 14, 30) data.frame(review = seq_along(gaps), days_after = gaps, date = start + gaps) ``` ```text review days_after date 1 1 1 2026-11-03 2 2 3 2026-11-05 3 3 7 2026-11-09 4 4 14 2026-11-16 5 5 30 2026-12-02 ``` ```python from datetime import date, timedelta import pandas as pd start = date(2026, 11, 2) gaps = [1, 3, 7, 14, 30] print(pd.DataFrame({"review": range(1, 6), "days_after": gaps, "date": [start + timedelta(days=g) for g in gaps]}).to_string(index=False)) ``` ```text review days_after date 1 1 2026-11-03 2 3 2026-11-05 3 7 2026-11-09 4 14 2026-11-16 5 30 2026-12-02 ``` | Topic | Learned | +1 day | +3 days | +7 days | +14 days | +30 days | |---|---|---|---|---|---|---| | *Example:* Sampling distributions and SE | 2026-11-02 | 2026-11-03 | 2026-11-05 | 2026-11-09 | 2026-11-16 | 2026-12-02 | | [Topic] | YYYY-MM-DD | YYYY-MM-DD | YYYY-MM-DD | YYYY-MM-DD | YYYY-MM-DD | YYYY-MM-DD | | [Topic] | YYYY-MM-DD | YYYY-MM-DD | YYYY-MM-DD | YYYY-MM-DD | YYYY-MM-DD | YYYY-MM-DD | ## Card bank: describing data **C1. Q:** What does the standard deviation measure, and in which units? **A:** The spread of observations around their mean (roughly, a typical distance from the mean), in the units of the data. **C2. Q:** Why does the sample variance divide by $n-1$? **A:** Using $\bar{x}$ in place of $\mu$ makes squared deviations slightly too small; dividing by $n-1$ makes the sample variance an unbiased estimate of the population variance. **C3. Q:** When should you prefer the median and IQR to the mean and SD? **A:** When the distribution is strongly skewed or has outliers. **C4. Q:** In a right-skewed distribution, where is the mean relative to the median? **A:** Usually above it: the long right tail pulls the mean up. **C5. Q:** Name the four measurement levels in order. **A:** Nominal, ordinal, interval, ratio. **C6. Q:** What is a z-score? **A:** The number of SDs an observation lies from the mean: $z = (x - \bar{x})/s$. **C7. Q:** What is the 1.5 IQR rule? **A:** Points below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$ are flagged as potential outliers. **C8. Q:** Which plot shows the shape of one numeric variable, and which compares it across groups? **A:** A histogram (or density plot) for shape; side-by-side box plots or dot plots for groups. ## Card bank: probability and distributions **C9. Q:** State the addition rule for $P(A \text{ or } B)$. **A:** $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. **C10. Q:** What does independence of $A$ and $B$ mean in a formula? **A:** $P(A \cap B) = P(A)\,P(B)$, equivalently $P(A \mid B) = P(A)$. **C11. Q:** Are mutually exclusive events with positive probabilities independent? **A:** No: if one happens the other cannot, so they are dependent. **C12. Q:** Write Bayes' theorem. **A:** $P(H \mid E) = P(E \mid H)P(H) / P(E)$, with $P(E) = P(E \mid H)P(H) + P(E \mid \neg H)P(\neg H)$. **C13. Q:** Why can a positive result from an accurate screening test still be probably a false positive? **A:** When the condition is rare, the many non-cases produce more false positives than the few cases produce true positives. **C14. Q:** Mean and variance of a binomial$(n, p)$ variable? **A:** Mean $np$, variance $np(1-p)$. **C15. Q:** Which distribution has mean equal to variance? **A:** The Poisson. **C16. Q:** What fractions of a normal distribution lie within 1, 2 and 3 SDs of the mean? **A:** About 68%, 95% and 99.7%. **C17. Q:** What are the conditions for a binomial setting? **A:** A fixed number of independent trials, two outcomes each, the same success probability on every trial. **C18. Q:** What happens to the t distribution as df increases? **A:** Its tails get lighter and it approaches the standard normal. ## Card bank: sampling distributions and confidence intervals **C19. Q:** What is a sampling distribution? **A:** The distribution of a statistic (such as $\bar{x}$) over all possible random samples of the same size. **C20. Q:** What is the standard error of the mean? **A:** The SD of the sampling distribution of $\bar{x}$, $\sigma/\sqrt{n}$; in practice it is estimated by $s/\sqrt{n}$. **C21. Q:** How much larger must a sample be to halve the SE? **A:** Four times as large. **C22. Q:** What does the central limit theorem say? **A:** For large enough $n$, the sampling distribution of the mean is approximately normal, whatever the shape of the population (with finite variance). **C23. Q:** Does a larger sample make the sample SD smaller? **A:** No, not systematically. The sample SD estimates the population spread, which does not depend on $n$; the SE shrinks in proportion to $1/\sqrt{n}$. **C24. Q:** General form of most confidence intervals? **A:** Estimate $\pm$ critical value $\times$ SE. **C25. Q:** Correct meaning of "95% confidence"? **A:** The method captures the true parameter in 95% of repeated samples; a single computed interval either does or does not contain it. **C26. Q:** Which CI is recommended for a proportion with small $n$ or $\hat{p}$ near 0 or 1? **A:** The Wilson score interval (not the Wald interval). **C27. Q:** What happens to a CI's width when you raise the confidence level from 95% to 99%? **A:** It gets wider. **C28. Q:** Do two overlapping 95% CIs prove the difference is not significant? **A:** No. They can overlap while the difference is significant; compute a CI for the difference itself. ## Card bank: hypothesis testing **C29. Q:** Define the p-value. **A:** The probability, computed assuming $H_0$ is true, of a test statistic at least as extreme as the one observed. **C30. Q:** Is the p-value the probability that $H_0$ is true? **A:** No. It is computed assuming $H_0$ is true. **C31. Q:** Define a Type I and a Type II error. **A:** Type I: rejecting a true $H_0$ (probability $\alpha$). Type II: failing to reject a false $H_0$ (probability $\beta$). **C32. Q:** What is power? **A:** $1 - \beta$: the probability of rejecting $H_0$ when a specified alternative is true. **C33. Q:** Name three things that increase power. **A:** Larger sample size, larger true effect, less variability (also a larger $\alpha$ or a justified one-sided test). **C34. Q:** Does p > .05 show there is no effect? **A:** No. The data may be compatible with both no effect and an important one; report the CI. **C35. Q:** How does a 95% CI relate to a two-sided test at $\alpha = 0.05$? **A:** The test rejects $H_0: \theta = \theta_0$ exactly when $\theta_0$ lies outside the 95% CI (for matching methods). **C36. Q:** What does the Bonferroni correction do? **A:** Tests each of $m$ hypotheses at $\alpha/m$ to keep the family-wise Type I error at most $\alpha$. **C37. Q:** Which procedure controls the false discovery rate? **A:** Benjamini-Hochberg. **C38. Q:** When may you use a one-sided test? **A:** Only when the direction was specified before seeing the data and an effect in the other direction would be treated the same as no effect. ## Card bank: choosing and running tests **C39. Q:** Default test for comparing means of two independent groups? **A:** Welch's t-test (does not assume equal variances). **C40. Q:** Same subjects measured twice, numeric outcome: which test? **A:** Paired t-test on the differences (Wilcoxon signed-rank if the differences are strongly non-normal). **C41. Q:** Rank-based alternative to the two-sample t-test? **A:** Mann-Whitney U (Wilcoxon rank-sum) test. It tests whether values in one group tend to be larger than in the other; it is a test of medians only if both groups have the same shape. **C42. Q:** Test for association between two categorical variables? **A:** Chi-square test of independence; Fisher's exact test when expected counts are small. **C43. Q:** Rule of thumb for chi-square expected counts? **A:** Most expected counts (at least 80%) should be 5 or more, and none below 1. **C44. Q:** Same students answered yes or no before and after: which test for a change in proportion? **A:** McNemar's test. **C45. Q:** Default `scipy.stats.ttest_ind` and R `t.test` for two groups: which tests? **A:** scipy: pooled (equal variances) unless `equal_var=False`; R: Welch. **C46. Q:** Main assumptions of the two-sample t-test? **A:** Independent observations; roughly normal sampling distribution of the means (normal data or large n without extreme skew); equal variances only for the pooled version. ## Card bank: ANOVA **C47. Q:** What does the F statistic in one-way ANOVA compare? **A:** Between-group mean square to within-group mean square. **C48. Q:** Degrees of freedom in one-way ANOVA with $k$ groups and $N$ observations? **A:** Between $k-1$, within $N-k$, total $N-1$. **C49. Q:** What does a significant one-way ANOVA tell you? **A:** At least one group mean differs; post hoc tests (Tukey HSD, Games-Howell) show which. **C50. Q:** Why not run all pairwise t-tests instead? **A:** The chance of at least one false positive grows with the number of tests. **C51. Q:** What is an interaction in a two-way ANOVA? **A:** The effect of one factor depends on the level of the other. ## Card bank: correlation and regression **C52. Q:** What does Pearson's $r$ measure? **A:** The strength and direction of a linear association, from -1 to 1. **C53. Q:** Can $r = 0$ coexist with a strong relationship? **A:** Yes, if the relationship is curved (for example U-shaped). **C54. Q:** When should you use Spearman instead of Pearson? **A:** For ordinal data, monotonic but non-linear relations, or outliers that dominate $r$. **C55. Q:** Formula for the least-squares slope in simple regression? **A:** $b_1 = S_{xy}/S_{xx} = r\,s_y/s_x$. **C56. Q:** Interpret $R^2 = 0.85$. **A:** 85% of the variance in $y$ is accounted for by the model (in the sample). **C57. Q:** The normality assumption in linear regression applies to what? **A:** The residuals (errors), not the raw outcome or predictors. **C58. Q:** Name two residual plots and what each checks. **A:** Residuals vs fitted values (linearity, constant variance); normal Q-Q plot of residuals (normality). **C59. Q:** Why is a regression slope from observational data not a causal effect? **A:** Confounders not in the model can create or hide the association. ## Card bank: generalised linear models **C60. Q:** Link function of logistic regression? **A:** The logit: $\ln(p/(1-p))$. **C61. Q:** How do you turn a logistic coefficient $b$ into an odds ratio? **A:** $e^{b}$. **C62. Q:** Link function of Poisson regression, and what $e^{b}$ means? **A:** The log link; $e^{b}$ is a rate ratio per unit increase in the predictor. **C63. Q:** What is overdispersion and one remedy? **A:** More variance in the counts than the Poisson model implies (variance larger than the mean, given the predictors); use a negative binomial (or quasi-Poisson) model. **C64. Q:** When is an odds ratio close to a relative risk? **A:** When the outcome is rare. ## Card bank: effect sizes, power and reporting **C65. Q:** Formula for Cohen's $d$ for two independent groups? **A:** $(\bar{x}_1 - \bar{x}_2)/s_p$, with $s_p$ the pooled SD. **C66. Q:** Why prefer Hedges' $g$ in small samples? **A:** $d$ overestimates the population effect size in small samples; $g$ multiplies it by a correction factor slightly below 1. **C67. Q:** Effect size for a chi-square test of independence? **A:** Cramér's $V = \sqrt{\chi^2/(N(\min(r,c)-1))}$. **C68. Q:** Approximate $n$ per group for $d = 0.5$, $\alpha = .05$ two-sided, power 0.80? **A:** About 64 per group (exact t-based answer 63.8, rounded up). **C69. Q:** Why is "observed power" after a study uninformative? **A:** For a given test and sample size it is a one-to-one function of the p-value, so it adds no new information. **C70. Q:** What should an APA-style result for a t-test include? **A:** Means and SDs, $t$(df), exact p (or p < .001), the effect size, and a 95% CI. **C71. Q:** How do you report a p-value of 0.0004? **A:** "p < .001". **C72. Q:** What do the ASA's 2016 p-value principles say about the 0.05 threshold? **A:** Scientific conclusions should not be based only on whether p passes a threshold; p does not measure effect size or importance, and proper inference needs full reporting and transparency. ## How to write good cards Five rules make cards fast to review and hard to fool yourself on: 1. **One idea per card.** A card that asks for three things is either all right or all wrong, so you cannot tell which part you know. 2. **Cue the retrieval.** The question should make you recall, not recognise: "What does a p-value measure?" rather than "Is the p-value the probability of $H_0$? yes/no". 3. **Short answers.** If the answer needs more than two sentences, split the card. 4. **Use your own example.** Tie the card to a concrete situation (Lakemoor University (fictional) exam scores, loaves at Brightwell Bakery (fictional)). 5. **Make cards from mistakes.** Every fix rule in academic/stats-error-log becomes a card. **Example A (fully worked).** Bad card: "Q: Explain hypothesis testing. A: (half a page)". Problems: many ideas, no cue, long answer. Rewritten as four cards: - Q: What does $H_0$ state in a two-sample t-test? A: The two population means are equal. - Q: Define the p-value. A: The probability of a result at least this extreme if $H_0$ were true. - Q: What is a Type I error? A: Rejecting a true $H_0$. - Q: What decides the probability of a Type I error? A: The significance level $\alpha$ you choose in advance. **Example B (half worked).** Bad card: "Q: Regression? A: $b_1 = S_{xy}/S_{xx}$, $R^2$, residuals, assumptions". Rewrite into cards: - Q: Formula for the least-squares slope? A: [ ]. - Q: What does $R^2$ measure? A: [ ]. - Q: [ ]? A: The residuals should look like random scatter around zero with constant spread. **Exercise C.** Turn this error-log fix rule into one or two good cards: "Same units measured twice: compute differences first, then a one-sample t-test on the differences." ## Export to Anki or another app Most flashcard apps, including Anki, can import a plain text file with one card per line and a separator between question and answer. Copy each card as one line in the form `question;answer`, save the lines as a `.txt` file, and in Anki use File, Import, then check that the field separator is set to "Semicolon" (Anki tries to detect it; tab and comma also work). Remove or replace any semicolons inside a card first (use a comma). Math: Anki shows `$...$` as plain text; its built-in MathJax expects `\(...\)` for inline math, so either rewrite formulas in that form or keep them as plain text. Example lines: ```text What is the standard error of the mean?;s divided by the square root of n: the SD of the sampling distribution of the mean Default test for two independent group means?;Welch t-test (does not assume equal variances) Link function of logistic regression?;The logit, ln(p/(1-p)) ``` ## Tracking table | Date | Cards reviewed | Correct | Accuracy | New cards added | Box counts (1/2/3/4/5) | Minutes | |---|---|---|---|---|---|---| | *Example:* 2026-11-03 | 24 | 18 | 75% | 8 | 30/10/0/0/0 | 15 | | *Example:* 2026-11-05 | 20 | 17 | 85% | 6 | 27/15/4/0/0 | 14 | | YYYY-MM-DD | [ ] | [ ] | [ ] | [ ] | [ ] | [ ] | | YYYY-MM-DD | [ ] | [ ] | [ ] | [ ] | [ ] | [ ] | The chart shows how cards should move up the boxes over a month of steady review (fictional counts). With the 1-3-7-14 day intervals, a new card answered correctly every time reaches box 5 on day 25 at the earliest, so box 5 fills only in week 4. If box 1 keeps growing, slow down new cards and split hard cards. ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Fictional number of cards in each Leitner box at the end of weeks 1 to 4.", "width": 360, "height": 200, "data": {"values": [ {"week": 1, "box": "Box 1", "cards": 30}, {"week": 1, "box": "Box 2", "cards": 15}, {"week": 1, "box": "Box 3", "cards": 5}, {"week": 1, "box": "Box 4", "cards": 0}, {"week": 1, "box": "Box 5", "cards": 0}, {"week": 2, "box": "Box 1", "cards": 18}, {"week": 2, "box": "Box 2", "cards": 20}, {"week": 2, "box": "Box 3", "cards": 14}, {"week": 2, "box": "Box 4", "cards": 6}, {"week": 2, "box": "Box 5", "cards": 0}, {"week": 3, "box": "Box 1", "cards": 12}, {"week": 3, "box": "Box 2", "cards": 14}, {"week": 3, "box": "Box 3", "cards": 18}, {"week": 3, "box": "Box 4", "cards": 22}, {"week": 3, "box": "Box 5", "cards": 0}, {"week": 4, "box": "Box 1", "cards": 9}, {"week": 4, "box": "Box 2", "cards": 10}, {"week": 4, "box": "Box 3", "cards": 15}, {"week": 4, "box": "Box 4", "cards": 20}, {"week": 4, "box": "Box 5", "cards": 18} ]}, "mark": "bar", "encoding": { "x": {"field": "week", "type": "ordinal", "title": "Week"}, "y": {"field": "cards", "type": "quantitative", "stack": "zero", "title": "Cards"}, "color": {"field": "box", "type": "nominal", "title": "Leitner box"} } } ``` ## Common mistakes with flashcards | Mistake | Better habit | |---|---| | Flipping the card before really trying to answer | say or write an answer first, even a guess | | Grading yourself generously ("I nearly had it") | nearly is wrong: back to box 1 | | Reviewing all cards every day | follow the boxes; well-known cards waste time | | Only definition cards | add method-choice and interpretation cards (C39 to C46 style) | | Cards copied from a textbook without understanding | write the answer in your own words with your own example | | Stopping once the exam is over | keep box 5 reviews for the next course that builds on this one | ## Self-test questions 1. A card in box 3 is answered wrongly. Where does it go, and when will you see it next? 2. Why do the review gaps grow (1, 3, 7, 14, 30 days) instead of staying the same? 3. What is wrong with the card "Q: What are the assumptions, formula and interpretation of ANOVA?" 4. What separator do you choose when importing `question;answer` lines into Anki, and what must you check first? 5. Your accuracy has been 65% for a week. Name two changes to make. ## My notes and next steps New cards this week (topics): [ ]. Cards I keep missing (numbers): [ ]. Daily review time and place: [ ]. Link to my error log entries turned into cards: [academic/stats-error-log]. ## Sources and further reading - Leitner, S. (1972). *So lernt man lernen*. Herder. - Ebbinghaus, H. (1885). *Über das Gedächtnis: Untersuchungen zur experimentellen Psychologie*. Duncker und Humblot. - Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., and Rohrer, D. (2006). Distributed practice in verbal recall tasks: a review and quantitative synthesis. *Psychological Bulletin*, 132(3), 354–380. - Karpicke, J. D., and Roediger, H. L. (2008). The critical importance of retrieval for learning. *Science*, 319(5865), 966–968. - Kornell, N. (2009). Optimising learning using flashcards: spacing is more effective than cramming. *Applied Cognitive Psychology*, 23(9), 1297–1317. - Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., and Willingham, D. T. (2013). Improving students' learning with effective learning techniques. *Psychological Science in the Public Interest*, 14(1), 4–58. - Wasserstein, R. L., and Lazar, N. A. (2016). The ASA statement on p-values: context, process, and purpose. *The American Statistician*, 70(2), 129–133. - Further reading (by name only): NIST/SEMATECH *e-Handbook of Statistical Methods*; de Smith, M. J. (2024). *Statistical Analysis Handbook*, 2024 edition. StatsRef, https://www.statsref.com/; *OpenIntro Statistics*. ## Answer key **Example B.** Slope: $b_1 = S_{xy}/S_{xx}$. $R^2$: the proportion of the variance in $y$ accounted for by the model. Third card question: "What should a residuals-versus-fitted plot look like if the linear model fits?" **Exercise C (model answer).** "Q: Same units measured twice, numeric outcome: what do you compute first? A: The within-unit differences." and "Q: Which test do you then run on the differences? A: A one-sample t-test against 0 (the paired t-test)." **Self-test.** 1. Back to box 1; you see it again tomorrow. 2. Each successful recall tends to make forgetting slower, so the next review can wait longer; growing gaps keep each review effortful but still successful, which gives more benefit per minute than fixed short gaps. The exact numbers are a practical convention, not a law. 3. It contains several ideas, so you cannot tell which part you know, and the answer is too long. Split it into one card per idea. 4. Semicolon (check that Anki detected it); first remove or replace semicolons inside questions and answers, and rewrite `$...$` math as `\(...\)` or plain text. 5. Add fewer new cards until accuracy recovers, and split or rewrite the cards you miss most (often by adding a concrete example). ## Revision history | Version | Date | Author | Change | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Name] | First version from template | | [0.2] | YYYY-MM-DD | [Name] | [Added 10 cards from week 6 error log] |