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Understand my data
Describe the response and check its distribution.
Descriptive statisticsData exploration
Available
Basic statistics for each numeric column, optionally by group: n, missing, mean, 5% trimmed mean, standard error, standard deviation, variance, CV, sum, SSQ (sum of squared values), min, quartiles, median, max, range, IQR, mode, skewness, kurtosis, confidence interval of the mean and an Anderson-Darling normality p-value. Discrete columns (whole numbers, at most 20 distinct values) also get a frequency table. Charts: histogram, box plot, time series (without a group) and, for discrete columns, a bar chart.
When to use
To describe the centre, spread and shape of a measurement before choosing a test, or to compare groups side by side without a formal test.
When not to
To decide whether groups differ (use t_test or one_way_anova), or for counts of categories (use pareto).
Data
Continuous. A numeric response, at least 1 row.
Also answers
Compare groups
Checks
Normality (for the confidence interval of the mean)
Formulas and sources
Mean and standard deviation
xˉ=n1i=1∑nxi,s=n−11i=1∑n(xi−xˉ)2
Normality testData exploration
Available
Tests whether a numeric column looks normally distributed, overall or per group: Anderson-Darling (used for the decision, as in Minitab), Shapiro-Wilk, skewness, excess kurtosis, and a normal probability plot.
When to use
Before a method that assumes normal data (capability_normal, t_test with small n, control charts for individuals), or when descriptive_stats flags non-normal data.
When not to
To compare groups (use t_test or one_way_anova). With very large n even trivial departures are significant; judge the probability plot and skewness too.
Summarizes every column: type, missing values, unique values, numeric range and spread, date range, top categories, suggested roles and data quality flags.
When to use
First step on any new dataset, before choosing an analysis: to see which columns are numeric responses, which are factors, and what needs cleaning.
When not to
To test a hypothesis or compare groups; use a test from list_methods.
Data
Continuous, Counts (defects), Pass/fail, Categories. At least 1 row.
Do groups (or paired measurements) differ in the response?
Chi-square testHypothesis tests
Available
Independence: tests whether the mix of categories (defect type, pass/fail) differs between groups, with Cramér's V, adjusted residuals per cell and Fisher's exact test for sparse 2 x 2 tables. Goodness of fit: tests counts against expected shares, with Cohen's w. Checks expected counts (Cochran's rule).
When to use
Category counts: does the defect mix or pass rate differ between shifts, lines or suppliers (3 or more groups too), or do counts match expected shares.
When not to
Numeric responses (use t_test or one_way_anova); exactly two groups of pass/fail data when a confidence interval for the difference is wanted (use proportion_test); repeated measurements of the same units (McNemar, not built).
Data
Categories, Pass/fail. A categorical response, a categorical or boolean factor, at least 2 rows.
Also answers
Find relationships, Fit a distribution
Checks
Expected counts: all at least 1 and at most 20% below 5 (Cochran); Independent units (reported, not tested)
Formulas and sources
Pearson chi-square
χ2=cells∑E(O−E)2,Eij=NRiCj,ν=(r−1)(c−1)
Find relationships
Do numeric variables move together with the response?
CorrelationData exploration
Available
Pearson or Spearman correlation for every pair of 2 to 10 numeric columns: coefficient, p-value, confidence interval, n per pair (missing values dropped pair by pair) and a Cohen 1988 magnitude label. Correlation matrix heatmap and binned scatter plots. States that correlation is not causation.
When to use
To screen which input variables move together with an output (or with each other) before regression or a designed experiment.
When not to
To prove that changing one variable changes another (use a designed experiment), or for categorical variables (use a test for groups).
Data
Continuous. A numeric response, at least 3 rows.
Checks
Normality of each variable (Pearson p-values and CIs); Linear relationship (Pearson)
Formulas and sources
Pearson r
r=∑i(xi−xˉ)2∑i(yi−yˉ)2∑i(xi−xˉ)(yi−yˉ)
Check stability over time
Is the process stable over time?
Control chartStatistical process control
Available
Shewhart control chart with phase I limits and tests for special causes. Variables data: chart i_mr for one measurement per time point, xbar_r for subgroups of 2 to 10, xbar_s for subgroups above 10 (up to 50). Attribute data, one row per sample with a count: p (fraction defective) and np (number defective) for pass/fail units, c (defects per sample) and u (defects per unit) for defect counts; p and u limits follow each sample's size. Reports limits, which tests fired at which points, and whether the process is in statistical control.
When to use
Is the process stable over time? i_mr: single readings; xbar_r, xbar_s: subgroups (small, large); p, np: defective units per lot; c, u: defects per sample.
When not to
Data without a meaningful time order, or to compare groups (use one_way_anova). Raw pass/fail rows: count the defectives per sample first, one row per sample.
Data
Continuous, Counts (defects), Pass/fail. A numeric response, at least 2 rows, rows in time order.
Also answers
Check capability against spec
Checks
Approximate normality of individual values (i_mr); Independence over time (no lag-1 autocorrelation); Binomial or Poisson variation between samples (p, np, c, u)
Formulas and sources
I chart limits
CL=xˉ,UCL,LCL=xˉ±3d2MR
Check capability against spec
Does the process meet its specification?
Process capability (normal)Process capability
Available
Capability of a normally distributed measurement against one or two specification limits: Cp, Cpk from within (short-term) sigma, Pp, Ppk from overall sigma, Cpm with a target, confidence intervals and expected parts per million out of spec. Checks stability with a control chart and normality first.
When to use
To judge whether a stable process meets its specification, for example 10 +/- 0.5, after a control chart shows no special causes.
When not to
Clearly non-normal data or pass/fail counts (non-normal and attribute capability are planned), or a process that is not in statistical control.
Data
Continuous. A numeric response, at least 10 rows, rows in time order.
Which categories account for most occurrences or cost?
Pareto analysisData exploration
Available
Ranks categories (defect types, causes, locations) by count or by a summed weight such as cost, with percent, cumulative percent and the vital few categories that together reach the cutoff (80% by default). Pareto chart.
When to use
To decide which defect types or causes to work on first, from a column of categories with one row per occurrence.
When not to
For a numeric measurement (use descriptive_stats), or to test whether categories differ significantly (a Pareto chart is descriptive, not a test).
Data
Categories. A categorical response, at least 1 row.
Also answers
Understand my data
Formulas and sources
Percent and cumulative percent
pk=100∑jwjwk,Pk=j≤k∑pj
Trust my measurements
Can the measurements be trusted (repeatability, reproducibility)?
Gage R&R study (crossed, ANOVA)msa
Available
Splits measurement variation into repeatability (same operator, same part), reproducibility (between operators, plus the operator x part interaction) and part-to-part variation with a two-way random-effects ANOVA. Reports % contribution, % study variation, % tolerance when spec limits or a tolerance are given, and the number of distinct categories (ndc), rated against the AIAG guidelines, with X-bar and R charts by operator, the interaction plot and a box plot by operator. % study variation and ndc are only meaningful when the parts span the process range.
When to use
Before trusting a measurement: 2 or more operators each measure the same parts (ideally 10) 2 or 3 times, in random order.
When not to
Destructive tests or parts that each operator cannot all measure (a nested study); pass/fail or rating data (attribute agreement); an unbalanced design with missing repeats; parts picked from a narrow band of the process.
Data
Continuous. A numeric response, a categorical or numeric or id factor, at least 8 rows.
Checks
Balanced crossed design; Consistent repeatability (R chart); Normality of residuals
m_k is the k-th central moment with denominator n.
Mean, sample variance (n - 1) and t-based confidence interval of the mean: Montgomery, Introduction to Statistical Quality Control, 8th ed., sections 3.1 and 4.3.
Quartiles: Hyndman and Fan (1996), Sample quantiles in statistical packages, The American Statistician 50(4), types 6 (weibull) and 7 (linear).
Skewness G1 and excess kurtosis G2: Joanes and Gill (1998), Comparing measures of sample skewness and kurtosis, The Statistician 47(1).
Anderson-Darling p-value: D'Agostino and Stephens (1986), Goodness-of-Fit Techniques, Table 4.9.
Box plot whiskers at 1.5 IQR: Tukey (1977), Exploratory Data Analysis.
Trimmed mean (5% from each end, rounded to whole values), mode and uncorrected sum of squares: Minitab, Display Descriptive Statistics, Methods and formulas.
z_(i) are the sorted standardized values; the p-value uses A*² = A²(1 + 0.75/n + 2.25/n²).
Shapiro-Wilk
W=∑i=1n(xi−xˉ)2(∑i=1naix(i))2
a_i are the Shapiro-Wilk coefficients for sample size n.
Plotting positions
pi=n+0.4i−0.3
For the normal probability plot.
Anderson-Darling A^2 and p-value: D'Agostino and Stephens (1986), Goodness-of-Fit Techniques, section 4.8 and Table 4.9.
Shapiro-Wilk W: Shapiro and Wilk (1965), Biometrika 52; p-value by Royston (1995), Applied Statistics 44 (scipy.stats.shapiro).
Skewness G1 and excess kurtosis G2: Joanes and Gill (1998), The Statistician 47(1).
Plotting positions (i - 0.3)/(n + 0.4): Benard and Bos-Levenbach (1953).
Standard deviation
s=n−11i=1∑n(xi−xˉ)2
Type and role rules: LSS Analyst PRD section 5 (data handling).
Sample standard deviation with n - 1 denominator.
Independence: R and C are row and column totals. Goodness of fit: E = N times the expected share and nu = k - 1. No continuity correction.
Likelihood ratio
G2=2cells∑OlnEO
Adjusted residual
rij=Eij(1−Ri/N)(1−Cj/N)Oij−Eij
About standard normal under independence; beyond 3 in size marks a cell that drives the association.
Effect size
V=N(min(r,c)−1)χ2,w=Nχ2
Cramér's V for independence, Cohen's w for goodness of fit.
Pearson chi-square, G^2 and adjusted residuals: Agresti, An Introduction to Categorical Data Analysis, 2nd ed. (2007), sections 2.4 and 2.5.
Effect sizes: Cramér (1946), Mathematical Methods of Statistics; Cohen (1988).
Reference data: Agresti (2007) Table 2.5 via R's chisq.test example; Fleiss (1981) smokers data via R's prop.test example; Mendel's pea data.
Kruskal-Wallis testHypothesis tests
Available
Rank-based comparison of two or more independent groups. Does not assume normal data. Reports H (tie-corrected), df, the p-value, epsilon squared and each group's median and mean rank.
When to use
Three or more machines, suppliers or shifts with small or skewed groups, outliers, or ordinal data, where one-way ANOVA's normality assumption is doubtful.
When not to
Roughly normal data (one_way_anova has more power and gives pairwise comparisons); paired or repeated measurements.
Data
Continuous. A numeric response, a categorical or boolean factor, at least 3 rows.
Checks
Similar distribution shapes (reported, not tested)
R_i is the rank sum of group i; the denominator corrects for ties. The p-value is from the chi-square distribution with k - 1 df.
Effect size
ε2=N−1H
Epsilon squared (Tomczak and Tomczak 2014).
Kruskal and Wallis (1952), JASA 47.
Tie correction and reference data: Hollander and Wolfe, Nonparametric Statistical Methods, as in R's kruskal.test examples.
Epsilon squared: Tomczak and Tomczak (2014), Trends in Sport Sciences 21(1).
Mann-Whitney testHypothesis tests
Available
Rank-based comparison of two independent groups (Wilcoxon rank-sum). Does not assume normal data. Reports U, the p-value (exact for small samples without ties), the Hodges-Lehmann shift estimate and the rank-biserial correlation.
When to use
Two groups (machines, shifts, suppliers) with small or skewed samples, outliers, or ordinal data, where a two-sample t-test's normality assumption is doubtful.
When not to
Paired measurements on the same units (use wilcoxon_signed_rank); three or more groups (use kruskal_wallis); large roughly normal samples (t_test has more power).
Data
Continuous. A numeric response, a categorical or boolean factor, at least 2 rows.
Checks
Similar distribution shapes (reported, not tested)
Formulas and sources
U statistic
U1=R1−2n1(n1+1)
R_1 is the sum of the ranks of the first group in the combined sample.
Tests whether the mean of a numeric response differs across the groups of one factor. Classic F test with Tukey HSD when variances are equal, Welch's F with Games-Howell when they are not (chosen by Brown-Forsythe unless set). Reports the ANOVA table, eta squared, omega squared (from Welch's F when Welch is used), all pairwise comparisons with simultaneous CIs, and normality of residuals.
When to use
Comparing the mean of one measurement across 3 or more machines, suppliers, shifts or settings (it also works for 2).
When not to
Two factors at once (use two_way_anova); paired or repeated measurements on the same units; strongly skewed small groups (use kruskal_wallis).
Data
Continuous. A numeric response, a categorical or boolean factor, at least 3 rows.
Checks
Equal variances (Brown-Forsythe); Normality of residuals
One-way ANOVA table: Montgomery, Design and Analysis of Experiments, 8th ed., section 3.3; reference data Example 3.1 (etch rate vs RF power).
Tukey HSD: Tukey (1949); scipy.stats.tukey_hsd (Tukey-Kramer for unequal n).
Welch's ANOVA: Welch (1951), Biometrika 38.
Games-Howell: Games and Howell (1976), Journal of Educational Statistics 1.
Effect sizes: eta squared and omega squared (Hays 1963), labels from Cohen (1988).
Omega squared from F: Carroll and Nordholm (1975), Educational and Psychological Measurement 35; applied to Welch's F as an approximation.
Equal variances: Brown and Forsythe (1974), JASA 69.
Proportion testHypothesis tests
Available
Tests a proportion of defectives (or any pass/fail event) against a target (one_sample) or compares two groups' proportions (two_sample). Uses the normal approximation when counts are large enough and the exact binomial or Fisher's exact test otherwise; reports both, Wilson or Newcombe confidence intervals and Cohen's h.
When to use
Pass/fail data: is the defective rate above a target, or do two machines, shifts or suppliers differ in their share of defectives (per-unit rows or counts per lot).
When not to
Three or more groups or more than two outcome categories (use chi_square); defects per unit where a unit can have several (a Poisson rate test, planned); proportions tracked over time (use control_chart chart p).
Data
Pass/fail. A categorical or boolean factor, at least 1 row.
Checks
Counts large enough for the normal approximation (exact test used otherwise); Independent units (reported, not tested)
Formulas and sources
One proportion, score test
z=p0(1−p0)/np^−p0,p^=x/n
Used when n p_0 and n (1 - p_0) are both at least 5; otherwise the exact binomial test, with P(X = k) = C(n, k) p_0^k (1 - p_0)^(n - k).
One- and two-proportion z tests: Montgomery, Introduction to Statistical Quality Control, 8th ed., section 4.3.4; Agresti, An Introduction to Categorical Data Analysis, 2nd ed., sections 1.4 and 2.2.
Wilson (1927), JASA 22; Clopper and Pearson (1934), Biometrika 26.
Newcombe (1998), Two-sided confidence intervals for the single proportion and Interval estimation for the difference between independent proportions, Statistics in Medicine 17 (method 10).
Reference data: Conover (1971) via R's binom.test example; Physicians' Health Study aspirin data in Agresti (2007).
t-testHypothesis tests
Available
Tests a mean against a target (one_sample), compares the means of two groups (two_sample, Welch's test by default) or compares two measurements on the same units (paired). Reports t, df, p-value, confidence interval of the difference, and Cohen's d or Hedges' g, with normality and equal-variance checks.
When to use
One numeric response: is the mean on target, did two machines, shifts or suppliers differ, or did a change shift the same parts' readings (before vs after).
When not to
Three or more groups (use one_way_anova); counts or proportions (use a proportion test); strongly skewed small samples (use mann_whitney or wilcoxon_signed_rank).
Data
Continuous. A numeric response, a categorical or boolean factor, at least 2 rows.
Checks
Normality of each group or of the paired differences; Equal variances (only when equal_variances is true)
Equal variances: Brown and Forsythe (1974), JASA 69.
Reference data: Student (1908), The probable error of a mean, Biometrika 6 (sleep data), as in R's t.test examples.
Wilcoxon signed-rank testHypothesis tests
Available
Rank-based test for paired measurements (response vs paired_with on the same units) or for one sample against a reference median. Does not assume normal data. Reports W+, the p-value (exact for small samples without ties), the Hodges-Lehmann estimate and the matched-pairs rank-biserial correlation.
When to use
Before vs after on the same parts, or a median against a target, when the differences are skewed, have outliers or the sample is small.
When not to
Two independent groups (use mann_whitney); large roughly normal samples (t_test has more power).
Exact and approximate p-values, and reference data (Hollander and Wolfe, depression scale): Hollander and Wolfe, Nonparametric Statistical Methods, as in R's wilcox.test examples.
Hodges and Lehmann (1963), Annals of Mathematical Statistics 34.
Fisher's z transformation; the Spearman variance is Bonett and Wright's.
Pearson and Spearman coefficients and p-values: scipy.stats.pearsonr and spearmanr (t distribution with n - 2 degrees of freedom).
Confidence interval: Fisher (1921) z transformation, SE 1/sqrt(n - 3); for Spearman SE sqrt((1 + r^2/2)/(n - 3)) from Bonett and Wright (2000), Psychometrika 65(1).
Magnitude labels for r: Cohen (1988), Statistical Power Analysis for the Behavioral Sciences, 2nd ed.: 0.1 small, 0.3 medium, 0.5 large.
Example values: Anscombe (1973), Graphs in Statistical Analysis, The American Statistician 27(1).
Linear regressionregression
Available
Simple or multiple linear regression (least squares) of a numeric response on 1 to 10 numeric predictors: equation, coefficients with CIs, standardized coefficients and VIFs, R², adjusted and predicted R², ANOVA table, Cohen's f². Residual checks for normality, constant variance, independence and multicollinearity, and the rows with large residuals, high leverage or high influence (Cook's distance, DFFITS). Fitted line plot and residual plots. States that coefficients are not proof of cause.
When to use
To estimate how a numeric output changes with one or more numeric inputs, predict it, or see which inputs matter after correlation has screened them.
When not to
Categorical inputs (use one_way_anova; a general linear model is planned), a pass/fail or categorical output (logistic regression is planned), a curved relationship without a transformed or squared term, or to prove cause from observational data.
Data
Continuous. A numeric response, a numeric factor, at least 4 rows.
Also answers
Model and optimise
Checks
Linear relationship (residuals vs fitted); Normality of residuals (Anderson-Darling); Constant variance (Breusch-Pagan, Koenker); Independence in row order (lag-1 autocorrelation, Durbin-Watson); Multicollinearity (VIF)
Least squares, tests, R², PRESS, leverage, standardized and deleted residuals, Cook's distance, DFFITS and VIF: Montgomery, Peck and Vining, Introduction to Linear Regression Analysis, 5th ed., chapters 3, 4, 6 and 9; reference data Example 3.1 (delivery time).
Numerical accuracy: NIST StRD linear regression datasets Norris and Longley (certified values).
Flag cutoffs: high leverage 2p/n (Hoaglin and Welsch 1978), Cook's D > 1 (Cook and Weisberg 1982), |DFFITS| > 2 sqrt(p/n) (Belsley, Kuh and Welsch 1980), VIF > 10 (Montgomery, Peck and Vining, section 9.4).
Independence: Durbin and Watson (1950), Biometrika 37; lag-1 autocorrelation tested against its large-sample standard error 1/sqrt(n) (Box and Jenkins).
D_i defectives among n_i units in sample i, summed over the baseline samples. A negative LCL is set to 0.
c and u chart limits
c:cˉ±3cˉ,uˉ=∑ni∑ci,u:uˉ±3niuˉ
c_i defects in sample i and n_i its units (area of opportunity). A negative LCL is set to 0.
Standardized point (attribute tests)
zi=σipi−pˉ,tests run on zi
sigma_i is the sample's own sigma from the limits above, so the tests for special causes work when the limits vary with the sample size.
Binomial or Poisson dispersion
X2=i=1∑mzi2∼χm−12,dispersion ratio=m−1X2
Over the m baseline samples. A ratio well above 1 means more variation between samples than binomial or Poisson chance.
Lag-1 autocorrelation
r1=∑i=1n(xi−xˉ)2∑i=2n(xi−xˉ)(xi−1−xˉ),flagged when ∣r1∣>n2
Computed on the plotted points (individuals, subgroup means or samples).
Step change
k^=argkmax∣tk∣,tk=spk1+n−k1xˉ1..k−xˉk+1..n,reported when ∣tk^∣≥4
Checked only when the chart signals and no baseline was given; each side has at least 8 points. s_p is the pooled standard deviation of the two parts.
I-MR chart: Montgomery, Introduction to Statistical Quality Control, 8th ed., section 6.4; reference data Example 6.5 (loan processing cost).
X-bar and R chart: Montgomery, section 6.2, limits X-double-bar +/- A2 R-bar, D3 R-bar and D4 R-bar, sigma R-bar / d2.
X-bar and S chart: Montgomery, section 6.3, limits X-double-bar +/- A3 S-bar, B3 S-bar and B4 S-bar, sigma S-bar / c4.
p, np, c and u charts: Montgomery, chapter 7; reference data the orange juice can example (p chart, Example 7.1), the printed circuit board example (c chart) and the personal computer assembly example (u chart). Variable limits for unequal sample sizes as in Montgomery section 7.2.
d2, d3, c4, A2, A3, D3, D4, B3, B4 computed from their definitions (see spc_constants).
Tests for special causes: Western Electric Statistical Quality Control Handbook (1956); Nelson (1984), Journal of Quality Technology 16(4). On attribute charts they run on standardized points, as in Nelson (1989), Journal of Quality Technology 21(4).
Dispersion check: Pearson chi-square test of homogeneity of proportions (binomial) or index of dispersion (Poisson); Cochran (1954), Biometrics 10. Overdispersion and the Laney p' chart: Laney (2002), Quality Engineering 14(4).
Lag-1 autocorrelation bound 2/sqrt(n): Box, Jenkins and Reinsel, Time Series Analysis.
Step change in the mean: maximum two-sample t over all splits, the likelihood-ratio statistic for one change in a normal mean (Hinkley 1970, Biometrika 57; Worsley 1979, JASA 74). Reported at max t >= 4.0, set by simulation (under 1% false alarms for in-control normal data up to n = 1000).
Planned: run chart
σ̂ within is MR-bar / d2(2) for single readings or R-bar / d2(n) for subgroups, as in the control chart.
Pp and Ppk (overall)
Pp=6sUSL−LSL,Ppk=min(3sUSL−μ^,3sμ^−LSL)
s is the sample standard deviation of all values (n - 1 denominator).
Cpm
Cpm=6s2+(μ^−T)2USL−LSL
T is the target.
Expected PPM out of spec
PPM=106[Φ(σ^LSL−μ^)+1−Φ(σ^USL−μ^)]
Φ is the standard normal CDF; σ̂ is the within or the overall sigma.
Confidence interval for Cp and Pp
Cpνχα/2,ν2,Cpνχ1−α/2,ν2
ν is the degrees of freedom of the sigma estimate.
Confidence interval for Cpk and Ppk
C^pk±z1−α/29n1+2(n−1)C^pk2
Bissell's normal approximation.
Cp, Cpk, Pp, Ppk, Cpm definitions and CIs: Montgomery, Introduction to Statistical Quality Control, 8th ed., chapter 8 (equations 8.19 and 8.20).
Cpk interval: Bissell (1990), How reliable is your capability index?, Applied Statistics 39(3).
Cpm: Chan, Cheng and Spiring (1988), Journal of Quality Technology 20(3).
Within sigma MR-bar / d2 or R-bar / d2, as in the control chart; d2 computed from its definition.
Categories sorted by w (count or summed weight), largest first.
Vital few
the first K categories, with K=min{k:Pk≥cutoff}
Pareto chart and vital few: Juran's Quality Handbook, 7th ed., and Montgomery, Introduction to Statistical Quality Control, 8th ed., section 5.4.
The vital few are the top categories up to and including the first one whose cumulative percent reaches the cutoff.
p parts, o operators, r repeats per part and operator. A negative estimate is set to 0. When the interaction is removed, MS_E and MS_OP are both replaced by the pooled (SS_OP + SS_E) / (df_OP + df_E).
Gage R&R
σGRR2=σrepeat2+σO2+σOP2,σtotal2=σGRR2+σP2
Reproducibility is the operator and interaction parts together.
k is the study variation multiplier, 6 by default.
Number of distinct categories
ndc=⌊1.41σGRRσP⌋
At least 1. AIAG asks for 5 or more.
AIAG, Measurement Systems Analysis Reference Manual, 4th ed. (2010), chapter III section B: ANOVA method, acceptance guidelines and ndc; reference data Table III-B 5 (10 parts, 3 appraisers, 3 trials).
Montgomery, Introduction to Statistical Quality Control, 8th ed., section 8.7 (gauge R&R with the two-factor random-effects ANOVA).
Minitab, Crossed Gage R&R Study, methods and formulas (interaction removed when its p-value is at least alpha, 0.05 by default).
X-bar and R chart constants: Montgomery, Introduction to Statistical Quality Control, 8th ed., Appendix VI.
Planned: attribute agreement
Used with ties or 50 or more values; t_j are the sizes of tied groups.
Rank-biserial correlation and the Hodges-Lehmann shift estimate.
Used when the variances differ.
omega squared uses the pooled MS_W, so it goes with the classic F test. When Welch's ANOVA is used, omega squared (Welch) comes from Welch's F instead; with the classic F the same formula gives the classic omega squared.
q is the studentized range quantile. Games-Howell uses each pair's own variances instead of MS_W.
z is the normal quantile for the confidence level. With the exact test the Clopper-Pearson interval is given instead.
Used when every expected count of the 2 x 2 table is at least 5; otherwise Fisher's exact test.
d = p1 - p2; l and u are each group's Wilson limits.
Cohen's h; for one sample p2 is p0.
d are the differences within pairs.
Welch-Satterthwaite degrees of freedom.
Cohen's d for one sample or paired differences, Hedges' g for two samples.
Used with ties, zeros or 50 or more pairs.
Matched-pairs rank-biserial correlation and the Hodges-Lehmann estimate.
h_ii is row i's leverage, the diagonal of the hat matrix X(X'X)^-1 X'.
r_i is the standardized residual, t_i the deleted (externally studentized) residual. Flags: |r_i| > 2, h_ii > 2p/n, D_i > 1 or |DFFITS| > 2 sqrt(p/n).
R_j^2 is from regressing predictor j on the other predictors.