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ProcessStat is a web statistics workshop where small business practitioners and students can begin routine data analysis and experiment planning. Select a goal and enter a number, and the results and graph will be displayed together. Expand detailed methods and settings when necessary. You can still use the existing Mellow Tools and ProcessLab.

My question is which function should I use?

Getting started

1. Select the question you want to know. Take a look at the screen with examples of packaging weight, work method, and learning time, or Blank data tableStart here. The examples are all fictitious data for practice purposes and will replace the current table when selected. revertYou can recover it with .
2. Name the column and enter a number. Paste tableaccepts a table separated by tabs, CSV, and spaces. You can use it without uploading a file. For CSV, use a decimal period.
3. Specify the columns to be used and conditions such as specifications and subgroups. Test method and confidence level More settingsYou can change it from . Current applied values ​​are displayed even when collapsed.
4. View results Or press Ctrl/⌘+Enter. If input changes, previous results cannot be saved until reanalyzed.
5. Save projectstores data and settings as JSON. Save reportsaves the results, graphs, and settings as independent HTML. Reports and prints also include detailed collapsed statistics. You can save as PDF from the Print menu.

One column is one group or variable. Same rows represent pairs of corresponding measurements or the same observation time point. Up to 5,000 rows, 20 columns, up to 8 regression explanatory variables. The table displays 30 rows at a time, but the entire row is used for analysis. Numbers can be entered up to the absolute value 10¹².

Frequently encountered terms

• Standard deviation: The extent to which the numbers are scattered around the mean. For example, if the weights are all gathered around 100g, it is small, and if they are significantly different from each other, it is large.
• Confidence interval: It expresses uncertainty in the estimated population mean or mean difference. A 95% confidence interval for the mean is not a range containing 95% of individual product weights.
• p-value: It is the probability that an outcome will be as extreme or more extreme than observed, assuming no differences. It is not the probability that the difference is real or that the result is correct.
• Paired comparison: Used when measuring the same object before and after. Put before and after values ​​for the same person/product in the same row. You should not pair two different groups by row number alone.
• Interaction: This is a phenomenon in which the effect of one condition varies depending on another condition. For example, the effect of temperature can vary depending on the heating time.
• Specifications and control limits: The standard is the allowable range determined by the customer/design, and the control limit is the standard for process variation calculated from data.

Analysis and calculation criteria

Calculation basis for additional analysis

Sample size · power

Power is calculated with the exact noncentral t distribution for 1-sample, paired and 2-sample t, the noncentral F for one-way ANOVA (Poisson-weighted incomplete beta series, λ = k·n·f²), the normal approximation without continuity correction for proportions (pooled variance for 2 samples), and the central F for the variance ratio. Choose two-sided or one-sided and solve for one of n, power or detectable difference. The required n is the smallest integer that reaches the target power; for 2 samples it is n per group (n₂ = r·n₁). For a 1-sample proportion with n ≤ 2000 the exact binomial power is also shown. The ANOVA effect is entered as the standard deviation of the group means (Cohen f = σ_m/σ) or as the largest mean difference (with the others in the middle). Take the standard deviation conservatively from previous data. References: R power.t.test and power.prop.test, the pwr package, Cohen (1988).

Acceptance sampling

Handles single attribute sampling plans (n, c). The acceptance probability is computed from the binomial distribution, from the hypergeometric distribution when the lot size N is given and N < 10n, or from the Poisson distribution when selected; the OC curve, AOQ/AOQL, ATI, the producer's risk at AQL and the consumer's risk at LTPD are shown. Plan design finds, for each c = 0–50, the smallest n (≤ 5000) with Pa ≤ β at LTPD, then picks the smallest of those n that also satisfies Pa ≥ 1 − α at AQL. AOQ and ATI assume that rejected lots are 100% inspected and defectives are replaced with good items; without N only the approximation AOQ = p·Pa is computed. The hypergeometric distribution takes the number of defectives in the lot as N·p rounded, so the OC curve is step-shaped. Reference: Montgomery, Introduction to Statistical Quality Control, Chapter 15.

Attribute agreement analysis

Evaluates a measurement system that rates items in categories, such as pass/fail or grades. The data consist of sample number, appraiser number, trial number and rating (numeric code) columns, plus an optional reference rating column, in a balanced design where every appraiser rates every sample the same number of times. The agreement rate is the proportion of samples for which all of the relevant ratings are identical (or all equal to the reference), and its confidence interval is the Clopper–Pearson exact binomial interval. Four kinds are computed: within appraiser (repeated ratings), each appraiser vs. reference, between appraisers, and all vs. reference. Fleiss κ = (P̄ − Pₑ)/(1 − Pₑ), where P̄ is the observed proportion of agreeing rating pairs per sample and Pₑ = Σpⱼ² is the chance agreement expected from the category proportions. The z test for κ uses the Fleiss (1971) standard error under H₀: κ = 0, and κ against the reference is computed with (rating, reference) as two raters for each sample × trial. When a reference exists, Cohen κ (rating × reference cross-table, standard error from Fleiss, Cohen, Everitt 1969) is also reported. For ordinal grades with 3 or more categories, the tie-corrected Kendall coefficient of concordance W (χ² = m(n−1)W, df = n−1) and Kendall τ-b against the reference are added. The criteria follow AIAG MSA 4th ed.: κ ≥ 0.75 good, 0.40–0.75 conditional, < 0.40 unacceptable; agreement rate ≥ 90% good, 80–90% conditional. The disagreement table shows, for each appraiser, the number of mixed cases where repeated ratings differ and the counts by reference→rating direction. References: AIAG MSA Reference Manual 4th ed. ch. III-C; Fleiss, J. L. (1971) Psychological Bulletin 76, 378–382; Cohen, J. (1960).

Equivalence test · TOST

A test that shows the difference between two means (or a mean and a target) lies within predefined equivalence limits. A large p-value from a classical t test is not evidence of “no difference”, so use this test to claim equality. The equivalence limit Δ (or asymmetric lower/upper limits) is set before the analysis from process or customer requirements, not from the data. Schuirmann’s (1987) two one-sided tests (TOST) reject H₀₁: difference ≤ lower limit and H₀₂: difference ≥ upper limit with separate one-sided t tests at significance level α, where t₁ = (d − lower)/SE and t₂ = (d − upper)/SE, and the TOST p-value is the larger of the two p-values. Rejecting both tests is equivalent to the (1 − 2α) confidence interval for the difference (90% when α = 0.05) lying entirely within the limits. The 2-sample case uses Welch (default) or Student pooled-variance standard errors and degrees of freedom, the paired case uses a 1-sample test on the pair differences, and the 1-sample case uses the difference from the target. It can also be computed from summary statistics alone (n, mean, standard deviation). Failing to show equivalence does not mean the difference is large; often the sample is small and the interval is wide. For reference, the per-group sample size needed to show equivalence with 80% power when the true difference is 0 is reported using the normal approximation n ≈ 2(z₁₋α + z₁₋β/₂)²σ²/Δ² (without the factor 2 for 1-sample and paired). References: Schuirmann, D. J. (1987) J. Pharmacokinetics and Biopharmaceutics 15, 657–680; Westlake (1976).

Proportions · contingency tables

For one proportion, the p-value comes from the exact binomial test. The two-sided p-value uses the minimum-likelihood method (R binom.test), summing all outcomes whose probability is less than or equal to that of the observed value; twice the one-sided sum is also shown. The confidence interval is the Wilson score interval by default, shown together with the Clopper–Pearson exact interval; the normal-approximation z is for reference. For two proportions, the pooled-proportion z test (no continuity correction) and Fisher's exact test are both computed; the interval for the difference in proportions uses the Newcombe score method (method 10), and the intervals for relative risk and odds ratio use the log-scale Wald method. If any cell is 0, 0.5 is added to every cell for the relative risk and odds ratio. Contingency tables use Pearson's chi-square test of independence to give χ², degrees of freedom, p-value, Cramér's V, expected counts and standardized residuals; for 2×2 tables the Yates-corrected and Fisher's exact p-values are added. A warning is shown if any cell has an expected count below 5; in that case Fisher's exact test or combining categories is recommended. The goodness-of-fit test is a chi-square test comparing observed counts with expected counts derived from the expected proportions (or uniform). References: Newcombe (1998) Stat Med 17:857–872; Agresti, Categorical Data Analysis; R binom.test / prop.test / fisher.test / chisq.test.

Nonparametric comparison

Compares values by rank without assuming a normal distribution. Mann–Whitney (Wilcoxon rank sum) computes U and W for two independent groups and shows an exact p-value by enumerating the full U distribution when there are no ties and each group has at most 20 values; otherwise it shows a normal-approximation p-value with tie and continuity corrections. The location shift is the Hodges–Lehmann estimate, the median of all pairwise differences; its interval comes from order statistics in the exact case (achieved confidence level shown) or from the Hollander–Wolfe large-sample index in the approximate case. Wilcoxon signed rank uses the positive rank sum V of the before/after differences in each row, excluding zeros; the p-value is exact when there are no ties or zeros and at most 25 pairs, otherwise a normal approximation. The estimate is the median of the Walsh averages (pseudo-median). The sign test is an exact binomial test that counts only the direction of the differences and can also compare one group with a reference median. Kruskal–Wallis approximates the tie-corrected H by chi-square and provides the effect size ε² = H/(N−1) and Dunn pairwise z tests (Holm correction). Mood’s median test builds a contingency table of counts above the overall median by group and compares it by chi-square. References: Hollander, Wolfe & Chicken, Nonparametric Statistical Methods; Dunn (1964) Technometrics 6:241–252; R wilcox.test / kruskal.test, scipy.stats.

Two-way ANOVA

The response Y is explained by two factors A and B and their interaction A×B with the fixed-effects model Y = μ + αᵢ + βⱼ + (αβ)ᵢⱼ + ε, fitted by least squares with effect coding (sum-to-zero constraints). Rows with the same (A, B) combination are treated as replicates; if each combination has only one observation, the additive model without the interaction is used. Sums of squares are reported as Type II by default: each main effect is the reduction after adjusting for the other main effect, and the interaction is the reduction after adjusting for both main effects. Type III is the value adjusted for the interaction as well under effect coding; for balanced data the two types coincide. F = mean square / error mean square, the p-value comes from the F distribution, and partial η² = SS(effect) / (SS(effect) + SS(error)). Post-hoc comparisons between the levels of each factor use Tukey HSD with the error mean square and degrees of freedom of the two-factor model, and the adjusted p-values and simultaneous confidence intervals come from the studentized range distribution (AS 190 numerical integration). When level sizes differ, the Tukey–Kramer method (pairwise harmonic-mean n) is applied. The assumptions are independence, normality and equal variance of the errors, checked with residual plots and the Brown–Forsythe test by combination. References: Montgomery, Design and Analysis of Experiments; Copenhaver & Holland (1988) J. Stat. Comput. Simul.; R anova/TukeyHSD, statsmodels anova_lm.

Measurement system analysis · Gage R&R

Estimates the variation of a measurement system from a fully crossed design in which o appraisers each measure p parts r times. The default method is a two-way random-effects ANOVA (Y = part + appraiser + part×appraiser), with variance components repeatability = MSE, part×appraiser = (MS_po − MSE)/r, appraiser = (MS_o − MS_po)/(p·r), part = (MS_p − MS_po)/(o·r), with negative values set to 0. If the interaction p-value is larger than the cutoff (default 0.25), the interaction is pooled into repeatability following the AIAG rule. The X̄-R method computes repeatability EV = R̄/d₂*, reproducibility AV = √((X̄_diff/d₂*)² − EV²/(p·r)), part variation PV = R_p/d₂*, where d₂* is taken from AIAG Appendix C (d₂ when there are more than 15 subgroups). Gage R&R = repeatability + reproducibility (appraiser + part×appraiser), and total variation = GRR + part variation. Study variation is 6σ, and %study variation = σ/σ_all, %tolerance = 6σ/tolerance, and %process = σ/σ_process. The number of distinct categories ndc = 1.41 × σ_Part/σ_GRR(rounded down, 1 if less than 1). The criteria are: %GRR below 10% good, 10–30% conditionally acceptable depending on use and cost, above 30% unacceptable, and ndc must be at least 5. Ranges beyond the control limits on the R chart indicate measurements that should be considered for re-measurement. Reference: AIAG, Measurement Systems Analysis Reference Manual, 4th ed. (2010), Chapter III.

Reliability and life data analysis

Uses failure times and a failure indicator (1 = failure, 0 = right-censored) to fit Weibull (2-parameter), lognormal, exponential and normal distributions by maximum likelihood (MLE). A censored unit contributes only the information that it had not failed by that time, so entering 1 for units that have not yet failed will underestimate life. Standard errors of the parameters come from the observed information matrix (second derivatives of the log-likelihood), and confidence intervals for positive parameters are computed on the log scale. Confidence intervals for percentile lives such as B10 life (the time by which 10% have failed), MTTF and R(t) use the delta method. Points on the probability plot use Johnson-adjusted ranks with Benard's median-rank approximation (i − 0.3)/(n + 0.4); the line is the MLE fit, not a rank regression. In the distribution selection table, a smaller AIC is better, and differences under 2 are hard to distinguish, so give priority to the distribution that matches the failure mechanism. The Anderson–Darling statistic is an approximation computed from failure times only, with no censoring adjustment or p-value. Kaplan–Meier survival probabilities are computed without a distribution assumption as S(t) = Π(1 − d/n), with confidence intervals from the Greenwood variance. If a group column is specified, the log-rank (Mantel–Cox) test (chi-square, df = number of groups − 1) and per-group distribution fits are shown as well. Reference: Meeker & Escobar, Statistical Methods for Reliability Data (1998); Abernethy, The New Weibull Handbook.

General regression · categorical factors and interactions

Continuous X are used as is; categorical X (codes 1, 2, 3 …) are entered with dummy (treatment) coding. The smallest code becomes the reference level, and the coefficient of each other level is its mean difference from the reference level. You can select squared terms X² and 2-factor interactions X_A×X_B(including categorical × continuous) can be selected. Coefficients are estimated by Householder QR least squares, and standard errors, t, p-values and confidence intervals use the t distribution with the residual degrees of freedom. The per-term ANOVA uses type II sums of squares: a term’s sum of squares is the reduction in residual sum of squares when that term is added to the model containing all other terms except the higher-order terms that contain it (squared and interaction terms); a categorical term has number of levels − 1 degrees of freedom. Stepwise selection (backward elimination) removes, one at a time, the term with the largest p-value above the threshold α, but keeps the main effects inside any remaining interaction or squared term by the hierarchy principle. Predicted R² comes from PRESS (sum of squared residual/(1 − leverage)), VIF is 1/(1−R²_j), and Cook's D is computed from the standardized residuals and leverage, flagging rows above 4/n as influential points. Rows with empty Y and filled X are treated as prediction rows and show both a confidence interval for the mean and a prediction interval for a single new observation. References: Montgomery, Peck & Vining, Introduction to Linear Regression Analysis; Belsley, Kuh & Welsch (1980).

Multiple response optimization · desirability function

For each response Y, the selected model (main effects, interactions, quadratic) is fitted in the same way as general regression. Each predicted value is converted to a Derringer–Suich desirability d (0–1): for maximize, 0 at or below the lower bound L, 1 at or above the upper bound T, and ((y−L)/(T−L)) in between^w; minimize is the reverse; target is 0 outside the lower/upper bounds, 1 at the target, and a power curve on each side; range is 1 inside and 0 outside. With weight w=1 the function is linear; with w>1 desirability is high only near the target. The overall desirability is the geometric mean with the importances r_i as exponents, D=(Π d_i^r_i)^1/Σr_i, so if even one response is 0, D is also 0. The optimal condition is found by a grid search within the min–max range of the data followed by coordinate-wise fine tuning; if it touches the boundary, you are notified that conditions outside the range may be better. The desirability map shows D when two conditions are varied and the rest are held at their optimal values. Reference: Derringer & Suich (1980), J. Quality Technology 12, 214–219; Montgomery, Design and Analysis of Experiments, 11.3.

Non-normal process capability · Box–Cox transformation and distribution fit

If the data are skewed and fail the normality test, a Box–Cox transformation (y = (x^λ − 1)/λ, or log when λ = 0) to make the data close to normal, then transforms the specification limits with the same λ and computes Cp and Cpk (Box & Cox 1964). λ is the value that maximizes the profile likelihood; it is usually rounded to the nearest 0.5, and if the confidence interval includes 1 the transformation is not really needed. Alternatively, a distribution such as Weibull or lognormal is fitted directly and Pp = (USL − LSL)/(X_99.865 − X_0.135), Ppk = min((USL − X₅₀)/(X_99.865 − X₅₀), (X₅₀ − LSL)/(X₅₀ − X_0.135)) (ISO 21747, AIAG SPC manual); the distribution with the smallest Anderson–Darling A² is chosen. Both methods are meaningful only when the process is stable (in control), and the expected PPM depends heavily on the tail of the fitted distribution, so interpret with care when the sample is small.

Tolerance interval

Where a confidence interval describes the location of the mean, a tolerance interval is a range for individual values: "with confidence γ, at least a proportion P of the population falls in this range". Under the normal assumption it is x̄ ± k·s, where the two-sided k uses the Howe (1969) approximation and the one-sided k is computed exactly from the noncentral t distribution (for n = 20, P = 99%, γ = 95%, k ≈ 3.62/3.30). If a distribution is hard to assume, use Wilks' nonparametric interval based on order statistics such as the minimum and maximum, but a large sample is needed to actually reach the requested confidence (e.g. 93 for 95/95, 473 for 99/95). If the tolerance interval falls inside the specification limits, that is statistical evidence that individual products meet the specification.

EWMA · CUSUM charts

Shewhart individuals charts are sensitive to large shifts but slow to detect small sustained shifts of 0.5–1.5σ, so the EWMA (z_i = λx_i + (1 − λ)z_i−1, typically λ = 0.1–0.3, L ≈ 2.7–3) and the cumulative sum CUSUM (C⁺/C⁻, k = 0.5, h = 4–5) are used (Montgomery, Introduction to Statistical Quality Control, Chapter 9). The EWMA limits are narrow at first and converge to μ ± Lσ√(λ/(2 − λ)); for CUSUM, the point where C⁺ or C⁻ exceeds H = hσ is the out-of-control signal, and FIR (starting at h/2) speeds up initial detection. σ is estimated as the average moving range of the reference period divided by 1.128, so the reference period must be in a stable state; if there is a target value, set the center line to the target. These charts are good at detecting small shifts, but a Shewhart chart is faster for large sudden changes, so it is best to look at both.

VIF · Durbin–Watson · prediction intervals · influential points

VIF = 1/(1 − R²_j) expresses how much predictor j is explained by the other predictors; values above 5–10 indicate multicollinearity that makes coefficient estimates unstable, so consider removing or combining variables (Belsley, Kuh & Welsch 1980). Durbin–Watson is an indicator of first-order autocorrelation in the residuals: around 2 is normal, and below 1.5 suggests positive autocorrelation (common in time-ordered data). Prediction at a new x has a confidence interval for the mean response (ŷ ± t·s√h₀) and a prediction interval for an individual observation (ŷ ± t·s√(1 + h₀)); the latter is always wider and grows sharply when extrapolating outside the data range. In the influence table, leverage (> 2p/n), standardized residuals (|r| > 2) and Cook's D (> 4/n) are criteria for flagging observations that drive the results, not criteria for automatic removal.

common assumption

Independence of observations, sampling methods, and measurement reliability cannot be automatically guaranteed by data alone. The user must check the normal error assumption of t-test/classical ANOVA, linearity/independence/homoscedasticity of regression, and stability/normality of process capability. The results were designed to determine not only the p value, but also the size of the difference, confidence interval, and process tolerance.

data storage

Measurement data is not transmitted from this page to the server. Calculations are performed within the browser, and the work is automatically saved to the current browser's local storage. If storage is blocked or capacity is insufficient, you will be notified on the screen. Deleting browser data may also result in the loss of automatic saves, so save long-term work as JSON. The report is a frozen result at the time of saving, and restoring measurements uses the JSON job file.

Verification and Reference Materials

Tests and regressions are contrasted with baseline data and analytically computable examples below. Input missing processing, constant/linear dependency, subgroup/defective product count verification are also automatically checked. This does not mean that all features have certified equivalence with commercial statistical programs.

• NIST — 2-sample t-test and verification example
• NIST StRD — Pontius regression baseline data
• NIST — Process capability index
• NIST — Variables control chart

We used the existing Mellow Tools statistical, graphing, and control chart calculation code as a copy exclusively for this service, and added common data linking, summary value testing, Holm correction, multiple regression, and reporting functions. DOE Experiment plan generatorlets you compare fractional factorial, PB, orthogonal array, CCD, Box–Behnken, and full factorial designs and create a run matrix. DOE response analysis, Gage R&R, and reliability analysis are not included in the current version.

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