{"id":100108,"date":"2026-02-15T20:16:59","date_gmt":"2026-02-15T20:16:59","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100108"},"modified":"2026-07-12T07:57:41","modified_gmt":"2026-07-12T07:57:41","slug":"gage-rr-calculator","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/ar\/calculators\/gage-rr-calculator\/","title":{"rendered":"ANOVA \u0644\u0642\u064a\u0627\u0633 \u062c\u0648\u062f\u0629 \u0627\u0644\u0642\u064a\u0627\u0633 \u0645\u0639 \u062a\u0623\u062b\u064a\u0631\u0627\u062a \u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0645\u062a\u0642\u0627\u0637\u0639\u0629"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Crossed Random-Effects Gage R&R ANOVA\",\"description\":\"Estimate repeatability,operator,operator-by-part,part and combined Gage R&R variance components from a balanced crossed replicated study using disclosed method-of-moments ANOVA.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/gage-rr-calculator\/\",\"applicationCategory\":\"Engineering Calculator\",\"operatingSystem\":\"Any (Web Browser)\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\",\"priceCurrency\":\"EUR\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"dateModified\":\"2026-07-12\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true}<\/script>\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Does this calculator certify AIAG MSA acceptance?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"No. It reports transparent crossed random-effects ANOVA estimates and makes no automatic acceptance decision. AIAG MSA criteria require the licensed manual and the applicable customer or quality-system requirements.\"}},{\"@type\":\"Question\",\"name\":\"What design is required?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"A balanced crossed study: every selected operator measures every selected part the same number of repeated trials. The model assumes random representative part and operator levels,independent approximately normal residuals and constant repeatability variance.\"}}]}<\/script>\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/vibromera.eu\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Calculators\",\"item\":\"https:\/\/vibromera.eu\/calculators\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Crossed random-effects Gage R&R ANOVA\",\"item\":\"https:\/\/vibromera.eu\/calculators\/gage-rr-calculator\/\"}]}<\/script>\r\n<link rel=\"preconnect\" 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20px;font-size:13px;font-weight:700;color:#fff;background:var(--vc-accent);border-radius:var(--vc-radius);text-decoration:none;white-space:nowrap;transition:background .15s}.vc-promo-link:hover{background:var(--vc-accent-hover)}\r\n.vc-footer{text-align:center;padding:32px 16px;font-size:13px;color:var(--vc-ink-muted)}.vc-footer a{color:var(--vc-accent);text-decoration:none}.vc-footer a:hover{text-decoration:underline}.vc-footer-links{margin-top:8px;font-size:12px}.vc-footer-links a{margin:0 8px}\r\n@media print{.vc-section-body{max-height:none!important}.vc-faq-a{max-height:none!important}.vc-results{max-height:none!important;padding:24px!important}.vc-copy-btn,.vc-section-chevron,.vc-faq-chevron,.vc-presets,.vc-promo{display:none!important}}\r\n\r\n.vc-form-grid-anova{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:18px}.vc-table-wrap{overflow:auto;margin-top:18px}.vc-grid-table{border-collapse:collapse;min-width:760px;width:100%}.vc-grid-table th,.vc-grid-table td,.vc-anova-table th,.vc-anova-table td{border:1px solid var(--vc-border);padding:8px;text-align:right}.vc-grid-table th:first-child,.vc-grid-table td:first-child,.vc-anova-table th:first-child,.vc-anova-table td:first-child{text-align:left}.vc-grid-input{width:100px;min-width:82px;padding:7px;border:1px solid var(--vc-border);border-radius:4px;font:inherit}.vc-anova-table{border-collapse:collapse;width:100%;min-width:720px}.vc-result-note{margin:0;color:var(--vc-ink-secondary);font-size:13px}.vc-method-note{font-size:13px;color:var(--vc-ink-secondary)}@media(max-width:600px){.vc-form-grid-anova{grid-template-columns:1fr}}\r\n<\/style>\r\n<div class=\"vc-calculator\" id=\"vc-grr-anova\"><header class=\"vc-header\"><p class=\"vc-header-eyebrow\">Balanced crossed replicated study<\/p><h1 class=\"vc-header-title\">Crossed Random-Effects Gage R&amp;R ANOVA<\/h1><p class=\"vc-header-subtitle\">Estimate variance components for part,operator,operator\u00d7part interaction and repeatability with a fully disclosed method-of-moments two-way ANOVA. No automatic acceptance decision is made.<\/p><div class=\"vc-badges\"><span class=\"vc-badge\">Two-way crossed ANOVA<\/span><span class=\"vc-badge\">Interaction retained<\/span><span class=\"vc-badge\">No AIAG conformance claim<\/span><\/div><\/header>\r\n<div class=\"vc-card\"><form class=\"vc-form\" id=\"vc-form\" autocomplete=\"off\"><div class=\"vc-form-grid-anova\">\r\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-parts\">Parts (random representative levels)<\/label><select class=\"vc-select\" id=\"vc-parts\"><option value=\"\">Select\u2026<\/option><option>3<\/option><option>4<\/option><option>5<\/option><option>6<\/option><option>7<\/option><option>8<\/option><option>9<\/option><option>10<\/option><\/select><\/div>\r\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-operators\">Operators (random representative levels)<\/label><select class=\"vc-select\" id=\"vc-operators\"><option value=\"\">Select\u2026<\/option><option>2<\/option><option>3<\/option><option>4<\/option><\/select><\/div>\r\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-trials\">Replicated trials per cell<\/label><select class=\"vc-select\" id=\"vc-trials\"><option value=\"\">Select\u2026<\/option><option>2<\/option><option>3<\/option><option>4<\/option><\/select><\/div>\r\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-unit\">Measurement unit<\/label><input class=\"vc-input\" id=\"vc-unit\" type=\"text\" placeholder=\"e.g. mm\"><\/div>\r\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-source\">Study provenance<\/label><input class=\"vc-input\" id=\"vc-source\" type=\"text\" placeholder=\"e.g. randomized study,report rev 2\"><\/div>\r\n<\/div><p id=\"vc-error\" role=\"alert\" style=\"margin:.75rem 0 0;color:#b42318\"><\/p><div class=\"vc-warning-box\"><p style=\"margin:0\"><strong>Required design and assumptions:<\/strong> every operator measures every part for the same number of trials; order should be randomized and measurements independent. Parts and operators are treated as random representative levels. Residuals are assumed approximately normal with constant variance. The calculation does not assess bias,linearity,stability,resolution,time effects or calibration uncertainty.<\/p><\/div><div class=\"vc-table-wrap\" id=\"vc-grid\"><\/div><\/form>\r\n<div class=\"vc-results\" id=\"vc-results\" aria-live=\"polite\"><div class=\"vc-results-head\"><h2 class=\"vc-results-title\">Variance-component estimates<\/h2><button type=\"button\" class=\"vc-copy-btn\" id=\"vc-copy-btn\">Copy<\/button><\/div><div class=\"vc-result-grid\">\r\n<div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Repeatability \u03c3<sub>e<\/sub><\/div><div class=\"vc-rcard-value\" id=\"vc-r-e\">\u2014<\/div><\/div>\r\n<div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Operator \u03c3<sub>O<\/sub><\/div><div class=\"vc-rcard-value\" id=\"vc-r-o\">\u2014<\/div><\/div>\r\n<div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Operator\u00d7part \u03c3<sub>PO<\/sub><\/div><div class=\"vc-rcard-value\" id=\"vc-r-po\">\u2014<\/div><\/div>\r\n<div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Reproducibility \u221a(\u03c3\u00b2<sub>O<\/sub>+\u03c3\u00b2<sub>PO<\/sub>)<\/div><div class=\"vc-rcard-value\" id=\"vc-r-repro\">\u2014<\/div><\/div>\r\n<div class=\"vc-rcard vc-rcard-primary\"><div class=\"vc-rcard-label\">Combined Gage R&amp;R \u03c3<sub>GRR<\/sub><\/div><div class=\"vc-rcard-value\" id=\"vc-r-grr\">\u2014<\/div><\/div>\r\n<div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Part-to-part \u03c3<sub>P<\/sub><\/div><div class=\"vc-rcard-value\" id=\"vc-r-p\">\u2014<\/div><\/div>\r\n<div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Total study \u03c3<sub>total<\/sub><\/div><div class=\"vc-rcard-value\" id=\"vc-r-total\">\u2014<\/div><\/div>\r\n<div class=\"vc-rcard\"><div class=\"vc-rcard-label\">GRR \/ total study variation<\/div><div class=\"vc-rcard-value\" id=\"vc-r-pct\">\u2014<\/div><\/div>\r\n<div class=\"vc-rcard\"><div class=\"vc-rcard-label\">GRR variance contribution<\/div><div class=\"vc-rcard-value\" id=\"vc-r-contrib\">\u2014<\/div><\/div>\r\n<\/div><p class=\"vc-result-note\" id=\"vc-r-note\"><\/p><div class=\"vc-table-wrap\"><table class=\"vc-anova-table\"><thead><tr><th>Source<\/th><th>SS<\/th><th>df<\/th><th>MS<\/th><th>Variance estimate<\/th><\/tr><\/thead><tbody><tr><td>Part<\/td><td id=\"vc-a-pss\">\u2014<\/td><td id=\"vc-a-pdf\">\u2014<\/td><td id=\"vc-a-pms\">\u2014<\/td><td id=\"vc-a-pvar\">\u2014<\/td><\/tr><tr><td>Operator<\/td><td id=\"vc-a-oss\">\u2014<\/td><td id=\"vc-a-odf\">\u2014<\/td><td id=\"vc-a-oms\">\u2014<\/td><td id=\"vc-a-ovar\">\u2014<\/td><\/tr><tr><td>Part\u00d7operator<\/td><td id=\"vc-a-iss\">\u2014<\/td><td id=\"vc-a-idf\">\u2014<\/td><td id=\"vc-a-ims\">\u2014<\/td><td id=\"vc-a-ivar\">\u2014<\/td><\/tr><tr><td>Repeatability<\/td><td id=\"vc-a-ess\">\u2014<\/td><td id=\"vc-a-edf\">\u2014<\/td><td id=\"vc-a-ems\">\u2014<\/td><td id=\"vc-a-evar\">\u2014<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div>\r\n<div class=\"vc-section vc-open\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\ud83d\udcd8<\/span><span class=\"vc-section-title\">Model,formulas and source boundary<\/span><\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner vc-theory\">\r\n<h3>Balanced crossed random-effects model<\/h3><div class=\"vc-formula-box\">y<sub>ijk<\/sub>=\u03bc+P<sub>i<\/sub>+O<sub>j<\/sub>+(PO)<sub>ij<\/sub>+e<sub>ijk<\/sub><br>\u03c3\u00b2<sub>e<\/sub>=MS<sub>e<\/sub><br>\u03c3\u00b2<sub>PO<\/sub>=max[(MS<sub>PO<\/sub>\u2212MS<sub>e<\/sub>)\/r,0]<br>\u03c3\u00b2<sub>O<\/sub>=max[(MS<sub>O<\/sub>\u2212MS<sub>PO<\/sub>)\/(pr),0]<br>\u03c3\u00b2<sub>P<\/sub>=max[(MS<sub>P<\/sub>\u2212MS<sub>PO<\/sub>)\/(or),0]<br>\u03c3\u00b2<sub>GRR<\/sub>=\u03c3\u00b2<sub>e<\/sub>+\u03c3\u00b2<sub>O<\/sub>+\u03c3\u00b2<sub>PO<\/sub><\/div><p>p,o and r are the numbers of parts,operators and trials. Components are in squared measurement units; displayed \u03c3 values are their square roots. Negative method-of-moments component estimates are reported in the note and truncated to zero for the combined nonnegative variance display.<\/p>\r\n<h3>Open statistical sources<\/h3><p><a href=\"https:\/\/www.itl.nist.gov\/div898\/handbook\/ppc\/section2\/ppc232.htm\" target=\"_blank\" rel=\"noopener\">NIST\/SEMATECH e-Handbook \u00a73.2.3.2<\/a> gives the balanced crossed two-way model and sums of squares for part,operator,interaction and residual. <a href=\"https:\/\/www.itl.nist.gov\/div898\/handbook\/prc\/section4\/prc44.htm\" target=\"_blank\" rel=\"noopener\">NIST\/SEMATECH e-Handbook \u00a77.4.4<\/a> explains random effects,expected mean squares and method-of-moments variance components.<\/p>\r\n<h3>Standards and acceptance boundary<\/h3><p><a href=\"https:\/\/www.iso.org\/standard\/69418.html\" target=\"_blank\" rel=\"noopener\">ISO5725-1:2023,edition2<\/a> provides general principles and definitions for trueness and precision and explicitly says calculation methods are in other parts; this calculator does not claim to implement ISO5725. <a href=\"https:\/\/www.aiag.org\/training-and-resources\/manuals\/details\/MSA-4\" target=\"_blank\" rel=\"noopener\">AIAG MSA 4th edition<\/a> is a separately licensed manual. Because its full licensed method was not available for this audit,the former 10\/30% labels,ndc coefficient\/threshold and minimum-study claims are not implemented or presented as verified.<\/p>\r\n<h3>Interpretation<\/h3><p>GRR\/total study variation and variance contribution are descriptive for the sampled part population; changing the selected parts can change them. They are not percentages of tolerance and no pass\/warn\/fail class is assigned. Confirm study design,assumptions,customer requirements and the governing licensed procedure before making a measurement-system decision.<\/p>\r\n<\/div><\/div><\/div><footer class=\"vc-footer\"><p>\u00a92024\u20132026 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a><\/p><p>Transparent method-of-moments crossed ANOVA,not AIAG or ISO certification. Scientific review:July2026.<\/p><\/footer><\/div>\r\n<script>(function(){'use strict';function $(id){return document.getElementById(id)}var resultIds=['vc-r-e','vc-r-o','vc-r-po','vc-r-repro','vc-r-grr','vc-r-p','vc-r-total','vc-r-pct','vc-r-contrib'];function num(raw){var s=String(raw).trim().replace(',','.');return \/^[+-]?(?:\\d+(?:\\.\\d*)?|\\.\\d+)(?:[eE][+-]?\\d+)?$\/.test(s)?Number(s):NaN}function fmt(x){return Number(x.toPrecision(12)).toString()}function clear(){resultIds.forEach(function(id){$(id).textContent='\u2014'});document.querySelectorAll('.vc-anova-table td[id]').forEach(function(x){x.textContent='\u2014'});$('vc-r-note').textContent='';$('vc-results').classList.remove('vc-visible')}function dims(){return {p:Number($('vc-parts').value),o:Number($('vc-operators').value),r:Number($('vc-trials').value)}}function setUrl(){var u=new 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