{"id":100151,"date":"2026-02-15T20:21:13","date_gmt":"2026-02-15T20:21:13","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100151"},"modified":"2026-07-12T20:08:11","modified_gmt":"2026-07-12T20:08:11","slug":"measurement-uncertainty","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/az\/calculators\/measurement-uncertainty\/","title":{"rendered":"Uncorrelated Standard-Uncertainty Combiner"},"content":{"rendered":"\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Uncorrelated Standard-Uncertainty Combiner\",\"description\":\"Combine documented standard-uncertainty contributions that are already expressed in the output unit and verified uncorrelated,then apply a user-justified coverage factor.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/measurement-uncertainty\/\",\"applicationCategory\":\"Engineering Reference\",\"operatingSystem\":\"Any\",\"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>\n<style>:root{--uc:#1c6480;--ucl:#e9f4f8;--ink:#181817;--sec:#595750;--mut:#817e76;--b:#d8d4cc;--bl:#e9e5dd;--alt:#f8f6f1;--red:#b42318;--yel:#fff8df;--amb:#8a6500;--font:'DM Sans',-apple-system,BlinkMacSystemFont,'Segoe UI',sans-serif}.uc-wrap{font-family:var(--font);color:var(--ink);max-width:1080px;margin:0 auto;border:1px solid var(--bl);border-radius:16px;overflow:hidden;background:#fff}.uc-wrap *,.uc-wrap *:before,.uc-wrap *:after{box-sizing:border-box}.uc-hero{padding:34px 38px 26px;background:linear-gradient(135deg,#f7fbfc,#eaf4f7)}.uc-kicker{text-transform:uppercase;letter-spacing:.1em;font-size:12px;font-weight:800;color:var(--uc)}.uc-hero 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h3{font-size:12px;text-transform:uppercase;letter-spacing:.06em;color:var(--mut);margin:0 0 8px}.uc-val{font-size:20px;font-weight:900;overflow-wrap:anywhere}.uc-unit{font-size:12px;color:var(--mut);margin-top:4px}.uc-alert{border-left:4px solid var(--amb);background:var(--yel);padding:14px 16px;margin-top:18px;font-size:14px;line-height:1.55}.uc-section{border-top:1px solid var(--bl);padding:28px 38px}.uc-section h2{font-size:24px;margin:0 0 10px}.uc-section h3{font-size:18px;margin:22px 0 8px}.uc-section p,.uc-section li{font-size:14px;line-height:1.65;color:var(--sec)}.uc-table-wrap{overflow-x:auto;border:1px solid var(--b);border-radius:10px}.uc-table{width:100%;border-collapse:collapse;min-width:780px}.uc-table th,.uc-table td{padding:12px 13px;text-align:left;vertical-align:top;border-bottom:1px solid var(--bl);font-size:13px;line-height:1.45}.uc-table th{background:var(--alt);font-weight:900}.uc-table tr:last-child td{border-bottom:0}.uc-code{font-family:ui-monospace,SFMono-Regular,Consolas,monospace}.uc-footer{padding:20px 38px;background:var(--alt);font-size:12px;color:var(--mut)}.uc-footer a,.uc-section a{color:var(--uc)}@media(max-width:700px){.uc-hero,.uc-main,.uc-section{padding-left:20px;padding-right:20px}.uc-grid,.uc-rgrid,.uc-row{grid-template-columns:1fr}.uc-wide{grid-column:auto}.uc-panel{padding:18px}.uc-table{min-width:700px}}<\/style>\n<div class=\"uc-wrap\"><header class=\"uc-hero\"><div class=\"uc-kicker\">Restricted first-order worksheet \u00b7 not a complete GUM budget<\/div><h1>Uncorrelated Standard-Uncertainty Combiner<\/h1><p class=\"uc-lead\">Combine standard-uncertainty contributions that have already been propagated through the measurement model,expressed in the output unit and verified uncorrelated. Apply only a documented coverage factor selected outside this worksheet.<\/p><div class=\"uc-badges\"><span class=\"uc-badge\">JCGM100:2008 restricted case<\/span><span class=\"uc-badge\">No automatic k=2\/95% claim<\/span><span class=\"uc-badge\">No ISO\/IEC17025 compliance claim<\/span><span class=\"uc-badge\">Signed measurand estimate supported<\/span><\/div><\/header>\n<main class=\"uc-main\"><section class=\"uc-panel\"><h2>Document the restricted calculation<\/h2><p class=\"uc-note\">This worksheet accepts final standard contributions <span class=\"uc-code\">q<sub>i<\/sub>=|c<sub>i<\/sub>|u(x<sub>i<\/sub>)<\/span> in the output unit. Determine the measurement model,sensitivity coefficients,standard uncertainties,correlations,degrees of freedom and coverage-factor policy before entering values.<\/p><form id=\"uc-form\" novalidate><div class=\"uc-grid\"><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-estimate\">Measurand estimate y<\/label><input class=\"uc-input\" id=\"uc-estimate\" inputmode=\"decimal\" autocomplete=\"off\"><span class=\"uc-help\">A signed finite value is allowed.<\/span><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-unit\">Output unit<\/label><input class=\"uc-input\" id=\"uc-unit\" maxlength=\"40\" autocomplete=\"off\"><span class=\"uc-help\">All entered contributions must already use this unit.<\/span><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-k\">Documented coverage factor k<\/label><input class=\"uc-input\" id=\"uc-k\" inputmode=\"decimal\" autocomplete=\"off\"><span class=\"uc-help\">No coverage probability is inferred from k alone.<\/span><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-relation\">Contribution relationship<\/label><select class=\"uc-select\" id=\"uc-relation\"><option value=\"\">Select\u2026<\/option><option value=\"uncorrelated\">Verified uncorrelated for this measurement model<\/option><\/select><\/div><div class=\"uc-field uc-wide\"><label class=\"uc-label\" for=\"uc-model\">Measurement model and contribution record<\/label><textarea class=\"uc-textarea\" id=\"uc-model\" maxlength=\"1800\" placeholder=\"Measurand definition;measurement function;input estimates and units;sensitivity coefficients;Type A\/B evaluation and distribution;degrees of freedom;correlation assessment;corrections;source documents\"><\/textarea><\/div><div class=\"uc-field uc-wide\"><label class=\"uc-label\" for=\"uc-k-source\">Coverage-factor justification and reporting policy<\/label><textarea class=\"uc-textarea\" id=\"uc-k-source\" maxlength=\"1200\" placeholder=\"Required coverage;effective degrees of freedom\/distribution or approved policy;source and revision;why the selected k applies\"><\/textarea><\/div><\/div>\n<div class=\"uc-components\"><h3>Standard contributions in the output unit<\/h3><p class=\"uc-note\">Leave both fields blank for an unused row. A named zero contribution may be recorded,but at least one contribution must be positive.<\/p>\n<div class=\"uc-row\"><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-name-1\">Contribution1 name\/source<\/label><input class=\"uc-input uc-name\" id=\"uc-name-1\" maxlength=\"160\"><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-q-1\">q<sub>1<\/sub> standard contribution<\/label><input class=\"uc-input uc-q\" id=\"uc-q-1\" inputmode=\"decimal\"><\/div><\/div>\n<div class=\"uc-row\"><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-name-2\">Contribution2 name\/source<\/label><input class=\"uc-input uc-name\" id=\"uc-name-2\" maxlength=\"160\"><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-q-2\">q<sub>2<\/sub> standard contribution<\/label><input class=\"uc-input uc-q\" id=\"uc-q-2\" inputmode=\"decimal\"><\/div><\/div>\n<div class=\"uc-row\"><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-name-3\">Contribution3 name\/source<\/label><input class=\"uc-input uc-name\" id=\"uc-name-3\" maxlength=\"160\"><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-q-3\">q<sub>3<\/sub> standard contribution<\/label><input class=\"uc-input uc-q\" id=\"uc-q-3\" inputmode=\"decimal\"><\/div><\/div>\n<div class=\"uc-row\"><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-name-4\">Contribution4 name\/source<\/label><input class=\"uc-input uc-name\" id=\"uc-name-4\" maxlength=\"160\"><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-q-4\">q<sub>4<\/sub> standard contribution<\/label><input class=\"uc-input uc-q\" id=\"uc-q-4\" inputmode=\"decimal\"><\/div><\/div>\n<div class=\"uc-row\"><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-name-5\">Contribution5 name\/source<\/label><input class=\"uc-input uc-name\" id=\"uc-name-5\" maxlength=\"160\"><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-q-5\">q<sub>5<\/sub> standard contribution<\/label><input class=\"uc-input uc-q\" id=\"uc-q-5\" inputmode=\"decimal\"><\/div><\/div>\n<div class=\"uc-row\"><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-name-6\">Contribution6 name\/source<\/label><input class=\"uc-input uc-name\" id=\"uc-name-6\" maxlength=\"160\"><\/div><div class=\"uc-field\"><label class=\"uc-label\" for=\"uc-q-6\">q<sub>6<\/sub> standard contribution<\/label><input class=\"uc-input uc-q\" id=\"uc-q-6\" inputmode=\"decimal\"><\/div><\/div><\/div><p class=\"uc-error\" id=\"uc-error\" role=\"alert\"><\/p><\/form>\n<section class=\"uc-results\" id=\"uc-results\" hidden data-json=\"\"><div class=\"uc-rhead\"><div class=\"uc-rtitle\">Restricted uncorrelated combination<\/div><div class=\"uc-rsub\" id=\"uc-summary\">\u2014<\/div><\/div><div class=\"uc-rgrid\"><article class=\"uc-card\"><h3>Reported expression<\/h3><div class=\"uc-val\" id=\"uc-r-expression\">\u2014<\/div><div class=\"uc-unit\">y\u00b1U;coverage interpretation comes from the documented policy<\/div><\/article><article class=\"uc-card\"><h3>Combined standard uncertainty u<sub>c<\/sub><\/h3><div class=\"uc-val\" id=\"uc-r-combined\">\u2014<\/div><div class=\"uc-unit\">positive square root of summed contribution variances<\/div><\/article><article class=\"uc-card\"><h3>Expanded uncertainty U<\/h3><div class=\"uc-val\" id=\"uc-r-expanded\">\u2014<\/div><div class=\"uc-unit\">k\u00d7u<sub>c<\/sub><\/div><\/article><article class=\"uc-card\"><h3>Coverage factor k<\/h3><div class=\"uc-val\" id=\"uc-r-k\">\u2014<\/div><div class=\"uc-unit\">user supplied and justified<\/div><\/article><article class=\"uc-card\"><h3>Relative expanded uncertainty<\/h3><div class=\"uc-val\" id=\"uc-r-relative\">\u2014<\/div><div class=\"uc-unit\">not applicable when y=0<\/div><\/article><article class=\"uc-card\"><h3>Recorded contributions<\/h3><div class=\"uc-val\" id=\"uc-r-count\">\u2014<\/div><div class=\"uc-unit\">all in the output unit and uncorrelated<\/div><\/article><\/div><\/section>\n<div class=\"uc-alert\"><strong>Applicability boundary:<\/strong> do not use this RSS worksheet when relevant contributions are correlated,when sensitivity propagation has not been performed,when the model is materially nonlinear,when distributions require propagation rather than a first-order approximation,or when k has not been justified. Use the full GUM\/JCGM method,Monte Carlo or an approved field-specific procedure.<\/div><\/section><\/main>\n<section class=\"uc-section\"><h2>Equations actually implemented<\/h2><div class=\"uc-table-wrap\"><table class=\"uc-table\"><thead><tr><th>Quantity<\/th><th>Equation<\/th><th>Scope<\/th><\/tr><\/thead><tbody><tr><td>Entered contribution<\/td><td class=\"uc-code\">q<sub>i<\/sub>=|c<sub>i<\/sub>|u(x<sub>i<\/sub>)<\/td><td>Must already be a standard uncertainty contribution in the output unit. The worksheet does not derive c<sub>i<\/sub> or u(x<sub>i<\/sub>).<\/td><\/tr><tr><td>Combined standard uncertainty<\/td><td class=\"uc-code\">u<sub>c<\/sub>=\u221a\u03a3q<sub>i<\/sub>\u00b2<\/td><td>Restricted to contributions verified uncorrelated in the stated first-order model.<\/td><\/tr><tr><td>Expanded uncertainty<\/td><td class=\"uc-code\">U=k u<sub>c<\/sub><\/td><td>k is supplied with an external justification;the worksheet does not map k to a coverage probability.<\/td><\/tr><tr><td>Relative expanded uncertainty<\/td><td class=\"uc-code\">100U\/|y|<\/td><td>Descriptive ratio only;undefined at y=0 and not a conformity rule.<\/td><\/tr><\/tbody><\/table><\/div><\/section>\n<section class=\"uc-section\"><h2>Standardizing common inputs before entry<\/h2><div class=\"uc-table-wrap\"><table class=\"uc-table\"><thead><tr><th>Available information<\/th><th>Standard uncertainty before sensitivity propagation<\/th><th>Boundary<\/th><\/tr><\/thead><tbody><tr><td>Mean of n independent repeat observations<\/td><td class=\"uc-code\">u(x\u0304)=s\/\u221an<\/td><td>Only when the estimator\/model is the mean and the repeatability design is applicable;record degrees of freedom.<\/td><\/tr><tr><td>Certificate expanded uncertainty U<sub>cert<\/sub> with stated k<sub>cert<\/sub><\/td><td class=\"uc-code\">u=U<sub>cert<\/sub>\/k<sub>cert<\/sub><\/td><td>Use the certificate&#8217;s exact statement,unit and coverage basis.<\/td><\/tr><tr><td>Symmetric rectangular bounds\u00b1a<\/td><td class=\"uc-code\">u=a\/\u221a3<\/td><td>If the entered value is the full width w=2a,use u=w\/\u221a12.<\/td><\/tr><tr><td>Symmetric triangular bounds\u00b1a<\/td><td class=\"uc-code\">u=a\/\u221a6<\/td><td>Use only when that distribution is justified.<\/td><\/tr><tr><td>Digital resolution increment \u03b4<\/td><td class=\"uc-code\">u=\u03b4\/\u221a12<\/td><td>JCGM100:2008 F.2.2.1 for equal probability over one indication interval;check for double counting with observed repeatability.<\/td><\/tr><\/tbody><\/table><\/div><\/section>\n<section class=\"uc-section\"><h2>Sources and standards boundary<\/h2><p><a href=\"https:\/\/www.bipm.org\/documents\/20126\/2071204\/JCGM_100_2008_E.pdf\/cb0ef43f-baa5-11cf-3f85-4dcd86f77bd6\" target=\"_blank\" rel=\"noopener\">JCGM100:2008(E),GUM1995 with minor corrections<\/a> distinguishes TypeA and TypeB evaluation,uses sensitivity coefficients in the first-order law of propagation,treats uncorrelated and correlated inputs separately,and defines <span class=\"uc-code\">U=k u<sub>c<\/sub><\/span>. Its AnnexG shows that k=2 corresponds to95.45% only for the stated normal-distribution case;finite effective degrees of freedom and other distributions change the relation.<\/p><p><a href=\"https:\/\/jcgm.bipm.org\/vim\/en\/2.33.html\" target=\"_blank\" rel=\"noopener\">VIM3 item2.33<\/a> says an uncertainty budget should include the measurement model,input estimates and uncertainties,covariances,probability distributions,degrees of freedom,evaluation type and coverage factor. That is why this page is labelled a restricted combiner rather than a complete uncertainty-budget calculator.<\/p><p><a href=\"https:\/\/www.iso.org\/standard\/66912.html\" target=\"_blank\" rel=\"noopener\">ISO\/IEC17025:2017<\/a> is current Edition3,published2017-11 and confirmed2023. The public ISO card establishes status and scope,but not the exact uncertainty-evaluation clauses;those requirements remain <strong>NEEDS_LICENSED_SOURCE<\/strong>. A numerical result from this page does not establish laboratory competence,method validation,traceability or ISO\/IEC17025 conformity.<\/p><\/section>\n<section class=\"uc-section\"><h2>What was corrected<\/h2><p>The former page automatically described its result as GUM\/ISO17025-compliant despite omitting the measurement model,sensitivity coefficients,correlations,degrees of freedom and coverage-factor justification. It hard-coded k=2 as \u201c95% confidence\u201d and k=3 as99.7%,which is only valid under specific distribution and degrees-of-freedom conditions. The replacement requires a documented k and makes no automatic probability statement.<\/p><p>The previous \u201cresolution\u201d preset divided the displayed increment by\u221a3;JCGM100:2008 F.2.2.1 gives \u03b4\/\u221a12 for a digital indication interval \u03b4,so the preset could double that component. All presets and pre-filled results were removed. Generic \u201cvalue\/divisor\u201d fields were replaced with final standard contributions in the output unit,strict parsing and source\/model records. User-entered units are now written with <span class=\"uc-code\">textContent<\/span>,not injected through <span class=\"uc-code\">innerHTML<\/span>.<\/p><\/section>\n<footer class=\"uc-footer\">\u00a92024\u20132026 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a> \u00b7 Scientific review July2026<\/footer><\/div>\n<script>(function(){'use strict';function $(id){return document.getElementById(id)}function num(s,signed){s=String(s).trim();if(!s||s.includes('.')&&s.includes(',')||!new RegExp('^'+(signed?'[+-]?':'[+]?')+'(?:\\\\d+(?:[.,]\\\\d*)?|[.,]\\\\d+)(?:[eE][+-]?\\\\d+)?$').test(s))return NaN;var n=Number(s.replace(',','.'));return Number.isFinite(n)&&Math.abs(n)<=1e100?n:NaN}function fmt(n){if(n===0)return'0';var a=Math.abs(n);return a>=1e9||a<1e-7?n.toExponential(9):Number(n.toPrecision(10)).toLocaleString('en-US',{maximumFractionDigits:20})}function clear(){['uc-r-expression','uc-r-combined','uc-r-expanded','uc-r-k','uc-r-relative','uc-r-count','uc-summary'].forEach(function(id){$(id).textContent='\u2014'});$('uc-results').hidden=true;$('uc-results').dataset.json=''}function fail(m){clear();$('uc-error').textContent=m}function calc(){var estimateRaw=$('uc-estimate').value,unit=$('uc-unit').value.trim(),kRaw=$('uc-k').value,relation=$('uc-relation').value,model=$('uc-model').value.trim(),ksource=$('uc-k-source').value.trim(),started=estimateRaw.trim()||unit||kRaw.trim()||relation||model||ksource;var names=[],rawq=[];for(var i=1;i<=6;i++){names.push($('uc-name-'+i).value.trim());rawq.push($('uc-q-'+i).value);if(names[i-1]||rawq[i-1].trim())started=true}if(!started){$('uc-error').textContent='';clear();return}var estimate=num(estimateRaw,true),k=num(kRaw,false);if(!Number.isFinite(estimate)||!Number.isFinite(k)||!(k>0&&k<=100)){fail('Enter a complete finite measurand estimate and a documented coverage factor greater than0 through100.');return}if(!unit||unit.length>40||\/[<>\\x00-\\x1F]\/.test(unit)){fail('Enter a short plain-text output unit without markup or control characters.');return}if(relation!=='uncorrelated'||!model||!ksource){fail('Confirm the uncorrelated restricted model,and document the measurement model and coverage-factor justification.');return}var rows=[],sumSq=0,positive=0;for(i=0;i<6;i++){var hasName=!!names[i],hasValue=!!rawq[i].trim();if(!hasName&&!hasValue)continue;if(hasName!==hasValue){fail('Each used contribution row requires both a name\/source and a numerical standard contribution.');return}var q=num(rawq[i],false);if(!Number.isFinite(q)||q<0){fail('Each contribution must be one complete finite non-negative number in the output unit.');return}if(q>0)positive++;sumSq+=q*q;rows.push({name:names[i],standard_contribution:q})}if(!rows.length||!positive||!Number.isFinite(sumSq)){fail('Record at least one positive standard contribution.');return}var combined=Math.sqrt(sumSq),expanded=k*combined,relative=estimate===0?null:expanded\/Math.abs(estimate)*100;if(![combined,expanded].every(Number.isFinite)||relative!==null&&!Number.isFinite(relative)){fail('The result exceeds the supported numerical range.');return}$('uc-error').textContent='';$('uc-r-expression').textContent=fmt(estimate)+' \u00b1 '+fmt(expanded)+' '+unit;$('uc-r-combined').textContent=fmt(combined)+' '+unit;$('uc-r-expanded').textContent=fmt(expanded)+' '+unit;$('uc-r-k').textContent=fmt(k);$('uc-r-relative').textContent=relative===null?'not applicable':fmt(relative)+' %';$('uc-r-count').textContent=String(rows.length);$('uc-summary').textContent='Restricted RSS of documented uncorrelated standard contributions;no coverage probability inferred.';$('uc-results').dataset.json=JSON.stringify({estimate:estimate,unit:unit,coverage_factor:k,combined_standard_uncertainty:combined,expanded_uncertainty:expanded,relative_expanded_percent:relative,relationship:relation,contributions:rows});$('uc-results').hidden=false}$('uc-form').addEventListener('input',calc);$('uc-form').addEventListener('change',calc);calc()})();<\/script>\n","protected":false},"excerpt":{"rendered":"<p>ISO 17025 \/ GUM-a uy\u011fun olaraq pulsuz \u00f6l\u00e7m\u0259 qeyri-m\u00fc\u0259yy\u0259nlik kalkulyatoru. 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