{"id":2663,"date":"2024-10-09T11:00:36","date_gmt":"2024-10-09T09:00:36","guid":{"rendered":"https:\/\/clooma.ai\/?post_type=ressource-pedagogiqu&#038;p=2663"},"modified":"2024-10-24T16:20:18","modified_gmt":"2024-10-24T14:20:18","slug":"pledge-rr-measure","status":"publish","type":"ressource-pedagogiqu","link":"https:\/\/clooma.ai\/en\/educational-resource\/pledge-rr-measure\/","title":{"rendered":"Variable Gage R&amp;R"},"content":{"rendered":"<p>The Gage R&amp;R (Repeatability and Reproducibility) method is a technique used in statistics to assess the reliability and precision of a measurement process. It is widely used in quality assurance and process improvement.&nbsp;<\/p>\n\n\n\n<p>The aim of the Gage R&amp;R method is to determine to what extent the total variability of a measurement is attributable to variability in the measurement process itself (repeatability) and to variability between different operators or measurement equipment (reproducibility).&nbsp;<\/p>\n\n\n\n<p>The Gage R&amp;R process generally involves the following steps:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Gage selection (measuring instrument)<\/strong> Select the measuring instrument to be evaluated.\u00a0<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Operator selection<\/strong> Select the operators who will perform the measurements. These operators may represent different members of staff, different teams, or different measuring equipment.\u00a0<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Defining tests<\/strong> Select a representative sample of the item to be measured and perform repeated measurements. This allows you to assess repeatability.\u00a0<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Repeat the tests<\/strong> Operators measure the same sample several times to assess repeatability.\u00a0<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Total change<\/strong> Analyze total measurement variation to determine the proportion due to repeatability (variability of the measurement process), reproducibility (variability between operators) and interaction (interaction between operators and parts).\u00a0<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Gage R&amp;R calculation<\/strong> Total variability is often expressed as a percentage, called the R&amp;R Gage, which indicates the proportion of total variability attributable to repeatability and reproducibility compared to the tolerance interval of the characteristic.\u00a0<\/li>\n<\/ul>\n\n\n\n<p>This method is particularly useful in industries where measurement accuracy is crucial, such as manufacturing. It enables sources of variability to be identified and quantified in order to improve the reliability of the measurement process.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">ANOVA Gage R&amp;R - Nested Methods for Destructive Testing<\/h2>\n\n\n\n<p>The nested Gage R&amp;R method is a variation of the standard Gage R&amp;R method which is used when each operator measures a specific sample of parts, and the operators do not necessarily measure the same parts. It is suited to situations where sample destruction is unavoidable, and where each operator is assigned to specific groups of parts from the same batch (homogeneous parts). This approach is particularly relevant in destructive testing, where sample production is limited.\u00a0<\/p>\n\n\n\n<p>To calculate GRR and Cpc using the <a href=\"https:\/\/clooma.ai\/en\/what-is-the-rr\/\">ANOVA method<\/a> we use the statistical analysis of Fisher's test :\u00a0<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Sources of variability<\/th><th>Sum of squares<\/th><th>Degree of freedom<\/th><th>Average square<\/th><th>F-Statistics<\/th><\/tr><\/thead><tbody><tr><td>Operator<\/td><td>SSA<\/td><td>a-1<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\text{MSA}=\\frac{\\text{SSA}}{\\text{a-1}}<\/span><\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\text{F}=\\frac{\\text{MSA}}{\\text{MSE}}<\/span><\/td><\/tr><tr><td>Parts<\/td><td>SSB<\/td><td>b-1<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\text{MSB}=\\frac{\\text{SSB}}{\\text{b-1}}<\/span><\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\text{F}=\\frac{\\text{MSB}}{\\text{MSE}}<\/span><\/td><\/tr><tr><td>Interaction (Operator\/part)<\/td><td>SSAB<\/td><td>(a-1)(b-1)<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\text{MSAB}=\\frac{\\text{SSAB}}{\\text{(a-1)(b-1)}}<\/span><\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\text{F}=\\frac{\\text{MSAB}}{\\text{MSE}}<\/span><\/td><\/tr><tr><td>Instrument<\/td><td>SSE<\/td><td>ab(n-1)<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\text{MSE}=\\frac{\\text{SSE}}{\\text{ab(n-1)}}<\/span><\/td><td><\/td><\/tr><tr><td>Total<\/td><td>TSS<\/td><td>N-1<\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>with:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>a = number of operators<\/li>\n\n\n\n<li>b = number of pieces<\/li>\n\n\n\n<li>n = number of repetitions<\/li>\n\n\n\n<li>N = total number of measurements = abn<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{SSA}=\\sum^{a}{\\frac{Y_{i}^{2}}{\\text{bn}}}-\\frac{Y_{**}^{2}}{N}<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{SSB}=\\sum^{b}{\\frac{Y_{i}^{2}}{\\text{an}}}-\\frac{Y_{**}^{2}}{N}<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{SSAB}=\\sum^{a}\\sum^{b}{\\frac{Y_{ij}^{2}}{n}}-\\frac{Y_{**}^{2}}{N}-\\text{SSA}-\\text{SSB}<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{TSS}=\\sum^{a}\\sum^{b}\\sum^{n}Y_{ijk}^{2}-\\frac{Y_{**}^{2}}{N}<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{SSE}=\\text{TSS}-\\text{SSA}-\\text{SSB}-\\text{SSAB}<\/span><\/p>\n\n\n\n<p>The repeatability of the measurement process is given by :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{R\u00e9p\u00e9tabilit\u00e9}=5.15\\sqrt{\\text{MSE}}<\/span><\/p>\n\n\n\n<p>The reproducibility of the measurement process is given by :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{Reproductibilit\u00e9}=5.15\\sqrt{\\frac{\\text{MSA}-\\text{MSAB}}{\\text{bn}}}<\/span><\/p>\n\n\n\n<p>The interaction is given by :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{Int\u00e9raction}=5.15\\sqrt{\\frac{\\text{MSAB}-\\text{MSE}}{\\text{n}}}<\/span><\/p>\n\n\n\n<p>The variability of the measurement process is given by<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{Dispersion}=5.15\\sqrt{\\text{Repeatability}^2+\\text{Reproducibility}^2+\\text{Interpretation}^2}<\/span><\/p>\n\n\n\n<p>Finally, we calculate :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{GRR}=frac{Dispersion}{IT}<\/span><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Nested ANOVA<\/h2>\n\n\n\n<p>The nested Gage R&amp;R method is a variation of the standard Gage R&amp;R method which is used when each operator measures a specific sample of parts, and the operators do not necessarily measure the same parts. It is suited to situations where sample destruction is unavoidable, and where each operator is assigned to specific groups of parts from the same batch (homogeneous parts). This approach is particularly relevant in destructive testing, where sample production is limited.&nbsp;<\/p>\n\n\n\n<p>To perform GRR and Cpc using the ANOVA method, we use the Fisher test:\u00a0<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>Source&nbsp;<\/td><td>Degree of freedom&nbsp;<\/td><td>Sum of squares (SS)&nbsp;<\/td><td>Square averages (MS)&nbsp;<\/td><td>F&nbsp;<\/td><\/tr><tr><td>Operator&nbsp;<\/td><td>b-1&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">SS_{op}=an\\sum_{j=1}^{b}(\\overline{yj}-\\overline{y})^{2}<\/span><\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{SS_{op}}{b-1}<\/span><\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{MS_{op}}{MS_{pi\u00e8ces(op\u00e9rateur)}}<\/span><\/td><\/tr><tr><td>Parts (operator)&nbsp;<\/td><td>b(a-1)&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">SS_{pi\u00e8ce(op\u00e9rateur)}=n\\sum_{i=1}^{a}\\sum_{j=1}^{b}(\\overline{yij}-\\overline{y})^{2}<\/span><\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{SS_{parts(operator))}}{b(a-1)}<\/span><\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{MS_{parts(operator)}}{MS_{repeatability}}<\/span><\/td><\/tr><tr><td>Repeatability&nbsp;<\/td><td>ab(n-1)&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">SS_{r\u00e9p\u00e9tabilit\u00e9}=\\sum_{i=1}^{a}\\sum_{j=1}^{b}\\sum_{k=1}^{n}(\\overline{yij}-\\overline{y})^{2}<\/span><\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{SS_{r\u00e9p\u00e9tabilit\u00e9}}{ab(n-1)}<\/span><\/td><td>&nbsp;<\/td><\/tr><tr><td>Total&nbsp;<\/td><td>abn-1&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">SS_{TOTAL}=SS_{operator}+SS_{parts(operator)}+SS_{repeatability}<\/span><\/td><td>&nbsp;<\/td><td>&nbsp;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>With :&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>b : number of operators\u00a0<\/li>\n\n\n\n<li>a: number of rooms\u00a0<\/li>\n\n\n\n<li>n: number of repetitions\u00a0<\/li>\n<\/ul>\n\n\n\n<p>We can therefore deduce the variabilities due to the different sources of variability:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>Source&nbsp;<\/td><td>Variances&nbsp;<\/td><\/tr><tr><td>Repeatability&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{repeatability}^{}2 = M S_{repeatability}<\/span><\/td><\/tr><tr><td>Reproducibility&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{repeatability}^{}2 = \\frac{MS_{operator}-MS_{part(operator)}}{an}<\/span><\/td><\/tr><tr><td>Part by part&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{part to part}}^{}2 = \\frac{MS_{part(operator))}-MS_{repeatability}}{n}<\/span><\/td><\/tr><tr><td>Measurement method&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma^2{<em>{\\text{measurement method}}}= \\sigma<\/em>{{repeatability}^{2}}+ \\sigma_{{reproducibility}^{2}}<\/span><\/td><\/tr><tr><td>Total&nbsp;<\/td><td><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma^2{<em>{\\text{total}}}= \\sigma<\/em>{\\text{measurement method}}^{2}}+ \\sigma_{\\text{part to part}}^{2}}<\/span><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><\/p>","protected":false},"featured_media":0,"template":"","meta":{"_acf_changed":false},"menu-ressource-pedagogique":[26],"class_list":["post-2663","ressource-pedagogiqu","type-ressource-pedagogiqu","status-publish","hentry","menu-ressource-pedagogique-2-gage-rr"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.9 - 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