{"id":1177,"date":"2024-09-05T22:36:30","date_gmt":"2024-09-05T20:36:30","guid":{"rendered":"https:\/\/site.ellistat.com\/?p=1177"},"modified":"2024-10-23T15:22:06","modified_gmt":"2024-10-23T13:22:06","slug":"how-to-calculate-cp-pp-ppk","status":"publish","type":"post","link":"https:\/\/clooma.ai\/en\/how-to-calculate-cp-pp-ppk\/","title":{"rendered":"How to calculate the capability indices Cp, Pp, Ppk"},"content":{"rendered":"<p>In this article, we'll look at how to calculate the various capability indices, in particular Cp, Pp and Ppk.<\/p>\n\n\n\n<p>To fully appreciate the capability of a process, the notion of time is particularly important, as two types of variability can be distinguished:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Short-term variability: when two parts are produced consecutively, these two parts will not be completely equivalent due to intrinsic machine variations. This short-term variability depends mainly on the machine.<\/li>\n\n\n\n<li>Long-term variability: when you produce on the same process over a longer period of time, the machine itself will go out of adjustment, and series changes, material batch changes, etc. will bring new sources of variation. This long-term variability depends not only on the machine itself, but also on numerous external sources of variability, and on the way in which the process is managed.<br><\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Example<\/h2>\n\n\n\n<p>To illustrate this point, let's look at the following production diagram:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/clooma.ai\/wp-content\/uploads\/2024\/09\/Court-et-long-terme.png\" alt=\"\" class=\"wp-image-1179\"\/><\/figure><\/div>\n\n\n<p>Production for the week is the result of production on the different days of the week. However, as the diagram above illustrates, production conditions on different days of the week are not equivalent. The centering of the process is not the same between production on Monday, Tuesday, etc...<\/p>\n\n\n\n<p>As a result, we can see that the variability over the whole week (red curve) is greater than the variability observed over a short period of time, as it takes into account fewer phenomena that can cause variability, such as a setting or a change of material.<\/p>\n\n\n\n<p>There are therefore two ways of characterizing process capability, depending on the type of variability observed: short-term cp and long-term pp.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Short-term Capability Index Cp<\/h2>\n\n\n\n<p><strong>Short-term Cp<\/strong>Short-term capability: Short-term capability characterizes the process's ability to produce good parts, taking into account only the process's intrinsic variability (the variability between two consecutive parts). Short-term capability is denoted Cp and is calculated by :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Cp= \\frac{\\text{tolerance interval}}{6*\\text{short term dispersion}}= \\frac{\\text{tolerance interval}}{6*\\sigma_{\\text{short term}}}<\/span><\/p>\n\n\n\n<p>In general, we want :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Cp &gt; 2<\/span><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Long-term capability index Pp<\/h2>\n\n\n\n<p><strong>Long-term Pp<\/strong>Long-term capability: Long-term capability characterizes the ability of the process to produce good parts over a long period of time, i.e. taking into account adjustments and process modifications that may occur. Long-term capability is denoted Pp and is calculated by :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Pp=\\frac{tolerance interval}{6\u2217long term dispersion}=\\frac{tolerance interval}{6\u2217\u03c3_{long term}}<\/span><\/p>\n\n\n\n<p>In general, we want :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Pp &gt; 1.33<\/span><\/p>\n\n\n\n<p>Since long-term capability takes into account more sources of variability than short-term capability, we necessarily have :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{long term dispersion} &lt; \\text{short term dispersion}<\/span><\/p>\n\n\n\n<p>And so<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Pp &lt; Cp<\/span><\/p>\n\n\n\n<p>If this is not the case, this does not mean that short-term dispersion is greater than long-term dispersion, but that dispersion is not stable over time.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Methods for calculating capability indices Cp, Pp, Ppk<\/h2>\n\n\n\n<p>As we've just seen, Cp and Pp are calculated using the same formula. Cp and Pp are differentiated by the time periods over which variability is calculated. There are several methods for calculating Cp and Pp.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Method 1: Taking several samples at regular intervals<\/h3>\n\n\n\n<p>The first method for calculating short- and long-term variability is to take several samples at regular intervals.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/clooma.ai\/wp-content\/uploads\/2024\/09\/Illustration.png\" alt=\"\" class=\"wp-image-1180\"\/><\/figure><\/div>\n\n\n<p><strong>Short-term variability :<\/strong>&nbsp;Short-term variability is calculated using the intra-series standard deviation of all samples:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{short term}}=\\sigma_{\\text{intra sample}}<\/span><\/p>\n\n\n\n<p><strong>Long-term variability :<\/strong>&nbsp;50 parts are sampled over a characteristic period of the process, to take into account multiple sources of process variation such as adjustments, tool changes, material changes, etc. Long-term variability is calculated by :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{long term}}=\\sigma_{\\text{all sample}}=\\sqrt{\\sum_{}^{}(\\frac{^{(x_{i}-\\mu)^{2}}}{n-1})}<\/span><\/p>\n\n\n\n<p>Example: The following sample is taken<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/clooma.ai\/wp-content\/uploads\/2024\/09\/Illustration-2.png\" alt=\"\" class=\"wp-image-1181\"\/><\/figure><\/div>\n\n\n<p>The tolerance interval is [1 ;10].<\/p>\n\n\n\n<p>We calculate the intra-series standard deviation :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{short term}}=\\sigma_{\\text{intra sample}}=1.7321<\/span><\/p>\n\n\n\n<p>The standard deviation of all parts is calculated:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{long term}}=\\sigma_{\\text{all sample}}=\\sqrt{\\sum_{}^{}(\\frac{^{(x_{i}-\\mu)^{2}}}{n-1})}=2.6904<\/span><\/p>\n\n\n\n<p>We therefore deduce :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Cp= \\frac{\\text{tolerance interval}}{6*\\text{short term dispersion}}= \\frac{\\text{tolerance interval}}{6*\\sigma_{\\text{short term}}= \\frac{9}{6*1.7321}=0.87<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Pp=\\frac{\\text{tolerance interval}}{6*\\text{long term dispersion}}= \\frac{\\text{tolerance interval}}{6*\\sigma_{\\text{long term}}=\\frac{9}{6*2.6904}=0.56<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Ppk=\\frac{Min(\\text{tolerance max}-\\mu,mu-\\text{tolerance min})}{3*\\text{long term dispersion}}= \\frac{4.33}{3*2.6904}=0.54<\/span><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Method 2: By taking two separate samples<\/h3>\n\n\n\n<p>The second method for calculating short- and long-term variability is to take two separate samples.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Short-term variability: 50 consecutive unadjusted parts are taken to calculate the short-term variability of the process. Short-term variability is calculated by :<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{short term}}=\\sigma_{\\text{all sample}}=\\sqrt{\\sum_{}^{}(\\frac{^{(x_{i}-\\mu)^{2}}}{n-1})}<\/span><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Long-term variability: 50 parts are sampled over a characteristic period of the process, to take into account multiple sources of process variation such as adjustments, tool changes, material changes, etc. Long-term variability is calculated by :<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{long term}}=\\sigma_{\\text{all sample}}=\\sqrt{\\sum_{}^{}(\\frac{^{(x_{i}-\\mu)^{2}}}{n-1})}<\/span><\/p>","protected":false},"excerpt":{"rendered":"<p>In this article, we'll look at how to calculate the various capability indices, in particular Cp, Pp and Ppk. To fully appreciate the capability of a process, the notion of time is particularly important, as two types of variability can be distinguished: Example To illustrate this, let's look at the following production diagram: [...] [...] [...] [...] [...] [...] [...] [...<\/p>","protected":false},"author":2,"featured_media":1178,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7,4],"tags":[],"class_list":["post-1177","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data-analysis","category-spc"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Comment calculer les indices de capabilit\u00e9 Cp, Pp, Ppk - Clooma<\/title>\n<meta name=\"description\" content=\"Quelles sont les diff\u00e9rentes m\u00e9thodes pour calculer les indices de capabilit\u00e9s Cp, Pp et Ppk \u00e0 partir de diff\u00e9rents types de pr\u00e9l\u00e8vements ?\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/clooma.ai\/en\/how-to-calculate-cp-pp-ppk\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Comment calculer les indices de capabilit\u00e9 Cp, Pp, Ppk - 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