\u26a0\ufe0f The batch adjustment range is not unrestricted: an off-center batch is mechanically constrained to be tighter, since it must still meet the minimum Cpk required for the factor. This coupling is not a calculation artifact; it is the behavior of a supplier managing its capability\u2014and it is directly visible on the control chart.<\/p>\n\n\n\n
Declare the X factors<\/h2>\n\n\n\n
The table in the first column is filled in row by row\u2014one factor per row\u2014using the \"+ factor\" button. The columns are organized into three groups:<\/p>\n\n\n\n
\u21d2 the identity of the factor: its name\u2014the one you will use in the formula\u2014and its distribution (Gaussian, Uniform, Triangular, Log-normal);<\/p>\n\n\n\n
- \n
- the tolerance of the drawing: Min and Max, that is, what is written on the drawing;<\/li>\n\n\n\n
- The requirement per lot: the minimum Cpk that the supplier agrees to maintain for this factor;<\/li>\n\n\n\n
- the permitted variation from one batch to another: the range within which the batch mean may vary (\u03bc min, \u03bc max) and the maximum standard deviation \u03c3 max.<\/li>\n<\/ul>\n\n\n\n
\ud83d\udca1 Distribution functions are not defined by their natural parameters but by (mean, standard deviation). This is what makes it possible to control any distribution function using a Cpk requirement and to compare distribution functions with each other when they have equal capability.<\/p>\n\n\n\n
The Y-output and its formula<\/h2>\n\n\n\n
Below the factors table, the Y-Output block contains the formula, the name of the calculated quantity, its tolerance (Min and Max), and, if necessary, a noise standard deviation\u2014anything not accounted for by the formula: unmodeled phenomena, measurement uncertainty on Y.<\/p>\n\n\n\n
The formula is written in plain text, using the names of the factors. Common operators are accepted, including ^ for exponentiation, as well as the functions sqrt, abs, exp, log, log10, sin, cos, tan, atan, min, max, pow, and the constant PI. A typing error is flagged at the exact location where it occurs, along with the name of the unknown term in question.<\/p>\n\n\n\n
\u26a0\ufe0f If any simulations produce an undefined value (division by zero, square root of a negative number), their count is displayed. This is not a minor issue: it means that your formula is not valid across the entire tolerance range of your factors.<\/p>\n\n\n\n
The Simulation Plan<\/h2>\n\n\n\n
The \"Intention\" column controls how the simulation is performed: the number of lots, the number of parts per lot, the sampling method, the mode of inter-lot variability, and the target Cpk per lot\u2014the value below which a lot will be declared non-capable.<\/p>\n\n\n\n
The three types of inter-batch variability address three different questions:<\/p>\n\n\n\n
Mode<\/strong><\/td> How does the batch average perform?<\/strong><\/td> What He Tells You<\/strong><\/td><\/tr><\/thead> Ideals<\/td> Each shot is perfectly centered on the target.<\/td> The optimistic scenario. This is the assumption underlying traditional tolerance calculation software: useful for comparison, but not for decision-making.<\/td><\/tr> Realists<\/td> The batch average fluctuates around the midpoint of the acceptable range, with large deviations being less common than small ones.<\/td> The typical industrial scenario, selected by default.<\/td><\/tr> At worst<\/td> The batch average can fall anywhere within the acceptable range, with no preference.<\/td> The worst-case scenario: what happens if the supplier takes full advantage of the leeway you've granted them.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n \u26a0\ufe0f The selected mode completely changes the results: switching from \u00abideal\u00bb to \u00abworst-case\u00bb can increase the nonconformance rate by several orders of magnitude, even though no tolerances have been changed. It is the adjustment drift\u2014not machine variation\u2014that dominates most real-world cases.<\/p>\n\n\n\n
The sampling method offers the Latin hypercube\u2014selected by default\u2014or simple Monte Carlo. The Latin hypercube distributes the samples more evenly across the domain and, given the same number of samples, yields a significantly more stable result\u2014especially since the formula is close to a sum, which is the case for all odds chains.<\/p>\n\n\n\n
The calculation is started by clicking the \"Simulate\" button. Any change to a parameter invalidates the displayed result, and the button becomes highlighted again: what you see always corresponds to the most recent calculation, never to the current input. The calculation time and the number of simulated parts are displayed in the title bar.<\/p>\n\n\n\n
The Five Charts<\/h2>\n\n\n\n
Distribution of Y. The histogram of the output characteristic, showing its specification limits. In \u00absingle lot\u00bb mode, the histogram of the selected lot is superimposed on that of the entire population (shown in faded gray): this makes it immediately clear where that lot falls within the overall distribution.<\/p>\n\n\n\n
![\"\"](\"https:\/\/ellistat.com\/wp-content\/uploads\/montecarlo1.png\")
<\/figure>\n\n\n\nDistributions of the X values. One panel per factor, and within each panel, one curve per batch, plotted based on the setting used for that batch. The curves are colored according to the batch\u2019s capability on Y: this makes it immediately clear what the failed batches have in common with respect to this factor (a shifted position or wider dispersion). A solid gray background represents the combination of all lots\u2014that is, the actual production distribution, which is often much wider than any single lot. The selected lot is plotted on top with a thick line, and the factor\u2019s tolerances are shown as a dashed line.<\/p>\n\n\n\n
Below the graph, a diagnostic statement identifies what distinguishes non-capable lots: the skewness of a factor, or its dispersion, along with the corresponding correlation.<\/p>\n\n\n\n
![\"\"](\"https:\/\/ellistat.com\/wp-content\/uploads\/montecarlo2.png\")
<\/figure>\n\n\n\nContribution to variance. Sorted horizontal bars, one per factor, that answer the question, \u00abIf I had to tighten just one tolerance, which one would it be?\u00bb Each bar is divided into two shades: the portion due to variation within lots and the portion due to variation between lots. A whisker indicates the confidence interval for the index, and two additional bars isolate interactions and noise.<\/p>\n\n\n\n
\ud83d\udca1 This is the most useful distinction on the screen. If a factor\u2019s contribution stems mainly from inter-batch variation, tightening up the machine won\u2019t make a difference: it\u2019s the centering that needs to be maintained. If it stems from intra-batch variation, it\u2019s the production process that needs to be improved. These two levers have neither the same cost nor the same person responsible.<\/p>\n\n\n\n
\u26a0\ufe0f When the sum of the contributions does not equal 100 %, the difference is accounted for by the interactions: the formula is not a simple sum, and a standard quadratic sum would underestimate the variance of Y.<\/p>\n\n\n\n
Conformity Triangle. One point per lot, positioned based on that lot\u2019s mean and standard deviation; green if the lot is capable, red otherwise. The contour lines are the iso-Cpk lines: the line corresponding to the target Cpk defines the acceptable region. The triangular shape is not a graphical choice; it is the model itself: the more a batch is offset from the mean, the less variation it is allowed. Each point is clickable and switches the results column to that batch.<\/p>\n\n\n\n
![\"\"](\"https:\/\/ellistat.com\/wp-content\/uploads\/montecarlo3.png\")
<\/figure>\n\n\n\nSelected batch. The histogram of the single selected batch, with out-of-tolerance parts shown in red, the fitted normal distribution, and the batch mean shown as a dotted line. This is the \u201cwhat-if\u201d view: this is what the customer would have received if this batch had been shipped.<\/p>\n\n\n\n
The results: the entire simulation, or a batch<\/h2>\n\n\n\n
The right-hand column has a two-position button, and that is a key distinction.<\/p>\n\n\n\n
The entire simulation provides the characteristics of the resulting population: mean, standard deviation, observed range, skewness and kurtosis, nonconformance rate in ppm, Pp, and Ppk. The \u00abBatch Risk\u00bb block supplements this by revealing what the mean hides: the median Cpk of the batches, the Cpk at the fifth percentile, the number of batches below the target, and the resulting statement\u2014\u201ca randomly selected batch has such-and-such a risk of being non-capable.\u201d.<\/p>\n\n\n\n
A batch provides details about a specific batch: its Cpk, mean, standard deviation, nonconformance rate, and, most importantly, the settings that were determined for it, factor by factor. This answers the question we ask ourselves when faced with an outlier batch: Why this one? A single sentence identifies the factor most responsible.<\/p>\n\n\n\n
Three ways to select a lot, all synchronized: click a point on the compliance triangle, select it from the drop-down list (each entry displays its Cpk), or use the \"Most Critical Lot\" button, which jumps directly to the worst one.<\/p>\n\n\n\n
\u26a0\ufe0f The non-conformance rate is an estimate, and its uncertainty is displayed alongside it: \u00ab710 ppm \u00b1 23 % at 95 % with 100,000 parts.\u00bb A low rate requires a large number of parts to be determined precisely (the number needed to achieve \u00b15 % is indicated). And if no parts fell outside the tolerance, this does not mean the rate is zero: the upper limit is noted.<\/p>\n\n\n\n
\ud83d\udca1 When the distribution of Y deviates significantly from the normal distribution\u2014which happens as soon as the formula is highly nonlinear (the Pp and Ppk indices calculated based on the standard deviation become misleading). Ellistat then switches to indices calculated using percentiles, in accordance with ISO 22514-2, and flags this change. This prevents the common inconsistency between a Ppk value indicating a few thousand ppm and a count that finds none.<\/p>\n\n\n\n
The variance statement<\/h2>\n\n\n\n
The bottom bar chart is the equivalent of the Sp\u00e9cigraphe\u2019s tolerance report, adapted to variances: one segment per factor, plus one segment for interactions and one for noise. A dot indicates the proportion of the total variance attributable to inter-batch variation, regardless of the cause.<\/p>\n\n\n\n
Below, the messages draw the following conclusions: the dominant factor and the source of its contribution; the factors with no measurable effect, for which tolerance limits can be relaxed\u2014and thus the money that can be recovered without risk; and the significance of interactions when they become significant.<\/p>\n\n\n\n
The Example Library<\/h2>\n\n\n\n
The title bar selector loads a complete case\u2014factors, formula, tolerances, simulation plan\u2014and starts the calculation. This is the fastest way to get started with the program, and it serves as the foundation for building your own models:<\/p>\n\n\n\n
- \n
- Compression spring: stiffness as a function of wire diameter, coil diameter, number of coils, and shear modulus. A typical case where one factor overshadows the others because it is raised to the fourth power;<\/li>\n\n\n\n
- Dimension chain: a set of assembly dimensions obtained by the difference of three dimensions. The strictly additive case, that of classical tolerancing;<\/li>\n\n\n\n
- Linear control problem: a model for which the exact solution is known, useful for verifying the mechanics;<\/li>\n\n\n\n
- Blank form: to start with your own equation.<\/li>\n<\/ul>\n\n\n\n
\ud83d\udca1 Your own templates are saved: the library is intended to become your company\u2019s catalog of transfer functions\u2014a spring, a fitting, or a guide assembly only needs to be defined once.<\/p>\n\n\n\n
Case Registration<\/h2>\n\n\n\n
A case is saved either when a simulation is started or by clicking the save icon next to the name. The icon is active only if there are unsaved changes.<\/p>","protected":false},"featured_media":0,"menu_order":0,"template":"","meta":{"_acf_changed":true},"menu-guide-dutilisateur":[42],"class_list":["post-4911","guide-dutilisateur","type-guide-dutilisateur","status-publish","hentry","menu-guide-dutilisateur-9-tolerancement"],"acf":[],"yoast_head":"\n
Simulation Monte-Carlo - Ellistat<\/title>\n<meta name=\"description\" content=\"Simulation Monte-Carlo : Le calcul d'une raideur de ressort, un jeu d'assemblage, un rapport de transmission.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ellistat.com\/en\/users-guide\/monte-carlo-simulation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Simulation Monte-Carlo - Ellistat\" \/>\n<meta property=\"og:description\" content=\"Simulation Monte-Carlo : Le calcul d'une raideur de ressort, un jeu d'assemblage, un rapport de transmission.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/ellistat.com\/en\/users-guide\/monte-carlo-simulation\/\" \/>\n<meta property=\"og:site_name\" content=\"Ellistat\" \/>\n<meta property=\"article:modified_time\" content=\"2026-10-14T13:09:09+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/ellistat.com\/wp-content\/uploads\/montecarlo1.png\" \/>\n\t<meta property=\"og:image:width\" content=\"908\" \/>\n\t<meta property=\"og:image:height\" content=\"634\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"11 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/ellistat.com\\\/guide-dutilisateur\\\/simulation-monte-carlo\\\/\",\"url\":\"https:\\\/\\\/ellistat.com\\\/guide-dutilisateur\\\/simulation-monte-carlo\\\/\",\"name\":\"Simulation Monte-Carlo - Ellistat\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/ellistat.com\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/ellistat.com\\\/guide-dutilisateur\\\/simulation-monte-carlo\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/ellistat.com\\\/guide-dutilisateur\\\/simulation-monte-carlo\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/ellistat.com\\\/wp-content\\\/uploads\\\/montecarlo1.png\",\"datePublished\":\"2026-10-14T13:06:03+00:00\",\"dateModified\":\"2026-10-14T13:09:09+00:00\",\"description\":\"Simulation Monte-Carlo : Le calcul d'une raideur de ressort, un jeu d'assemblage, un rapport de transmission.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/ellistat.com\\\/guide-dutilisateur\\\/simulation-monte-carlo\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/ellistat.com\\\/guide-dutilisateur\\\/simulation-monte-carlo\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/ellistat.com\\\/guide-dutilisateur\\\/simulation-monte-carlo\\\/#primaryimage\",\"url\":\"https:\\\/\\\/ellistat.com\\\/wp-content\\\/uploads\\\/montecarlo1.png\",\"contentUrl\":\"https:\\\/\\\/ellistat.com\\\/wp-content\\\/uploads\\\/montecarlo1.png\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/ellistat.com\\\/guide-dutilisateur\\\/simulation-monte-carlo\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\\\/\\\/ellistat.com\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Simulation Monte-Carlo\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/ellistat.com\\\/#website\",\"url\":\"https:\\\/\\\/ellistat.com\\\/\",\"name\":\"Ellistat\",\"description\":\"La suite qualit\u00e9 de l'industrie m\u00e9canique\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/ellistat.com\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Monte Carlo Simulation - Ellistat","description":"Monte Carlo Simulation: Calculating spring stiffness, an assembly, and a gear ratio.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/ellistat.com\/en\/users-guide\/monte-carlo-simulation\/","og_locale":"en_US","og_type":"article","og_title":"Simulation Monte-Carlo - Ellistat","og_description":"Simulation Monte-Carlo : Le calcul d'une raideur de ressort, un jeu d'assemblage, un rapport de transmission.","og_url":"https:\/\/ellistat.com\/en\/users-guide\/monte-carlo-simulation\/","og_site_name":"Ellistat","article_modified_time":"2026-10-14T13:09:09+00:00","og_image":[{"width":908,"height":634,"url":"https:\/\/ellistat.com\/wp-content\/uploads\/montecarlo1.png","type":"image\/png"}],"twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"11 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/ellistat.com\/guide-dutilisateur\/simulation-monte-carlo\/","url":"https:\/\/ellistat.com\/guide-dutilisateur\/simulation-monte-carlo\/","name":"Monte Carlo Simulation - Ellistat","isPartOf":{"@id":"https:\/\/ellistat.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/ellistat.com\/guide-dutilisateur\/simulation-monte-carlo\/#primaryimage"},"image":{"@id":"https:\/\/ellistat.com\/guide-dutilisateur\/simulation-monte-carlo\/#primaryimage"},"thumbnailUrl":"https:\/\/ellistat.com\/wp-content\/uploads\/montecarlo1.png","datePublished":"2026-10-14T13:06:03+00:00","dateModified":"2026-10-14T13:09:09+00:00","description":"Monte Carlo Simulation: Calculating spring stiffness, an assembly, and a gear ratio.","breadcrumb":{"@id":"https:\/\/ellistat.com\/guide-dutilisateur\/simulation-monte-carlo\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/ellistat.com\/guide-dutilisateur\/simulation-monte-carlo\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/ellistat.com\/guide-dutilisateur\/simulation-monte-carlo\/#primaryimage","url":"https:\/\/ellistat.com\/wp-content\/uploads\/montecarlo1.png","contentUrl":"https:\/\/ellistat.com\/wp-content\/uploads\/montecarlo1.png"},{"@type":"BreadcrumbList","@id":"https:\/\/ellistat.com\/guide-dutilisateur\/simulation-monte-carlo\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/ellistat.com\/"},{"@type":"ListItem","position":2,"name":"Simulation Monte-Carlo"}]},{"@type":"WebSite","@id":"https:\/\/ellistat.com\/#website","url":"https:\/\/ellistat.com\/","name":"Ellistat","description":"The quality suite for the mechanical engineering industry","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/ellistat.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"}]}},"_links":{"self":[{"href":"https:\/\/ellistat.com\/en\/wp-json\/wp\/v2\/guide-dutilisateur\/4911","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ellistat.com\/en\/wp-json\/wp\/v2\/guide-dutilisateur"}],"about":[{"href":"https:\/\/ellistat.com\/en\/wp-json\/wp\/v2\/types\/guide-dutilisateur"}],"wp:attachment":[{"href":"https:\/\/ellistat.com\/en\/wp-json\/wp\/v2\/media?parent=4911"}],"wp:term":[{"taxonomy":"menu-guide-dutilisateur","embeddable":true,"href":"https:\/\/ellistat.com\/en\/wp-json\/wp\/v2\/menu-guide-dutilisateur?post=4911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}