tag:blogger.com,1999:blog-40045377405495098112014-10-04T18:00:22.382-07:00Probability Concepts for Statistical Process ControlLean Managementnoreply@blogger.comBlogger5125tag:blogger.com,1999:blog-4004537740549509811.post-56293875595609298752007-11-15T04:49:00.001-08:002007-11-15T04:57:00.791-08:00Confidence Interval (CI)A confidence interval (CI) is an interval estimate of a population parameter. Instead of estimating the parameter by a single value, an interval of likely estimates is given.<br /><br />More precisely a CI for a population parameter is an interval with an associated probability p that is generated from a random sample of an underlying population such that if the sampling was repeated numerous times and the confidence interval recalculated from each sample according to the same method, a proportion p of the confidence intervals would contain the population parameter in question.Lean Managementnoreply@blogger.com0tag:blogger.com,1999:blog-4004537740549509811.post-76191847037021636072007-11-05T19:43:00.000-08:002007-11-05T19:44:48.040-08:00Independency in Probability<p style="font-style: italic;">The odds against there being a bomb on a plane are a million to one, and against two bombs a million times a million to one. Next time you fly, cut the odds and take a bomb.</p> <p align="right"><i>— Benny Hill</i></p><br />Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other.<br /><br />Independency is a requirement in <a href="http://probability-concepts.blogspot.com/2007/11/central-limit-theorem.html" target="_blank">Central Limit Theorem</a>.Lean Managementnoreply@blogger.com0tag:blogger.com,1999:blog-4004537740549509811.post-21153672748771505272007-11-05T19:41:00.000-08:002007-11-05T20:32:05.372-08:00Student t-distributionSuppose we have a simple random sample of size <i>n</i> drawn from a <a href="http://statistical-process-control.blogspot.com/2007/11/normal-law-distribution.html" target="_blank">Normal population</a> with mean <img src="http://www-stat.stanford.edu/%7Enaras/jsm/TDensity/img1.gif" align="middle" height="18" width="10" /> and standard deviation <img src="http://www-stat.stanford.edu/%7Enaras/jsm/TDensity/img2.gif" align="bottom" height="9" width="9" /> . Let <img src="http://www-stat.stanford.edu/%7Enaras/jsm/TDensity/img3.gif" align="bottom" height="11" width="12" /> denote the sample mean and <i>s</i>, the sample standard deviation. Then the quantity<br /><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 514px; height: 44px;" src="http://www-stat.stanford.edu/%7Enaras/jsm/TDensity/img4.gif" alt="" border="0" /> has a <i>t</i> distribution with <i>n</i>-1 degrees of freedom.<br /><br />The <i>t</i> density curves are symmetric and bell-shaped like the <a href="http://probability-concepts.blogspot.com/2007/11/normal-law-distribution.html" target="_blank">normal distribution</a> and have their peak at 0. However, the spread is more than that of the standard <a href="http://probability-concepts.blogspot.com/2007/11/normal-law-distribution.html" target="_blank">normal distribution</a>. This is due to the fact that in formula (1), the denominator is <i>s</i> rather than <img src="http://www-stat.stanford.edu/%7Enaras/jsm/TDensity/img2.gif" align="bottom" height="9" width="9" /> . Since <i>s</i> is a random quantity varying with various samples, the variability in <i>t</i> is more, resulting in a larger spread. <p> The larger the degrees of freedom, the closer the <i>t</i>-density is to the <a href="http://probability-concepts.blogspot.com/2007/11/normal-law-distribution.html" target="_blank">normal density</a>. This reflects the fact that the standard deviation <i>s</i> approaches <img src="http://www-stat.stanford.edu/%7Enaras/jsm/TDensity/img2.gif" align="bottom" height="9" width="9" /> for large sample size <i>n</i>.</p><p>Articles which reference Student t-distribution:</p><ul><li><a href="http://statistical-process-control.blogspot.com/2007/11/students-t-distribution.html" target="_blank">The misabuse of Student t-distribution according to the Founder of Statistical Process Control</a><br /></li></ul>Lean Managementnoreply@blogger.com0tag:blogger.com,1999:blog-4004537740549509811.post-56448005946085718822007-11-03T05:49:00.001-07:002008-12-09T22:33:59.814-08:00Normal Law DistributionNormal Law Distribution, known popularly as the "<span style="font-weight: bold;">Bell Curve</span>" or mathematically as the "<span style="font-weight: bold;">Gaussian Law</span>" seems to be a "common sense" law:<br /><br />It describes mathematically the idea that extremes are rare and elements around averages more and more numerous. So Normal Law could obviously not be a half-circle nor a square but a bell curve.<br /><br />To characterize the <span style="font-weight: bold;">Bell Curve</span>, we only need two parameters: a mean around which most of the population will be found and a standard deviation which impacts the distance between the average and the queues of the extremes.<br /><br />The existence of <span style="font-weight: bold;">Normal Law</span> is based on the <a style="font-weight: bold;" href="http://probability-concepts.blogspot.com/2007/11/central-limit-theorem.html" target="_blank">Central Limit Theorem.</a><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_UozXMvG5llc/Ryn5RDyNG9I/AAAAAAAAACE/_D9_ihSnpZQ/s1600-h/bell_curve.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://1.bp.blogspot.com/_UozXMvG5llc/Ryn5RDyNG9I/AAAAAAAAACE/_D9_ihSnpZQ/s320/bell_curve.jpg" alt="" id="BLOGGER_PHOTO_ID_5127903722206993362" border="0" /></a><br />Articles which reference Normal Law Distribution:<br /><ul><li><a href="http://statistical-process-control.blogspot.com/2007/11/non-normal-distributions-in-real-world.html" target="_blank">Non-normal distribution in the Real World</a><br /></li><li><a href="http://deming-lean-management.blogspot.com/2007/11/becoming-black-belt-without-six-sigma.html" target="_blank">Becoming a Blackbelt without Six Sigma</a></li></ul>Lean Managementnoreply@blogger.com0tag:blogger.com,1999:blog-4004537740549509811.post-22888652492674092252007-11-03T05:42:00.001-07:002007-12-20T16:37:11.059-08:00Central Limit TheoremThe <span style="font-weight: bold;">Central Limit Theorem</span> states that if a sum of <a href="http://probability-concepts.blogspot.com/2007/11/independency-in-probability.html" target="_blank">independent</a> and identically-distributed random variables has a finite variance, then it will be approximately <a href="http://statistical-process-control.blogspot.com/2007/11/normal-law-distribution.html" target="_blank">normally distributed</a> and the sampling distribution will have the same mean as the population, but the variance divided by the sample size.<br /><br />In short the <span style="font-weight: bold;">Central Limit Theorem</span> states that the sum of a number of random variables with finite variances will tend to a <a href="http://statistical-process-control.blogspot.com/2007/11/normal-law-distribution.html" target="_blank">normal distribution</a> as the number of variables grows.<br /><br />The smaller variance is intuitivally understandable if one just imagines that one size variation in the sample can compensate another.Lean Managementnoreply@blogger.com0