{"id":8,"date":"2024-04-03T23:09:22","date_gmt":"2024-04-03T20:09:22","guid":{"rendered":"https:\/\/sisu.ut.ee\/measurement\/32-mean-standard-deviation-and-standard-uncertainty\/"},"modified":"2024-08-29T16:09:59","modified_gmt":"2024-08-29T13:09:59","slug":"32-mean-standard-deviation-and-standard-uncertainty","status":"publish","type":"page","link":"https:\/\/sisu.ut.ee\/measurement\/32-mean-standard-deviation-and-standard-uncertainty\/","title":{"rendered":"3.2. Mean, standard deviation and standard uncertainty"},"content":{"rendered":"<p><strong>Brief summary:<\/strong> the lecture explains calculation of <strong>mean<\/strong> (<em>V<\/em><sub>m<\/sub>) and <strong>standard deviation<\/strong> (<em>s<\/em>). Illustrates again the 68% probability of <em>s<\/em>. Explains how the <strong>standard uncertainty<\/strong> of repeatability <em>u\u00a0<\/em>(<em>V<\/em>, REP) can be estimated as standard deviation of parallel measurement results. Stresses the importance of standard uncertainty as the key parameter in carrying out uncertainty calculations: uncertainties corresponding to different sources (not only to repeatability) and to different distribution functions are converted to standard uncertainties when uncertainty calculations are performed.<\/p>\n<p style=\"text-align: center;\"><\/p><div class=\"ratio ratio-16x9 mb-3\"><div class=\"video-placeholder-wrapper video-placeholder-wrapper--16x9\">\n\t\t\t    <div class=\"video-placeholder d-flex justify-content-center align-items-center\">\n\t\t\t        <div class=\"overlay text-white p-2 w-100 text-center d-block justify-content-center align-items-center\">\n\t\t\t            <div>To view third-party content, please accept cookies.<\/div>\n\t\t\t            <button class=\"btn btn-secondary btn-sm mt-1 consent-change\">Change consent<\/button>\n\t\t\t        <\/div>\n\t\t\t    <\/div>\n\t\t\t<\/div>\n<\/div>\n<h4 style=\"text-align: center;\"><strong>Mean, standard deviation and standard uncertainty<br>\n<\/strong><a style=\"line-height: 1.6em;\" href=\"http:\/\/www.uttv.ee\/naita?id=17554\" target=\"_blank\" rel=\"noopener\">http:\/\/www.uttv.ee\/naita?id=17554<\/a><\/h4>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ND3iryaVQ68\" target=\"_blank\" rel=\"noopener\">https:\/\/www.youtube.com\/watch?v=ND3iryaVQ68<\/a><\/p>\n<p>One of the most common approaches for improving the reliability of measurements is making replicate measurements of the same quantity. In such a case very often the measurement result is presented as the <strong style=\"line-height: 1.6em;\">mean value<\/strong> of the replicate measurements. In the case of pipetting <em style=\"line-height: 1.6em;\">n\u00a0<\/em> times with the same pipette volumes <em style=\"line-height: 1.6em;\">V<\/em><sub>1<\/sub>, <em style=\"line-height: 1.6em;\">V<\/em><sub>2<\/sub>, \u2026, <em style=\"line-height: 1.6em;\">V<sub>n<\/sub><\/em> are obtained and the mean value <em style=\"line-height: 1.6em;\">V<\/em><sub>m<\/sub> is calculated as follows:<\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" width=\"258\" height=\"81\" class=\"alignnone wp-image-176\" title=\"valem3-2.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem3-2.png\" alt=\"valem3-2.png\"><\/td>\n<td>(3.2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As explained in section 3.1, the mean value calculated this way is an estimate of the true mean value (which could be obtained if it were possible to make an infinite number of measurements).<\/p>\n<p>The scatter of values obtained from repeated measurements is characterized by <strong style=\"line-height: 1.6em;\">standard deviation<\/strong> of pipetted volumes, which for the same case of pipetting is calculated as follows:<\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" width=\"192\" height=\"88\" class=\"alignnone wp-image-177\" title=\"valem3-3.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem3-3.png\" alt=\"valem3-3.png\"><\/td>\n<td>(3.3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The <em>n<\/em> \u2013 1 in the denominator is often called <strong>number of degrees of freedom<\/strong>. We will see later that this is an important characteristic of a set or repeated measurements. The higher it is the more reliable mean and standard deviation can be from the set.<\/p>\n<p>Two important interpretations of the standard deviation:<\/p>\n<ol>\n<li><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">If <\/span><em style=\"line-height: 1.6em;\">V<\/em><sub>m<\/sub><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\"> and <\/span><em style=\"line-height: 1.6em;\">s\u00a0<\/em><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">(<\/span><em style=\"line-height: 1.6em;\">V\u00a0<\/em><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">) have been found from a sufficiently large number of measurements (usually 10-15 is enough) then the probability of every next measurement (performed under the same conditions) falling within the range <\/span><em style=\"line-height: 1.6em;\">V<\/em><sub>m<\/sub><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\"> \u00b1 <\/span><em style=\"line-height: 1.6em;\">s\u00a0<\/em><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">(<\/span><em style=\"line-height: 1.6em;\">V\u00a0<\/em><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">) is roughly 68.3%.<\/span><\/li>\n<li><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">If we make a number of repeated measurements under the same conditions then the standard deviation of the obtained values characterized the uncertainty due to non-ideal repeatability (often called as repeatability standard uncertainty) of the measurement: <\/span><em style=\"line-height: 1.6em;\">u\u00a0<\/em><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">(<\/span><em style=\"line-height: 1.6em;\">V<\/em><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">, REP) = <\/span><em style=\"line-height: 1.6em;\">s<\/em><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">(<\/span><em style=\"line-height: 1.6em;\">V<\/em><span style=\"line-height: 1.6em;\" data-mce-mark=\"1\">). Non-ideal repeatability is one of the uncertainty sources in all measurements.\u00a0<a href=\"#\" data-bs-toggle=\"modal\" data-bs-target=\"#popup-modal\" data-title=\"[1]\" data-content=\"We will see later that standard deviation of measurements repeated under conditions that change in predefined way (i.e. it is not repeatability) is also extremely useful in uncertainty calculation, as it enables taking a number of uncertainty sources into account simultaneously.\">[1]<\/a><\/span><\/li>\n<\/ol>\n<p>Standard deviation is the basis of defining <strong>standard uncertainty<\/strong> \u2013 uncertainty at standard deviation level, denoted by small <em>u<\/em>. Three important aspects of standard uncertainty are worth stressing here:<\/p>\n<ol>\n<li><span style=\"line-height: 1.6em;\">Standard deviation can be calculated also for quantities that are not normally distributed. This enables to obtain for them standard uncertainty estimates.<\/span><\/li>\n<li><span style=\"line-height: 1.6em;\">Furthermore, also uncertainty sources that are systematic by their nature and cannot be evaluated by repeating measurements can still be expressed numerically as standard uncertainty estimates.<\/span><\/li>\n<li><span style=\"line-height: 1.6em;\">Converting different types of uncertainty estimates to standard uncertainty is very important, because as we will see in section 4, most of the calculations in uncertainty evaluation, especially combining the uncertainties corresponding to different uncertainty sources, are carried out using standard uncertainties.<\/span><\/li>\n<\/ol>\n<p>Standard uncertainty of a quantity (in our case volume <em>V<\/em>) expressed in the units of that quantity is sometimes also called absolute standard uncertainty. Standard uncertainty of a quantity divided by the value of that quantity is called <strong>relative standard uncertainty<\/strong>, <em>u<\/em><sub>rel<\/sub> (similarly to eq 1.1). In the case of volume <em>V<\/em>:<\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr align=\"center\">\n<td><img loading=\"lazy\" decoding=\"async\" width=\"119\" height=\"45\" class=\"alignnone wp-image-242\" title=\"valem32-34.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem32-34.png\" alt=\"valem32-34.png\"><\/td>\n<td>(3.4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/sisu.ut.ee\/measurement\/self-test-3-2\/\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-60\" style=\"margin-right: auto; margin-left: auto;\" title=\"selftest.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/selftest.png\" alt=\"selftest.png\" width=\"104\" height=\"41\"><\/a><\/p>\n<div>***\n<div>\n<p>[1] We will see later that standard deviation of measurements repeated under conditions that changer in predefined way (i.e. it is not repeatability) is also extremely useful in uncertainty calculation, as it enables taking a number of uncertainty sources into account simultaneously.<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Brief summary: the lecture explains calculation of mean (Vm) and standard deviation (s). Illustrates again the 68% probability of s. Explains how the standard uncertainty of repeatability u\u00a0(V, REP) can be estimated as standard deviation of parallel measurement results. Stresses &#8230;<\/p>\n","protected":false},"author":14,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"class_list":["post-8","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/pages\/8","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/users\/14"}],"replies":[{"embeddable":true,"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/comments?post=8"}],"version-history":[{"count":3,"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/pages\/8\/revisions"}],"predecessor-version":[{"id":813,"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/pages\/8\/revisions\/813"}],"wp:attachment":[{"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/media?parent=8"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}