{"id":13,"date":"2024-04-03T23:09:23","date_gmt":"2024-04-03T20:09:23","guid":{"rendered":"https:\/\/sisu.ut.ee\/measurement\/42-combining-uncertainty-components-combined-standard-uncertainty\/"},"modified":"2024-06-19T15:45:05","modified_gmt":"2024-06-19T12:45:05","slug":"42-combining-uncertainty-components-combined-standard-uncertainty","status":"publish","type":"page","link":"https:\/\/sisu.ut.ee\/measurement\/42-combining-uncertainty-components-combined-standard-uncertainty\/","title":{"rendered":"4.2. Calculating the combined standard uncertainty"},"content":{"rendered":"<p>The uncertainty components that were quantified in the previous lecture are now combined into the <strong>combined standard uncertainty<\/strong> (<em>u<\/em><sub>c<\/sub>) \u2013 standard uncertainty that takes into account contributions from all important uncertainty sources by combining the respective uncertainty components. The concept of <strong>indirect measurement<\/strong> \u2013 whereby the value of the <strong>output quantity<\/strong> (measurement result) is found by some function (model) from several <strong>input quantities<\/strong> \u2013 is introduced and explained. The majority of chemical measurements are indirect measurements. The general case of combining the uncertainty components into combined standard uncertainty as well as several specific cases are presented and explained.<\/p>\n<p><span style=\"line-height: 1.6em;\">The first video lecture explains in a simple way how the uncertainty components are combined in the particular example of pipetting. The second video lecture presents the general overview of combining the uncertainty components.<\/span><\/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>Combining the uncertainty components into the combined standard uncertainty in the case of pipetting<br><\/strong><a style=\"line-height: 1.6em;\" href=\"http:\/\/www.uttv.ee\/naita?id=17556\" target=\"_blank\" rel=\"noopener\">http:\/\/www.uttv.ee\/naita?id=17556<\/a><\/h4>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.youtube.com\/watch?v=S5v58VQ4zSg\" target=\"_blank\" rel=\"noopener\">https:\/\/www.youtube.com\/watch?v=S5v58VQ4zSg<\/a><\/p>\n<p>In all cases where combined standard uncertainty is calculated from uncertainty components all the uncertainty components have to be converted to standard uncertainties.<\/p>\n<p><span style=\"line-height: 1.6em;\">In the example of pipetting the combined standard uncertainty is calculated from the uncertainty components found in the previous section as follows:<\/span><\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td><img loading=\"lazy\" decoding=\"async\" width=\"423\" height=\"81\" class=\"alignnone wp-image-245\" title=\"valem42-44b.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-44b.png\" alt=\"valem42-44b.png\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-44b.png 423w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-44b-300x57.png 300w\" sizes=\"auto, (max-width: 423px) 100vw, 423px\"><\/td>\n<td>(4.4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This is the typical way of calculating combined standard uncertainty if all the uncertainty components refer to the same quantity and are expressed in the same units. It is often used in the case of <strong>direct measurements<\/strong> \u2013 measurements whereby the measurement instrument (pipette in this case) gives immediately the value of the result, without further calculations needed.<\/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;\">\u00a0\u00a0<strong>Combining the uncertainty components into the combined standard uncertainty: simple cases and the general case<\/strong><br><a href=\"http:\/\/www.uttv.ee\/naita?id=17826\" target=\"_blank\" rel=\"noopener\">http:\/\/www.uttv.ee\/naita?id=17826<\/a><\/h4>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FJ4hn9LgGmw\" target=\"_blank\" rel=\"noopener\">https:\/\/www.youtube.com\/watch?v=FJ4hn9LgGmw<\/a><\/p>\n<p>An indirect measurement is one where the <strong>output quantity<\/strong> (result) is found ba calculation (using a <strong>model equation<\/strong>) from several input quantities. A typical example is titration. In case of titration with 1:1 mole ratio the analyte concentration in the sample solution <em>C<\/em><sub>S<\/sub> (the output quantity) is expressed by the input quantities \u2013 volume of sample solution taken for titration (<em>V<\/em><sub>S<\/sub>), titrant concentration (<em>C<\/em><sub>T<\/sub>) and titrant volume consumed for titration (<em>V<\/em><sub>T<\/sub>) \u2013 as follows:<\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td><img loading=\"lazy\" decoding=\"async\" width=\"102\" height=\"52\" class=\"alignnone wp-image-246\" title=\"valem42-45.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-45.png\" alt=\"valem42-45.png\"><\/td>\n<td>(4.5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"line-height: 1.6em;\">In the general case if the output quantity <\/span><em style=\"line-height: 1.6em;\">Y<\/em><span style=\"line-height: 1.6em;\"> is found from input quantities <\/span><em style=\"line-height: 1.6em;\">X<\/em><sub>1<\/sub><span style=\"line-height: 1.6em;\">, <\/span><em style=\"line-height: 1.6em;\">X<\/em><sub>2<\/sub><span style=\"line-height: 1.6em;\">, \u2026 <\/span><em style=\"line-height: 1.6em;\">X<\/em><sub>n<\/sub><span style=\"line-height: 1.6em;\"> according to some function <\/span><em style=\"line-height: 1.6em;\">F<\/em><span style=\"line-height: 1.6em;\"> as follows<\/span><\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td><img loading=\"lazy\" decoding=\"async\" width=\"178\" height=\"27\" class=\"alignnone wp-image-247\" title=\"valem42-46.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-46.png\" alt=\"valem42-46.png\"><\/td>\n<td>\u00a0(4.6)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\" align=\"right\">then the combined standard uncertainty of the output quantity <em>u<\/em><sub>c<\/sub>(<em>y<\/em>) can be expressed via the standard uncertainties of the input quantities <em>u<\/em>(<em>x<\/em><sub>i<\/sub>) as follows:<\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td><img loading=\"lazy\" decoding=\"async\" width=\"465\" height=\"68\" class=\"alignnone wp-image-248\" title=\"valem42-47.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-47.png\" alt=\"valem42-47.png\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-47.png 465w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-47-300x44.png 300w\" sizes=\"auto, (max-width: 465px) 100vw, 465px\"><\/td>\n<td>(4.7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\" align=\"right\">The terms <img loading=\"lazy\" decoding=\"async\" width=\"95\" height=\"59\" class=\"alignnone wp-image-249\" style=\"vertical-align: middle;\" title=\"valem42-47a.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-47a.png\" alt=\"valem42-47a.png\">\u00a0 are the uncertainty components. The terms\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"34\" height=\"52\" class=\"alignnone wp-image-250\" style=\"vertical-align: middle;\" title=\"valem42-47b.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-47b.png\" alt=\"valem42-47b.png\">are partial derivatives.\u00a0At first sight the eq 4.7 may seem very complex but it is in fact not too difficult to use \u2013 the uncertainty components can be calculated numerically using the Kragten\u2019s spreadsheet method (as is demonstrated in section 9.7).<\/p>\n<p>In specific cases simpler equations hold. If the output quantity is expressed via the input quantities as follows<\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td><img loading=\"lazy\" decoding=\"async\" width=\"170\" height=\"26\" class=\"alignnone wp-image-251\" title=\"valem42-48.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-48.png\" alt=\"valem42-48.png\"><\/td>\n<td>\u00a0(4.8)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td>then\u00a0<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" width=\"296\" height=\"34\" class=\"alignnone wp-image-252\" title=\"valem42-49.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-49.png\" alt=\"valem42-49.png\"><\/td>\n<td>\u00a0(4.9)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\" align=\"right\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/p>\n<p>Importantly, irrespective of whether the input quantities are added or subtracted, the squared standard uncertainties under the square root are always added.<\/p>\n<p>This way of combining uncertainty components is in principle the same as used above for the case of pipetting.<\/p>\n<p>If the measurement model is<\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td><img loading=\"lazy\" decoding=\"async\" width=\"101\" height=\"52\" class=\"alignnone wp-image-253\" title=\"valem42-491.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-491.png\" alt=\"valem42-491.png\"><\/td>\n<td>\u00a0(4.10)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td>then\u00a0<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" width=\"418\" height=\"73\" class=\"alignnone wp-image-254\" title=\"valem42-492.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-492.png\" alt=\"valem42-492.png\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-492.png 418w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-492-300x52.png 300w\" sizes=\"auto, (max-width: 418px) 100vw, 418px\"><\/td>\n<td>\u00a0(4.11)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\" align=\"right\">As can be seen, here it is the relative standard uncertainties that are combined and the squared summing gives us the relative combined standard uncertainty of the output quantity. The absolute combined standard uncertainty of the output quantity is found as follows:<\/p>\n<table class=\"table table-hover\" border=\"0\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td><img loading=\"lazy\" decoding=\"async\" width=\"438\" height=\"73\" class=\"alignnone wp-image-255\" title=\"valem42-493.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-493.png\" alt=\"valem42-493.png\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-493.png 438w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/valem42-493-300x50.png 300w\" sizes=\"auto, (max-width: 438px) 100vw, 438px\"><\/td>\n<td>\u00a0(4.12)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\" align=\"right\"><a href=\"https:\/\/sisu.ut.ee\/measurement\/self-test-4-2\/\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-60\" 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<p style=\"text-align: left;\" align=\"right\">The file used in second video can be downloaded from here.\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/p>\n<p>\u00a0<\/p>\n\n\n<div class=\"wp-block-group attached-files-group is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-861358d0-43a9-4a9b-a046-8906b3f9a042\" href=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/18\/combining_u_components.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">combining_u_components.pdf<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The uncertainty components that were quantified in the previous lecture are now combined into the combined standard uncertainty (uc) \u2013 standard uncertainty that takes into account contributions from all important uncertainty sources by combining the respective uncertainty components. The concept &#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-13","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/pages\/13","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=13"}],"version-history":[{"count":3,"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/pages\/13\/revisions"}],"predecessor-version":[{"id":666,"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/pages\/13\/revisions\/666"}],"wp:attachment":[{"href":"https:\/\/sisu.ut.ee\/measurement\/wp-json\/wp\/v2\/media?parent=13"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}