{"id":1416,"date":"2023-07-26T11:32:19","date_gmt":"2023-07-26T11:32:19","guid":{"rendered":"https:\/\/statorials.org\/pl\/propagacja-bledu\/"},"modified":"2023-07-26T11:32:19","modified_gmt":"2023-07-26T11:32:19","slug":"propagacja-bledu","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/propagacja-bledu\/","title":{"rendered":"Co to jest propagacja b\u0142\u0119d\u00f3w? (definicja \u2013 przyk\u0142ad)"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\"><strong>Propagacja b\u0142\u0119d\u00f3w<\/strong> ma miejsce, gdy mierzysz pewne wielko\u015bci <em>a<\/em> , <em>b<\/em> , <em>c<\/em> , \u2026 z niepewno\u015bci\u0105 <em>\u03b4a<\/em> , <em>\u03b4b<\/em> , <em>\u03b4c<\/em> \u2026 a nast\u0119pnie chcesz obliczy\u0107 inn\u0105 wielko\u015b\u0107 <em>Q<\/em> , korzystaj\u0105c z pomiar\u00f3w <em>a<\/em> , <em>b<\/em> , <em>c<\/em> itd.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Okazuje si\u0119, \u017ce niepewno\u015bci \u03b4 <em>a<\/em> , \u03b4 <em>b<\/em> , <em>\u03b4c<\/em> b\u0119d\u0105 <strong>si\u0119 propagowa\u0107<\/strong> (tzn. \u201erozprzestrzenia\u0107\u201d) a\u017c do niepewno\u015bci <em>Q.<\/em><\/span><\/p>\n<p> <span style=\"color: #000000;\">Aby obliczy\u0107 niepewno\u015b\u0107 <em>Q<\/em> , oznaczon\u0105 \u03b4 <em>Q<\/em> , mo\u017cemy skorzysta\u0107 z nast\u0119puj\u0105cych wzor\u00f3w.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Uwaga:<\/strong> Dla ka\u017cdego z poni\u017cszych wzor\u00f3w przyjmuje si\u0119, \u017ce wielko\u015bci <em>a<\/em> , <em>b<\/em> , <em>c<\/em> itd. zawieraj\u0105 b\u0142\u0119dy <em>losowe<\/em> i <em>nieskorelowane<\/em> .<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Dodawanie lub odejmowanie<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Je\u015bli <em>Q<\/em> = a + b + \u2026 + c \u2013 (x + y + \u2026 + z)<\/span><\/p>\n<p> <span style=\"color: #000000;\">Wtedy \u03b4 <em>Q<\/em> = \u221a <span style=\"border-top: 1px solid black;\">(\u03b4a) <sup>2<\/sup> + (\u03b4b) <sup>2<\/sup> + \u2026 + (\u03b4c) <sup>2<\/sup> + (\u03b4x) <sup>2<\/sup> + (\u03b4y) <sup>2<\/sup> + \u2026 + (\u03b4z) <sup>2<\/sup><\/span><\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Przyk\u0142ad:<\/strong> Za\u0142\u00f3\u017cmy, \u017ce mierzysz d\u0142ugo\u015b\u0107 osoby od ziemi do talii i wynosi 40 cali \u00b1 0,18 cala. Nast\u0119pnie mierzysz d\u0142ugo\u015b\u0107 osoby od pasa do czubka g\u0142owy i wynosi ona 30 cali \u00b1 0,06 cala.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Za\u0142\u00f3\u017cmy, \u017ce nast\u0119pnie u\u017cyjesz tych dw\u00f3ch pomiar\u00f3w do obliczenia ca\u0142kowitego wzrostu osoby. Wysoko\u015b\u0107 zostanie obliczona w nast\u0119puj\u0105cy spos\u00f3b: 40 cali + 30 cali = <strong>70<\/strong> cali. Niepewno\u015b\u0107 tego oszacowania mo\u017cna obliczy\u0107 w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">\u03b4 <em>Q<\/em> = \u221a <span style=\"border-top: 1px solid black;\">(\u03b4a) <sup>2<\/sup> + (\u03b4b) <sup>2<\/sup> + \u2026 + (\u03b4c) <sup>2<\/sup> + (\u03b4x) <sup>2<\/sup> + (\u03b4y) <sup>2<\/sup> + \u2026 + (\u03b4z) <sup>2<\/sup><\/span><\/span><\/li>\n<li> <span style=\"color: #000000;\">\u03b4 <em>Q<\/em> = \u221a <span style=\"border-top: 1px solid black;\">(0,18) <sup>2<\/sup> + (0,06) <sup>2<\/sup><\/span><\/span><\/li>\n<li> <span style=\"color: #000000;\"><em>\u03b4Q<\/em> = <strong>0,1897<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Daje nam to ostateczny pomiar <strong>70 \u00b1 0,1897<\/strong> cala.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Mno\u017cenie lub dzielenie<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Je\u015bli <em>Q<\/em> = (ab\u2026c) \/ (xy\u2026z)<\/span><\/p>\n<p> <span style=\"color: #000000;\">Wtedy <em>\u03b4Q<\/em> = |Q| <span style=\"border-top: 1px solid black;\">* \u221a (\u03b4a\/a) <sup>2<\/sup> + (\u03b4b\/b) <sup>2<\/sup> + \u2026 + (\u03b4c\/c) <sup>2<\/sup> + (\u03b4x\/x) <sup>2<\/sup> + (\u03b4y\/y) <sup>2<\/sup> + \u2026 + (\u03b4z\/z) <sup>2<\/sup><\/span><\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Przyk\u0142ad:<\/strong> Za\u0142\u00f3\u017cmy, \u017ce chcesz zmierzy\u0107 stosunek d\u0142ugo\u015bci elementu <em>a<\/em> do d\u0142ugo\u015bci elementu <em>b<\/em> . D\u0142ugo\u015b\u0107 <em>a<\/em> wynosi 20 cali <span style=\"color: #000000;\">\u00b1 0,34 cala, a d\u0142ugo\u015b\u0107 <em>b<\/em> wynosi 15 cali \u00b1 0,21 cala.<\/span><\/span><\/p>\n<p> <span style=\"color: #000000;\">Stosunek zdefiniowany jako <em>Q<\/em> = <em>a\/b<\/em> zostanie obliczony jako: 20\/15 = <strong>1,333<\/strong> . Niepewno\u015b\u0107 tego oszacowania mo\u017cna obliczy\u0107 w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><em>\u03b4Q<\/em> = |Q| <span style=\"border-top: 1px solid black;\">* \u221a (\u03b4a\/a) <sup>2<\/sup> + (\u03b4b\/b) <sup>2<\/sup> + \u2026 + (\u03b4c\/c) <sup>2<\/sup> + (\u03b4x\/x) <sup>2<\/sup> + (\u03b4y\/y) <sup>2<\/sup> + \u2026 + (\u03b4z\/z) <sup>2<\/sup><\/span><\/span><\/li>\n<li> <span style=\"color: #000000;\"><em>\u03b4Q<\/em> = |1,333| * \u221a <span style=\"border-top: 1px solid black;\">(0,34\/20) <sup>2<\/sup> + (0,21\/15) <sup>2<\/sup><\/span><\/span><\/li>\n<li> <span style=\"color: #000000;\"><em>\u03b4Q<\/em> = <strong>0,0294<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Daje nam to ostateczny stosunek <strong>1,333 \u00b1 0,0294<\/strong> cala.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Zmierzona ilo\u015b\u0107 pomno\u017cona przez dok\u0142adn\u0105 liczb\u0119<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Je\u015bli <em>A<\/em> jest znane dok\u0142adnie i <em>Q<\/em> = <em>A<\/em> x<\/span><\/p>\n<p> <span style=\"color: #000000;\">Wtedy \u03b4 <em>Q<\/em> = |A|\u03b4x<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Przyk\u0142ad:<\/strong> Za\u0142\u00f3\u017cmy, \u017ce mierzysz \u015brednic\u0119 ko\u0142a jako 5 metr\u00f3w \u00b1 0,3 metra. Nast\u0119pnie u\u017cyj tej warto\u015bci do obliczenia obwodu ko\u0142a <em>c = \u03c0d<\/em> .<\/span><\/p>\n<p> <span style=\"color: #000000;\">Obw\u00f3d zostanie obliczony jako <em>c = \u03c0d<\/em> = <em>\u03c0*5<\/em> = <strong>15,708<\/strong> . Niepewno\u015b\u0107 tego oszacowania mo\u017cna obliczy\u0107 w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><em>\u03b4Q<\/em> = |A|\u03b4x<\/span><\/li>\n<li> <span style=\"color: #000000;\">\u03b4 <em>Q<\/em> = | <em>\u03c0<\/em> | * 0,3<\/span><\/li>\n<li> <span style=\"color: #000000;\"><em>\u03b4Q<\/em> = <strong>0,942<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Zatem obw\u00f3d ko\u0142a wynosi <strong>15,708 \u00b1 0,942<\/strong> metra.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Niepewno\u015b\u0107 w pot\u0119dze<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Je\u015bli <em>n<\/em> jest liczb\u0105 dok\u0142adn\u0105 i <em>Q<\/em> = <em>x <sup>n<\/sup><\/em><\/span><\/p>\n<p> <span style=\"color: #000000;\">Wtedy \u03b4 <em>Q<\/em> = | <em>P<\/em> | * | <em>n<\/em> | * (\u03b4x <em>\/x<\/em> )<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Przyk\u0142ad:<\/strong> Za\u0142\u00f3\u017cmy, \u017ce mierzysz bok sze\u015bcianu jako <em>s<\/em> = 2 cale <strong>\u00b1<\/strong> 0,02 cala. Nast\u0119pnie u\u017cyjesz tej warto\u015bci do obliczenia obj\u0119to\u015bci sze\u015bcianu <em>v<\/em> = <em>s<\/em> <sup>3<\/sup> .<\/span><\/p>\n<p> <span style=\"color: #000000;\">Obj\u0119to\u015b\u0107 zostanie obliczona w nast\u0119puj\u0105cy spos\u00f3b: <em>v<\/em> = <em>s<\/em> <sup>3<\/sup> = 2 <sup>3<\/sup> = <strong>8 cali <sup>3<\/sup><\/strong> . Niepewno\u015b\u0107 tego oszacowania mo\u017cna obliczy\u0107 w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">\u03b4 <em>Q<\/em> = | <em>P<\/em> | * | <em>n<\/em> | * (\u03b4x <em>\/x<\/em> )<\/span><\/li>\n<li> <span style=\"color: #000000;\"><em>\u03b4Q<\/em> = |8| * |3| * (.02\/2)<\/span><\/li>\n<li> <span style=\"color: #000000;\"><em>\u03b4Q<\/em> = <strong>0,24<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Zatem obj\u0119to\u015b\u0107 sze\u015bcianu wynosi <strong>8 \u00b1 0,24 cala. <sup>3<\/sup><\/strong> .<\/span><\/p>\n<h3> <strong>Og\u00f3lny wz\u00f3r propagacji b\u0142\u0119du<\/strong><\/h3>\n<p> <span style=\"color: #000000;\">Je\u017celi <em>Q<\/em> = <em>Q(x)<\/em> jest funkcj\u0105 <em>x<\/em> , to og\u00f3lny wz\u00f3r na propagacj\u0119 b\u0142\u0119du mo\u017cna zdefiniowa\u0107 w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong><em>\u03b4Q = |dQ<\/em> \/ <em>dX<\/em> |\u03b4x<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Pami\u0119taj, \u017ce rzadko b\u0119dziesz musia\u0142 wyprowadza\u0107 te wzory od zera, ale pomocna mo\u017ce by\u0107 znajomo\u015b\u0107 og\u00f3lnego wzoru u\u017cytego do ich wyprowadzenia.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Propagacja b\u0142\u0119d\u00f3w ma miejsce, gdy mierzysz pewne wielko\u015bci a , b , c , \u2026 z niepewno\u015bci\u0105 \u03b4a , \u03b4b , \u03b4c \u2026 a nast\u0119pnie chcesz obliczy\u0107 inn\u0105 wielko\u015b\u0107 Q , korzystaj\u0105c z pomiar\u00f3w a , b , c itd. Okazuje si\u0119, \u017ce niepewno\u015bci \u03b4 a , \u03b4 b , \u03b4c b\u0119d\u0105 si\u0119 propagowa\u0107 (tzn. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-1416","post","type-post","status-publish","format-standard","hentry","category-przewodnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Co to jest propagacja b\u0142\u0119d\u00f3w? 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