{"id":1054,"date":"2021-04-10T23:08:10","date_gmt":"2021-04-10T22:08:10","guid":{"rendered":"https:\/\/agripedia.ch\/terminologie\/?page_id=1054"},"modified":"2023-06-26T13:08:44","modified_gmt":"2023-06-26T12:08:44","slug":"anhang-12-statistik","status":"publish","type":"page","link":"https:\/\/agripedia.ch\/terminologie\/anhaenge\/anhang-12-statistik\/","title":{"rendered":"Anhang 12: Statistik"},"content":{"rendered":"<div class=\"fusion-fullwidth fullwidth-box fusion-builder-row-1 fusion-flex-container nonhundred-percent-fullwidth non-hundred-percent-height-scrolling\" style=\"--awb-border-radius-top-left:0px;--awb-border-radius-top-right:0px;--awb-border-radius-bottom-right:0px;--awb-border-radius-bottom-left:0px;--awb-flex-wrap:wrap;\" ><div class=\"fusion-builder-row fusion-row fusion-flex-align-items-flex-start fusion-flex-content-wrap\" style=\"max-width:1144px;margin-left: calc(-4% \/ 2 );margin-right: calc(-4% \/ 2 );\"><div class=\"fusion-layout-column fusion_builder_column fusion-builder-column-0 fusion_builder_column_1_1 1_1 fusion-flex-column\" style=\"--awb-bg-size:cover;--awb-width-large:100%;--awb-margin-top-large:0px;--awb-spacing-right-large:1.92%;--awb-margin-bottom-large:20px;--awb-spacing-left-large:1.92%;--awb-width-medium:100%;--awb-order-medium:0;--awb-spacing-right-medium:1.92%;--awb-spacing-left-medium:1.92%;--awb-width-small:100%;--awb-order-small:0;--awb-spacing-right-small:1.92%;--awb-spacing-left-small:1.92%;\"><div class=\"fusion-column-wrapper fusion-column-has-shadow fusion-flex-justify-content-flex-start fusion-content-layout-column\"><div class=\"fusion-title title fusion-title-1 fusion-sep-none fusion-title-text fusion-title-size-two\"><h2 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:40;line-height:1.5;\">Anhang 12: Statistik<\/h2><\/div><div class=\"fusion-title title fusion-title-2 fusion-sep-none fusion-title-text fusion-title-size-four\"><h4 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:24;--minFontSize:24;line-height:1.5;\">Mittelwerte und Streuung<\/h4><\/div><div class=\"fusion-text fusion-text-1\"><p>Beispiel: Merkmale Weizenfl\u00e4che und Weizenertrag, erhoben auf 8 Betrieben (<a href=\"https:\/\/agripedia.ch\/terminologie\/begriffe\/?term=Reihe,%20statistische\">statistische Reihe<\/a>).<\/p>\n<\/div>\n<div class=\"table-1\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n<thead>\n<tr valign=\"top\">\n<th style=\"text-align: center;\">Betrieb<br \/>\nNr.<\/th>\n<th style=\"text-align: center;\">Weizen Fl\u00e4che<br \/>\nha<\/th>\n<th style=\"text-align: center;\">Weizenertrag<br \/>\ndt\/Betrieb<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">1,0<\/td>\n<td style=\"text-align: center;\">50<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">2<\/td>\n<td style=\"text-align: center;\">1,2<\/td>\n<td style=\"text-align: center;\">60<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">3<\/td>\n<td style=\"text-align: center;\">3,0<\/td>\n<td style=\"text-align: center;\">132<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">4<\/td>\n<td style=\"text-align: center;\">2,0<\/td>\n<td style=\"text-align: center;\">96<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">5<\/td>\n<td style=\"text-align: center;\">1,0<\/td>\n<td style=\"text-align: center;\">66<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">6<\/td>\n<td style=\"text-align: center;\">0,7<\/td>\n<td style=\"text-align: center;\">35<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">7<\/td>\n<td style=\"text-align: center;\">1,2<\/td>\n<td style=\"text-align: center;\">54<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">8<\/td>\n<td style=\"text-align: center;\">1,0<\/td>\n<td style=\"text-align: center;\">52<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">Summe<\/td>\n<td style=\"text-align: center;\">11,1<\/td>\n<td style=\"text-align: center;\">545<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">\\(xy^*\\)<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">49,10<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">\\(\\bar{x}\\)<\/td>\n<td style=\"text-align: center;\">1,39<\/td>\n<td style=\"text-align: center;\">68,13<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">\\(v_x\\)<\/td>\n<td style=\"text-align: center;\">0,57<\/td>\n<td style=\"text-align: center;\">973,27<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">\\(s_x\\)<\/td>\n<td style=\"text-align: center;\">0,75<\/td>\n<td style=\"text-align: center;\">31,20<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">\\(vk_x\\)<\/td>\n<td style=\"text-align: center;\">0,54<\/td>\n<td style=\"text-align: center;\">0,46<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"fusion-text fusion-text-2\"><ul>\n<li>\u00a0<strong>Einfaches arithmetisches Mittel<\/strong> (Durchschnitt, Mittel, Mittelwert):<br \/>\nSumme einer statistischen Reihe, dividiert durch die Zahl der Reihenglieder.In mathematischer Schreibweise f\u00fcr das Merkmal x:<\/p>\n<p class=\"\">\\(\\bar = \\frac = \\sum\\limits_^n (X_i)\/n\\)<\/p>\n<p>Ergebnisse Beispiel (e.a.M.):<br \/>\n\u2013 Weizenfl\u00e4che: \\(\\bar\\) = 1,39 ha<br \/>\n\u2013 Weizenertrag: \\(\\bar\\) = 68,13 dt\/ha<\/li>\n<li><strong>Varianz<\/strong>: Mittlere quadratische Abweichung aller Einzelwerte einer statistischen Reihe von zugeh\u00f6rigen einfachen arithmetischen Mitteln.<br \/>\nErgebnisse Beispiel:<br \/>\n\u2013 Weizenfl\u00e4che: \\(v_x\\) = 0,57 ha<br \/>\n\u2013 Weizenertrag: \\(v_x\\) = 973 dt\/ha<\/p>\n<p class=\"\">\\(v_x = \\frac \\sum\\limits_^n (x_i\u00a0 \u2013 \\bar)^2\\)<\/p>\n<\/li>\n<li><strong>Standardabweichung<\/strong>: Wurzel der Varianz.<br \/>\nIn mathematischer Schreibweise f\u00fcr das Merkmal x:<\/p>\n<p class=\"\">\\(s_x = \\sqrt{\\frac \\sum\\limits_^n (x_i\u00a0 \u2013 \\bar)^2} \\)<\/p>\n<p>Die Definitionsformel kann durch arithmetische Umformungen in eine Form gebracht werden, die rechentechnisch einfacher zu handhaben ist:<\/p>\n<p class=\"\">\\(s_x = \\sqrt{\\frac \\sum\\limits_^n (x_i ^2) \u2013 n_\\bar^2} \\)<\/p>\n<p>Ergebnisse Beispiel:<br \/>\n\u2013 Weizenfl\u00e4che: \\(s_x\\) = 0,75 ha<br \/>\n\u2013 Weizenertrag: \\(s_x\\) = 9,73 dt\/ha<\/li>\n<li><strong>Variationskoeffizient<\/strong>: Verh\u00e4ltnis der Standardabweichung zum arithmetischen Mittel.\n<p class=\"\">\\(Vk_x = \\frac{\\bar}\\)<\/p>\n<p>Ergebnisse Beispiel:<br \/>\n\u2013 Weizenfl\u00e4che: \\(vk_x\\) = 0,54 ha<br \/>\n\u2013 Weizenertrag: \\(vk_x\\) = 0,56 dt\/ha<\/li>\n<li><strong>Gewogenes arithmetisches Mittel<\/strong>: Mit Gewichtungsfaktoren korrigierte Summe der Mess- bzw. Beobachtungswerte einer statistischen Reihe dividiert durch die Summe der Gewichtungsfaktoren.<br \/>\nIn mathematischer Schreibweise f\u00fcr das Merkmal x und den Gewichtungsfaktor g:<\/p>\n<p class=\"\">\\(\\bar^* = \\frac = \\sum\\limits_^n g_ix_i \/ \\sum\\limits_^n g_i\\)<\/p>\n<p>Bei einer Gewichtung geht man davon aus, dass nicht jede Messung bzw. Beobachtung die gleiche Bedeutung aufweist. In der betriebswirtschaftlichen Praxis ist das gewogene arithmetische Mittel von Verh\u00e4ltniszahlen bedeutsam. Will man beispielsweise ber\u00fccksichtigen, dass die Weizenertr\u00e4ge pro Fl\u00e4cheneinheit (Merkmal y) auf unterschiedlich grossen Fl\u00e4chen (Merkmal x) erzielt worden sind, so kann obige Formel in folgenden Ausdruck \u00fcberf\u00fchrt werden:<\/p>\n<p class=\"\">\\(\\bar\/\\bar^* = \\frac \\)<\/p>\n<p>Diese Beziehung kann algebraisch vereinfacht werden. Man erh\u00e4lt f\u00fcr das gewogene arithmetische Mittel von Verh\u00e4ltniszahlen die Formel:<\/p>\n<p class=\"\">\\(\\bar\/\\bar^* = \\frac = \\sum\\limits_^n y_i \/ \\sum\\limits_^n x_i \\)<\/p>\n<p>Ergebnis Beispiel: Weizenertrag: \\(xy^*\\) = 49,1 dt\/ha<\/li>\n<\/ul>\n<\/div><div class=\"fusion-title title fusion-title-3 fusion-sep-none fusion-title-text fusion-title-size-four\"><h4 class=\"fusion-title-heading title-heading-left fusion-responsive-typography-calculated\" style=\"margin:0;--fontSize:24;--minFontSize:24;line-height:1.5;\">Median, Modus, Quartilen<\/h4><\/div><div class=\"fusion-text fusion-text-3\"><ul>\n<li><strong>Median<\/strong>: Wert, der bei der Abz\u00e4hlung einer Reihe von der Gr\u00f6sse nach geordneten Merkmalswerten (z.B. Messreihe) in der Mitte liegt.<\/li>\n<li><strong>Modus<\/strong>: H\u00e4ufigster Wert innerhalb einer statistischen Reihe.<\/li>\n<li><strong>Quartilen<\/strong>: Aufteilung einer der Gr\u00f6sse nach geordneten statistischen Reihe in vier Teile.<\/li>\n<\/ul>\n<p>Beispiel Weizenfl\u00e4che (ha)<\/p>\n<table style=\"border: 1px solid #333;\" cellspacing=\"0\" cellpadding=\"0\">\n<thead>\n<tr>\n<th style=\"border-bottom: 2px solid #333 !important;\" colspan=\"2\">Unteres Viertel<br \/>\n(Unteres Quartil)<\/th>\n<th style=\"border-bottom: 2px solid #333 !important; border-left: 1px solid #333 !important;\" colspan=\"4\">Mittlere 50 %<\/th>\n<th style=\"border-bottom: 2px solid #333 !important; border-left: 1px solid #333 !important;\" colspan=\"2\">Oberes Viertel<br \/>\n(Oberes Quartil)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td height=\"4\"><\/td>\n<td><\/td>\n<td style=\"border-left: 1px solid #333;\"><\/td>\n<td><\/td>\n<td style=\"border-left: 1px solid #333;\"><\/td>\n<td><\/td>\n<td style=\"border-left: 1px solid #333;\"><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">x<sub>6<\/sub><\/td>\n<td style=\"text-align: center;\" width=\"53\">x<sub>1<\/sub><\/td>\n<td style=\"border-left: 1px solid #333333; text-align: center;\" width=\"53\">x<sub>5<\/sub><\/td>\n<td style=\"text-align: center;\" width=\"53\">x<sub>8<\/sub><\/td>\n<td style=\"border-left: 1px solid #333333; text-align: center;\" width=\"45\">x<sub>2<\/sub><\/td>\n<td style=\"text-align: center;\" width=\"45\">x<sub>7<\/sub><\/td>\n<td style=\"border-left: 1px solid #333333; text-align: center;\" width=\"49\">x<sub>4<\/sub><\/td>\n<td style=\"text-align: center;\" width=\"55\">x<sub>3<\/sub><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" width=\"44\" height=\"13\">0,7<\/td>\n<td style=\"text-align: center;\" width=\"53\">1,0<\/td>\n<td style=\"border-left: 1px solid #333333; text-align: center;\" width=\"53\">1,0<\/td>\n<td style=\"text-align: center;\" width=\"53\">1,0<\/td>\n<td style=\"border-left: 1px solid #333333; text-align: center;\" width=\"45\">1,2<\/td>\n<td style=\"text-align: center;\" width=\"45\">1,2<\/td>\n<td style=\"border-left: 1px solid #333333; text-align: center;\" width=\"49\">2,0<\/td>\n<td style=\"text-align: center;\" width=\"55\">3,0<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td width=\"44\" height=\"1\"><\/td>\n<td width=\"53\"><\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"53\"><\/td>\n<td width=\"53\"><\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"45\"><\/td>\n<td width=\"45\"><\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"49\"><\/td>\n<td width=\"55\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" colspan=\"2\" width=\"101\" height=\"13\">0,85<\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"53\"><\/td>\n<td style=\"text-align: center;\" colspan=\"2\" width=\"102\">1,1<\/td>\n<td width=\"45\"><\/td>\n<td style=\"border-left: 1px solid #333333; text-align: center;\" colspan=\"2\" width=\"108\">2,5<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" colspan=\"2\" width=\"101\" height=\"13\">Mittelwert<\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"53\"><\/td>\n<td style=\"text-align: center;\" colspan=\"2\" width=\"102\">Median<\/td>\n<td width=\"45\"><\/td>\n<td style=\"text-align: center; border-left: 1px solid #333;\" colspan=\"2\" width=\"108\">Mittelwert<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td width=\"44\" height=\"4\"><\/td>\n<td width=\"53\"><\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"53\"><\/td>\n<td width=\"53\"><\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"45\"><\/td>\n<td width=\"45\"><\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"49\"><\/td>\n<td width=\"55\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td width=\"44\" height=\"13\"><\/td>\n<td style=\"text-align: center;\" colspan=\"2\" width=\"109\">1,0<\/td>\n<td width=\"53\"><\/td>\n<td style=\"border-left: 1px solid #333;\" width=\"45\"><\/td>\n<td style=\"text-align: center;\" colspan=\"2\" width=\"98\">1,6<\/td>\n<td width=\"55\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" colspan=\"4\" width=\"214\" height=\"12\">untere Quartilgrenze<\/td>\n<td style=\"text-align: center; 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id=\"vorname\" value=\"\"  class=\"fusion-form-input\" required=\"true\" aria-required=\"true\" placeholder=\"Ihr Vorname *\" data-holds-private-data=\"false\" minlength=\"0\"\/><\/div><div class=\"fusion-form-field fusion-form-text-field fusion-form-label-above\" style=\"\" data-form-id=\"1491\"><div class=\"fusion-form-label-wrapper\"><label for=\"organisation\">Organisation<\/label><\/div><div class=\"fusion-form-input-with-icon\"><i class=\"awb-form-icon fa-building fas\"><\/i><input type=\"text\" autocomplete=\"off\" name=\"organisation\" id=\"organisation\" value=\"\"  class=\"fusion-form-input\" placeholder=\"Ihre Organisation \" data-holds-private-data=\"false\" minlength=\"0\"\/><\/div><\/div><div class=\"fusion-form-field fusion-form-email-field fusion-form-label-above\" style=\"\" data-form-id=\"1491\"><div class=\"fusion-form-label-wrapper\"><label for=\"e-mail\">E-Mail-Adresse <abbr class=\"fusion-form-element-required\" title=\"required\">*<\/abbr><\/label><\/div><div class=\"fusion-form-input-with-icon\"><i class=\"awb-form-icon fa-at fas\"><\/i><input type=\"email\" autocomplete=\"off\" name=\"e-mail\" id=\"e-mail\" value=\"\"  class=\"fusion-form-input\" required=\"true\" aria-required=\"true\" placeholder=\"Ihre E-Mail-Adresse *\" data-holds-private-data=\"false\"\/><\/div><\/div><div class=\"fusion-form-field fusion-form-text-field fusion-form-label-above\" data-form-id=\"1373\">\n\t<div id=\"agri-term-wrapper\" class=\"fusion-form-label-wrapper\">\n\t\t<label for=\"Begriff\">\n\t\t\tBegriff \n\t\t<\/label>\n\t<\/div>\n<\/div>\n<script>\n\t\/\/ input element\n    const input = document.createElement('input');\n    input.setAttribute('id', 'terme');\n    input.setAttribute('name', 'Begriff');\n    input.setAttribute('type', 'text');\n    input.setAttribute('value', getTerm());\n    input.setAttribute('class', 'fusion-form-input');\n    input.setAttribute('aria-required', 'true');\n    input.setAttribute('required', 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