{"id":43617,"date":"2023-09-12T21:01:16","date_gmt":"2023-09-12T15:31:16","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=43617"},"modified":"2025-09-13T18:44:53","modified_gmt":"2025-09-13T13:14:53","slug":"unit-iv-statistics-new-scheme-2023","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=43617","title":{"rendered":"UNIT \u2013 IV STATISTICS (NEW SCHEME \u2013 2023)"},"content":{"rendered":"\n<p>SYLLABUS:&nbsp; Statistical data \u2013 Ungrouped data \u2013 Grouped data \u2013 Discrete data \u2013 Continuous data \u2013 Arithmetic mean \u2013 Variance \u2013 Standard deviation \u2013 Fitting a straight line using the method of least squares \u2013 Simple problems \u2013 Engineering applications (not for examinations).<\/p>\n\n\n\n<p class=\"has-text-align-center has-vivid-red-color has-text-color has-link-color has-medium-font-size wp-elements-22326fe742a91873f20f4738ca1ac9c3\">Arithmetic Mean:<\/p>\n\n\n\n<p class=\"has-vivid-red-color has-text-color has-link-color wp-elements-0667b3a5aed7dad40c2afd5500ac025c\"><strong>Data<\/strong><\/p>\n\n\n\n<p class=\"has-black-color has-text-color has-link-color wp-elements-d37f9084f49868262b613edd8f3d95a6\">Data is a collection of information assembled by observations, research measurements, or analysis.  This data is divided into Ungrouped and Grouped.<\/p>\n\n\n\n<p class=\"has-vivid-red-color has-text-color has-link-color wp-elements-4cd249bdac06258a35ca7d15a7c5e671\"><strong>Ungrouped data<\/strong><\/p>\n\n\n\n<p>Ungrouped data is a type of data that has not been organized or grouped into specific&nbsp;categories. It is also known as raw data or unorganized data. It is simply a collection of observations or measurements without any structure or organization.<\/p>\n\n\n\n<p class=\"has-vivid-red-color has-text-color has-link-color wp-elements-a33ac1fb887947c2df3f48904e2cc16e\">To find Arithmetic Mean <\/p>\n\n\n\n<p class=\"has-small-font-size\">      The arithmetic mean of n observations x<sub>1, <\/sub>x<sub>2, <\/sub>. . . . x<sub>n <\/sub>is defined as<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ \\frac{x_1\\ +\\ x_2\\ +\\ &#8230;&#8230;&#8230;.\\ +\\ x_n}{n}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{\\Sigma_{i=1}^{n} x_i}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1\\ .}\\ \\color{red}{Calculate\\ the\\ arithmetic mean\\ of}\\  14,\\ 26,\\ 28,\\ 20,\\ 32,\\ 30\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 6\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 14\\ +\\ 26\\ +\\ +\\ 28\\ +\\ 20\\ +\\ 32\\ +\\ 30\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 150\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ \\frac{\\Sigma x_i}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{150}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2\\ .}\\ \\color{red}{Calculate\\ the\\ arithmetic mean\\ of}\\  14,\\ 26,\\ 28,\\ 20,\\ 35,\\ and\\ 33\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 6\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 14\\ +\\ 26\\ +\\ +\\ 28\\ +\\ 20\\ +\\ 35\\ +\\ 33\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 156\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ \\frac{\\Sigma x_i}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{156}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ 26\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3\\ .}\\ \\color{red}{Calculate\\ the\\ arithmetic\\ mean\\ of}\\  10,\\ 12,\\ 14,\\ 16\\ and\\ 18\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024,\\ October\\ 2024,\\ (Supp)\\ June\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 5\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 10\\ +\\ 12\\ +\\ 14\\ +\\ 16\\ +\\ 18\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 70\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ \\frac{\\Sigma x_i}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{70}{5}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ 14\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4\\ .}\\ The\\ arithmetic\\ mean\\ of\\ 10\\ numbers\\ is\\ 20\\ ,\\  \\color{red}{find\\ the\\ sum\\ of\\ the}\\ \\hspace{10cm}\\]\\[\\color{red}{values}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\ October\\ 2024\\ Supp(June)\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 10\\ and\\ \\bar{x}\\ =\\ 20\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x} \\ =\\ \\frac{\\Sigma X}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20\\ =\\ \\frac{\\Sigma X}{10}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\Sigma X\\ =\\ 200\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5\\ .}\\ The\\ arithmetic\\ mean\\ of\\ 10\\ numbers\\ is\\ 30\\ ,\\  \\color{red}{find\\ the\\ total\\ of\\ the}\\ \\hspace{10cm}\\]\\[\\color{red}{numbers.}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 10\\ and\\ \\bar{x}\\ =\\ 30\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x} \\ =\\ \\frac{\\Sigma X}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[30\\ =\\ \\frac{\\Sigma X}{10}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\Sigma X\\ =\\ 300\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6\\ .}\\ The\\ arithmetic\\ mean\\ of\\ 6\\ values\\ is\\ 45\\ .\\  If\\ 3\\ is\\ added\\ to\\ each\\ of\\ the\\ \\hspace{5cm}\\\\ numbers,\\ \\color{red}{then\\ find\\ the\\ arithmetic\\ mean\\ of\\ the\\ new\\ set\\ of\\ values.}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 6\\ and\\ \\bar{x}\\ =\\ 45\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x} \\ =\\ \\frac{\\Sigma X}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[45\\ =\\ \\frac{\\Sigma X}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\Sigma X\\ =\\ 270\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[corrected\\ \\Sigma X\\ =\\ 270\\ +\\ 18\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[corrected\\ \\Sigma X\\ =\\ 288\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[correct\\ mean\\ =\\ \\frac{288}{6}\\ =\\ 48\\]<\/div>\n\n\n\n<p class=\"has-vivid-red-color has-text-color has-link-color wp-elements-2791157384a988e8220e5c2238dfdc55\">Grouped data<\/p>\n\n\n\n<p>In ungrouped form, a discrete data can be represented as&nbsp;<em>x<\/em>1,&nbsp;<em>x<\/em>2,&nbsp;<em>x<\/em>3, &#8230;..&nbsp;<em>xn&nbsp;<\/em>where each&nbsp;<em>xi&nbsp;<\/em>is an element of the data and&nbsp;<em>n&nbsp;<\/em>is the number of elements in the data. The number of times an element&nbsp;<em>xi&nbsp;<\/em>occurs in a data is called the frequency of the element and it is denoted by&nbsp;<em>fi<\/em><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_i\\]<\/mathml><\/td><td><mathml>\\[f_i\\]<\/mathml><\/td><td><mathml>\\[f_i\\ x_i\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_1\\]<\/mathml><\/td><td><mathml>\\[f_1\\]<\/mathml><\/td><td><mathml>\\[f_1\\ x_1\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_2\\]<\/mathml><\/td><td><mathml>\\[f_2\\]<\/mathml><\/td><td><mathml>\\[f_2\\ x_2\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_3\\]<\/mathml><\/td><td><mathml>\\[f_3\\]<\/mathml><\/td><td><mathml>\\[f_3\\ x_3\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8230;<\/td><td>&#8230;<\/td><td>&#8230;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_m\\]<\/mathml><\/td><td><mathml>\\[f_m\\]<\/mathml><\/td><td><mathml>\\[f_m\\ x_m\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td><mathml>\\[N\\ =\\ \\Sigma f_i\\] <\/mathml><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The arithmetic mean of a grouped frequency distribution is defined as&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ \\frac{\\Sigma f_i\\ x_i}{N}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<p>Where   f<sub>i<\/sub>   is frequency,  x<sub>i<\/sub>   = 1,2,.. n is the mid values of the each of the class intervals and N = \u03a3f<sub>i<\/sub>  , total frequency.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7\\ .}\\ Find\\ the\\ arithmetic\\ mean\\ of\\ the\\ following\\ frequency\\ distribution\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>x<\/td><td>1<\/td><td>2<\/td><td>3<\/td><td>4<\/td><td>5<\/td><td>6<\/td><td>7<\/td><\/tr><tr><td>f<\/td><td>5<\/td><td>9<\/td><td>12<\/td><td>17<\/td><td>14<\/td><td>10<\/td><td>6<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>x<\/td><td>f<\/td><td>fx<\/td><\/tr><tr><td>1<\/td><td>5<\/td><td>5<\/td><\/tr><tr><td>2<\/td><td>9<\/td><td>18<\/td><\/tr><tr><td>3<\/td><td>12<\/td><td>36<\/td><\/tr><tr><td>4<\/td><td>17<\/td><td>68<\/td><\/tr><tr><td>5<\/td><td>14<\/td><td>70<\/td><\/tr><tr><td>6<\/td><td>10<\/td><td>60<\/td><\/tr><tr><td>7<\/td><td>6<\/td><td>42<\/td><\/tr><tr><td><\/td><td>N = 73<\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 299\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\bar{x}\\ =\\ \\frac{\\Sigma f_i\\ x_i}{N}\\ =\\ \\frac{299}{73}\\ =\\ 4.09\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/qn8jG7_vqC0\" title=\"Arithmetic mean - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p>A common way to group data is by creating a frequency distribution table. This involves dividing the data into intervals or classes, and then counting how many observations fall into each interval. This can be useful for identifying patterns and trends in the data, and for creating graphs such as histograms to visualize the distribution of the data.<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Class Interval<\/td><td class=\"has-text-align-center\" data-align=\"center\">No. of elements<br>in the Class Interval<br>f<sub>i<\/sub><\/td><td class=\"has-text-align-center\" data-align=\"center\">Mid-value <br>      x<sub>i<\/sub><\/td><td class=\"has-text-align-center\" data-align=\"center\">f<sub>i<\/sub>x<sub>i<\/sub><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[I_1\\ -\\ I_2\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[f_1\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_1\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[f_1\\ x_1\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[I_2\\ -\\ I_3\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[f_2\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_2\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[f_2\\ x_2\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[I_3\\ -\\ I_4\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[f_3\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_3\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[f_3\\ x_3\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8230;<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8230;<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8230;<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8230;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[I_m\\ -\\ I_n\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[f_n\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[x_n\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[f_n\\ x_n\\]<\/mathml><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[N\\ =\\ \\Sigma\\ f_i\\]<\/mathml><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><mathml>\\[\\Sigma f_i\\ x_i\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ \\frac{\\Sigma f_i\\ x_i}{N}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8\\ .}\\ Find\\ the\\ arithmetic\\ mean\\ of\\ the\\ following\\ data\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Class interval<\/td><td class=\"has-text-align-center\" data-align=\"center\">0-10<\/td><td class=\"has-text-align-center\" data-align=\"center\">10-20<\/td><td class=\"has-text-align-center\" data-align=\"center\">20-30<\/td><td class=\"has-text-align-center\" data-align=\"center\">30-40<\/td><\/tr><tr><td>Frequency<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2024,\\ Supp\\ June\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>Marks<\/td><td>No. of students<br>f<sub>i<\/sub><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mid-value <br>      x<sub>i<\/sub><\/td><td>f<sub>i<\/sub>x<sub>i<\/sub><\/td><\/tr><tr><td>0-10<\/td><td>2<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td>10<\/td><\/tr><tr><td>10-20<\/td><td>5<\/td><td class=\"has-text-align-left\" data-align=\"left\">15<\/td><td>75<\/td><\/tr><tr><td>20-30<\/td><td>1<\/td><td class=\"has-text-align-left\" data-align=\"left\">25<\/td><td>25<\/td><\/tr><tr><td>30-40<\/td><td>3<\/td><td class=\"has-text-align-left\" data-align=\"left\">35<\/td><td>105<\/td><\/tr><tr><td><\/td><td>N = 11<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 215\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\bar{x}\\ =\\ \\frac{\\Sigma f_i\\ x_i}{N}\\ =\\ \\frac{215}{11}\\ =\\ 19.54\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 9\\ .}\\ Find\\ the\\ arithmetic\\ mean\\ of\\ the\\ following\\ data\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Class interval<\/td><td>0-10<\/td><td>10-20<\/td><td>20-30<\/td><td>30-40<\/td><td>40-50<\/td><td>50-60<\/td><\/tr><tr><td>Frequency<\/td><td>2<\/td><td>6<\/td><td>9<\/td><td>7<\/td><td>4<\/td><td>2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>Marks<\/td><td>No. of students<br>f<sub>i<\/sub><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mid-value <br>      x<sub>i<\/sub><\/td><td>f<sub>i<\/sub>x<sub>i<\/sub><\/td><\/tr><tr><td>0-10<\/td><td>2<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td>10<\/td><\/tr><tr><td>10-20<\/td><td>6<\/td><td class=\"has-text-align-left\" data-align=\"left\">15<\/td><td>90<\/td><\/tr><tr><td>20-30<\/td><td>9<\/td><td class=\"has-text-align-left\" data-align=\"left\">25<\/td><td>225<\/td><\/tr><tr><td>30-40<\/td><td>7<\/td><td class=\"has-text-align-left\" data-align=\"left\">35<\/td><td>245<\/td><\/tr><tr><td>40-50<\/td><td>4<\/td><td class=\"has-text-align-left\" data-align=\"left\">45<\/td><td>180<\/td><\/tr><tr><td>50-60<\/td><td>2<\/td><td class=\"has-text-align-left\" data-align=\"left\">55<\/td><td>110<\/td><\/tr><tr><td><\/td><td>N = 30<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 860\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\bar{x}\\ =\\ \\frac{\\Sigma f_i\\ x_i}{N}\\ =\\ \\frac{860}{30}\\ =\\ 28.67\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10\\ .}\\ Find\\ the\\ arithmetic\\ mean\\ of\\ the\\ following\\ data\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Class-interval<\/td><td>0-10<\/td><td>10-20<\/td><td>20-30<\/td><td>30-40<\/td><td>40-50<\/td><td>50-60<\/td><td>60-70<\/td><\/tr><tr><td>Frequency<\/td><td>3<\/td><td>5<\/td><td>16<\/td><td>18<\/td><td>12<\/td><td>7<\/td><td>4<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>Marks<\/td><td>No. of students<br>f<sub>i<\/sub><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mid-value <br>      x<sub>i<\/sub><\/td><td>f<sub>i<\/sub>x<sub>i<\/sub><\/td><\/tr><tr><td>0-10<\/td><td>3<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td>15<\/td><\/tr><tr><td>10-20<\/td><td>5<\/td><td class=\"has-text-align-left\" data-align=\"left\">15<\/td><td>75<\/td><\/tr><tr><td>20-30<\/td><td>16<\/td><td class=\"has-text-align-left\" data-align=\"left\">25<\/td><td>400<\/td><\/tr><tr><td>30-40<\/td><td>18<\/td><td class=\"has-text-align-left\" data-align=\"left\">35<\/td><td>630<\/td><\/tr><tr><td>40-50<\/td><td>12<\/td><td class=\"has-text-align-left\" data-align=\"left\">45<\/td><td>540<\/td><\/tr><tr><td>50-60<\/td><td>7<\/td><td class=\"has-text-align-left\" data-align=\"left\">55<\/td><td>385<\/td><\/tr><tr><td>60-70<\/td><td>4<\/td><td class=\"has-text-align-left\" data-align=\"left\">65<\/td><td>260<\/td><\/tr><tr><td><\/td><td>N = 65<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 2305\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\bar{x}\\ =\\ \\frac{\\Sigma f_i\\ x_i}{N}\\ =\\ \\frac{2305}{65}\\ =\\ 35.46\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11\\ .}\\ Find\\ the\\ arithmetic\\ mean\\ of\\ the\\ following\\ data\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Class-interval<\/td><td>0-10<\/td><td>10-20<\/td><td>20-30<\/td><td>30-40<\/td><td>40-50<\/td><td>50-60<\/td><td>60-70<\/td><\/tr><tr><td>Frequency<\/td><td>11<\/td><td>14<\/td><td>15<\/td><td>20<\/td><td>15<\/td><td>13<\/td><td>12<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>Marks<\/td><td>No. of students<br>f<sub>i<\/sub><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mid-value <br>      x<sub>i<\/sub><\/td><td>f<sub>i<\/sub>x<sub>i<\/sub><\/td><\/tr><tr><td>0-10<\/td><td>11<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td>55<\/td><\/tr><tr><td>10-20<\/td><td>14<\/td><td class=\"has-text-align-left\" data-align=\"left\">15<\/td><td>210<\/td><\/tr><tr><td>20-30<\/td><td>15<\/td><td class=\"has-text-align-left\" data-align=\"left\">25<\/td><td>375<\/td><\/tr><tr><td>30-40<\/td><td>20<\/td><td class=\"has-text-align-left\" data-align=\"left\">35<\/td><td>700<\/td><\/tr><tr><td>40-50<\/td><td>15<\/td><td class=\"has-text-align-left\" data-align=\"left\">45<\/td><td>675<\/td><\/tr><tr><td>50-60<\/td><td>13<\/td><td class=\"has-text-align-left\" data-align=\"left\">55<\/td><td>715<\/td><\/tr><tr><td>60-70<\/td><td>12<\/td><td class=\"has-text-align-left\" data-align=\"left\">65<\/td><td>780<\/td><\/tr><tr><td><\/td><td>N = 100<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 3510\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\bar{x}\\ =\\ \\frac{\\Sigma f_i\\ x_i}{N}\\ =\\ \\frac{3510}{100}\\ =\\ 35.10\\]<\/div>\n\n\n\n<p class=\"has-vivid-red-color has-text-color has-link-color has-medium-font-size wp-elements-bd5c26d59cedb306254d469feae836c9\"><strong>Variance<\/strong>:<\/p>\n\n\n\n<p>The mean of the squared differences of all elements from their arithmetic mean is called the&nbsp;variance.&nbsp;It is denoted by <strong>\u03c3\u00b2<\/strong>&nbsp; and is defined as<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma^2\\ =\\ \\frac{\\Sigma x_i^2}{n}\\ -\\ (\\frac{\\Sigma x_i}{n})^2\\ (For\\ ungrouped\\ Data)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma^2\\ =\\ \\frac{\\Sigma x_i^2\\ f_i}{n}\\ -\\ (\\frac{\\Sigma x_i\\ f_i}{n})^2\\ (For\\ grouped\\ Data)\\]<\/div>\n\n\n\n<p class=\"has-vivid-red-color has-text-color has-link-color has-medium-font-size wp-elements-2f67ae58d3a6979495c8a6fbf69abeef\"><strong>Standard Deviation<\/strong><\/p>\n\n\n\n<p>A positive square root of the variance is called standard deviation. It is denoted by <strong>\u03c3<\/strong>.&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma x_i^2}{n}\\ -\\ (\\frac{\\Sigma x_i}{n})^2}\\ (For\\ ungrouped\\ Data)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ x_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ x_i}{N}})^2\\ \\ (For\\ grouped\\ Data)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12\\ .}\\ \\text{If the standard deviation of a data is 6.8,}\\ \\color{red}{find\\ its\\ variance}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ \\sigma\\ =\\ 6.8\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ W.\\ K.\\ T\\ variance\\ =\\ \\sigma^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ (6.8)^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{variance\\ =\\ 46.24}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 13\\ .}\\ \\text{If the standard deviation of a data is 8,}\\ \\color{red}{find\\ its\\ variance}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ \\sigma\\ =\\ 8\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ W.\\ K.\\ T\\ variance\\ =\\ \\sigma^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ (8)^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{variance\\ =\\ 64}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 14\\ .}\\ \\text{If the variance of a data is 100,}\\ \\color{red}{then\\ find\\ its\\ standard\\ deviation}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ variance\\ =\\ 100\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ W.\\ K.\\ T\\ variance\\ =\\ \\sigma^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{variance}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{100}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ 10\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{standard\\ deviation\\ =\\ 10}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 15\\ .}\\ Find\\ the\\ variance\\ of\\ -3,\\ -2,\\ -1,\\ 0,\\ 1,\\ 2,\\ 3\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 8\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table class=\"has-fixed-layout\" style=\"border-width:1px\"><tbody><tr><td>x<sub>i<\/sub><\/td><td>x<sub>i<\/sub><sup> 2<\/sup><\/td><\/tr><tr><td>-3<\/td><td>9<\/td><\/tr><tr><td>-2<\/td><td>4<\/td><\/tr><tr><td>-1<\/td><td>1<\/td><\/tr><tr><td>0<\/td><td>0<\/td><\/tr><tr><td>1<\/td><td>1<\/td><\/tr><tr><td>2<\/td><td>4<\/td><\/tr><tr><td>3<\/td><td>9<\/td><\/tr><tr><td><mathml>\\[\\Sigma x_i\\ =\\ 0\\]<\/mathml><\/td><td><mathml>\\[\\Sigma x_i^2\\ =\\ 28\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\n\\[\\sigma^2\\ =\\ \\frac{\\Sigma x_i^2}{n}\\ -\\ (\\frac{\\Sigma x_i}{n})^2\\ (For\\ ungrouped\\ Data)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{7}\\ (28)\\ =\\ 4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma^2\\ =\\ 4\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/94fmd91u5D0\" title=\"Standard Deviation - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; we\nb-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 16\\ .}\\ Find\\ the\\ standard\\ deviation\\ of\\ 2,\\ 7,\\ 3,\\ 12\\ and\\ 9\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2024\\ Supp(June)\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 5\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table class=\"has-fixed-layout\" style=\"border-width:1px\"><tbody><tr><td>x<sub>i<\/sub><\/td><td>x<sub>i<\/sub><sup> 2<\/sup><\/td><\/tr><tr><td>2<\/td><td>4<\/td><\/tr><tr><td>7<\/td><td>49<\/td><\/tr><tr><td>3<\/td><td>9<\/td><\/tr><tr><td>12<\/td><td>144<\/td><\/tr><tr><td>9<\/td><td>81<\/td><\/tr><tr><td><mathml>\\[\\Sigma x_i\\ =\\ 33\\]<\/mathml><\/td><td><mathml>\\[\\Sigma x_i^2\\ =\\ 287\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma x_i^2}{n}\\ -\\ (\\frac{\\Sigma x_i}{n})^2}\\ (For\\ ungrouped\\ Data)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{\\frac{287}{5}\\ -\\ (\\frac{33}{5})^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{57.4\\ -\\ (6.6)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{57.4\\ -\\ 43.56}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{13.84}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ 3.72\\ (approximately)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 17\\ .}\\ \\text{Calculate the standard deviation for the following data}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Items<\/td><td>5<\/td><td>15<\/td><td>25<\/td><td>35<\/td><\/tr><tr><td>Frequency<\/td><td>2<\/td><td>1<\/td><td>1<\/td><td>3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023,\\ April\\ 2024\\ April\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ n\\ =\\ 4\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ x_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ x_i}{N}})^2\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>x<sub>i<\/sub><\/td><td>f<sub>i <\/sub><\/td><td><mathml>\\[f_i\\ x_i\\]<\/mathml><\/td><td>x<sub>i <\/sub><sup>2<\/sup><\/td><td>f<sub>i<\/sub> x<sub>i<\/sub><sup> 2<\/sup><\/td><\/tr><tr><td>5<\/td><td>2<\/td><td>10<\/td><td>25<\/td><td>50<\/td><\/tr><tr><td>15<\/td><td>1<\/td><td>15<\/td><td>225<\/td><td>225<\/td><\/tr><tr><td>25<\/td><td>1<\/td><td>25<\/td><td>625<\/td><td>625<\/td><\/tr><tr><td>35<\/td><td>3<\/td><td>105<\/td><td>1225<\/td><td>3675<\/td><\/tr><tr><td><\/td><td>N=7<\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 155\\]<\/mathml><\/td><td><\/td><td><mathml>\\[\\Sigma f_i\\ d_i^2\\ =\\ 4575\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ x_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ x_i}{N}})^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{\\frac{4575}{7}\\ -\\ (\\frac{155}{7}})^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{653.57\\ -\\ 490.18}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{163.3904}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ 12.78\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 18\\ .}\\ Find\\ the\\ standard\\ deviation\\ of\\ the\\ following\\ data:\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Class interval<\/td><td>4 &#8211; 8<\/td><td>8 &#8211; 12<\/td><td>12 &#8211; 16<\/td><td>16 &#8211; 20<\/td><\/tr><tr><td>frequency<\/td><td>3<\/td><td>6<\/td><td>4<\/td><td>7<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2024\\ June(Supp)\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ x_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ x_i}{N})^2}, \\  where\\ N\\ =\\ \\Sigma {f_i}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>Marks<\/td><td>f<sub>i<\/sub><\/td><td>x<sub>i <\/sub><\/td><td>f<sub>i<\/sub>x<sub>i<\/sub><\/td><td>x<sub>i<\/sub><sup> 2<\/sup><\/td><td>f<sub>i<\/sub>xi<sup>2<\/sup><\/td><\/tr><tr><td>4 -8<\/td><td>3<\/td><td>6<\/td><td>18<\/td><td>36<\/td><td>108<\/td><\/tr><tr><td>8 -12<\/td><td>6<\/td><td>10<\/td><td>60<\/td><td>100<\/td><td>600<\/td><\/tr><tr><td>12 &#8211; 16<\/td><td>4<\/td><td>14<\/td><td>56<\/td><td>196<\/td><td>784<\/td><\/tr><tr><td>16 &#8211; 20<\/td><td>7<\/td><td>18<\/td><td>126<\/td><td>324<\/td><td>2268<\/td><\/tr><tr><td><\/td><td>N = 20<\/td><td><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 260\\]<\/mathml><\/td><td><\/td><td><mathml>\\[\\Sigma\\ f_i\\ x_i^2\\ =\\ 3760\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ x_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ x_i}{N})^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{\\frac{3760}{20}\\ -\\ (\\frac{-260}{20})^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{188\\ -\\ (13)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{188\\ -\\ 169}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{19}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ 4.36\\ (approximately)\\]<\/div>\n\n\n\n<p class=\"has-text-align-center has-vivid-red-color has-text-color has-link-color has-medium-font-size wp-elements-029b2390ae721ac3ff6f7dc6e9e31886\">CURVE FITTING<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Fitting a straight line using the method of least squares<\/h4>\n\n\n\n<h4 class=\"wp-block-heading\">Statement of principle of least squares<\/h4>\n\n\n\n<p>Let y = f(x) be the functional relationship sought between the variables (x,y).&nbsp; Let (x<sub>i<\/sub>,y<sub>i<\/sub>), <sub>i = 1,2,3,&#8230;,n<\/sub>.&nbsp; be the observed set of values of the variables (x,y).&nbsp; d<sub>i<\/sub> = y<sub>i<\/sub> \u2013 f(x<sub>i<\/sub>) which is the difference between the observed value of y and the value of y.&nbsp; The principle of least squares states that the parameters involved in y = f(x) should be chosen in such a way that \u03a3<img decoding=\"async\" src=\"blob:https:\/\/yanamtakshashila.com\/d272751e-13fa-434b-a63e-14f283873036\" alt=\"equation.pdf\"> is minimum.<\/p>\n\n\n\n<p>Let us consider the fitting of the straight line y = ax + b &#8212; (1) to a set of \u2018n\u2019 points&nbsp;<\/p>\n\n\n\n<p>(x<sub>i<\/sub>,y<sub>i<\/sub>), <sub>i = 1,2,3,&#8230;,n<\/sub>.&nbsp; Here a and b are arbitrary constants and can be found by using Normal equations.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i\\ +\\ nb\\ =\\ \\Sigma y_i\\ &#8212;&#8212;&#8211;\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i^2\\ +\\ b\\ \\Sigma\\ x_i\\ =\\ \\Sigma x_i\\ y_i\\ &#8212;&#8212;&#8211;\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 19\\ .}\\ Write\\ down\\ the\\ normal\\ equations\\ to\\ fit\\ a\\ straight\\ line\\ y\\ =\\ a\\ x\\ +\\ b.\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023,\\ April\\ 2024,\\ October\\ 2024,\\ April\\ 25,\\ Supp(June)\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p>                      The Normal equations are<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i\\ +\\ nb\\ =\\ \\Sigma y_i\\ &#8212;&#8212;&#8211;\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i^2\\ +\\ b\\ \\Sigma\\ x_i\\ =\\ \\Sigma x_i\\ y_i\\ &#8212;&#8212;&#8211;\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 20\\ .}\\ Fit\\ a\\ straight\\ line\\ to\\ the\\ following\\ data\\ by\\ the\\ method\\ of\\ least\\ squares.\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>x<\/td><td>0<\/td><td>1<\/td><td>2<\/td><td>3<\/td><td>4<\/td><\/tr><tr><td>y<\/td><td>1<\/td><td>1<\/td><td>3<\/td><td>4<\/td><td>6<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024,\\ April\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ the\\ line\\ be\\ y\\ =\\ a\\ x\\ +\\ b\\ &#8212;&#8212;&#8211;\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Normal\\ equations\\ are\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i\\ +\\ nb\\ =\\ \\Sigma y_i\\ &#8212;&#8212;&#8211;\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i^2\\ +\\ b\\ \\Sigma\\ x_i\\ =\\ \\Sigma x_i\\ y_i\\ &#8212;&#8212;&#8211;\\ (3)\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>x<sub>i<\/sub><\/td><td>y<sub>i <\/sub><\/td><td>x<sub>i<\/sub><sup> 2<\/sup><\/td><td>x<sub>i<\/sub> y<sub>i<\/sub><\/td><\/tr><tr><td>0<\/td><td>1<\/td><td>0<\/td><td>0<\/td><\/tr><tr><td>1<\/td><td>1<\/td><td>1<\/td><td>1<\/td><\/tr><tr><td>2<\/td><td>3<\/td><td>4<\/td><td>6<\/td><\/tr><tr><td>3<\/td><td>4<\/td><td>9<\/td><td>12<\/td><\/tr><tr><td>4<\/td><td>6<\/td><td>16<\/td><td>24<\/td><\/tr><tr><td><mathml>\\[\\Sigma x_i\\ =\\ 10\\]<\/mathml><\/td><td><mathml>\\[\\Sigma y_i\\ =\\ 15\\]<\/mathml><\/td><td><mathml>\\[\\Sigma {x_i}^2\\ =\\ 30\\]<\/mathml><\/td><td><mathml>\\[\\Sigma x_i\\ y_i\\ =\\ 43\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(2)\\ becomes\\ a\\ (10)\\ +\\ 5b\\ =\\ 15\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 3\\ &#8212;&#8212;&#8211;\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(3)\\ becomes\\ a\\ (30)\\ +\\ b\\ (10)\\ =\\ 43\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 4.3\\ &#8212;&#8212;&#8211;\\ (5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Solving\\ (4)\\ and\\ (5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 4.3\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ =\\ 1.3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ a\\ =\\ 1.3\\ in\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(1.3)\\ +\\ b\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.6\\ +\\ b\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 3\\ -\\ 2.6\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 0.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Eqn\\ (1)\\ becomes\\ y\\ =\\ 1.3\\ x\\ +\\ 0.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 21\\ .}\\ Using\\ the\\ method\\ of\\ least\\ square\\ ,\\  fit\\ a\\ straight\\ line\\ to\\ the\\ following\\ data:\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>x<\/td><td>0<\/td><td>1<\/td><td>2<\/td><td>3<\/td><td>4<\/td><\/tr><tr><td>y<\/td><td>10<\/td><td>14<\/td><td>19<\/td><td>26<\/td><td>30<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ the\\ line\\ be\\ y\\ =\\ a\\ x\\ +\\ b\\ &#8212;&#8212;&#8211;\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Normal\\ equations\\ are\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i\\ +\\ nb\\ =\\ \\Sigma y_i\\ &#8212;&#8212;&#8211;\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i^2\\ +\\ b\\ \\Sigma\\ x_i\\ =\\ \\Sigma x_i\\ y_i\\ &#8212;&#8212;&#8211;\\ (3)\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>x<sub>i<\/sub><\/td><td>y<sub>i <\/sub><\/td><td>x<sub>i<\/sub><sup> 2<\/sup><\/td><td>x<sub>i<\/sub> y<sub>i<\/sub><\/td><\/tr><tr><td>0<\/td><td>10<\/td><td>0<\/td><td>0<\/td><\/tr><tr><td>1<\/td><td>14<\/td><td>1<\/td><td>14<\/td><\/tr><tr><td>2<\/td><td>19<\/td><td>4<\/td><td>38<\/td><\/tr><tr><td>3<\/td><td>26<\/td><td>9<\/td><td>78<\/td><\/tr><tr><td>4<\/td><td>30<\/td><td>16<\/td><td>120<\/td><\/tr><tr><td><mathml>\\[\\Sigma x_i\\ =\\ 10\\]<\/mathml><\/td><td><mathml>\\[\\Sigma y_i\\ =\\ 99\\]<\/mathml><\/td><td><mathml>\\[\\Sigma {x_i}^2\\ =\\ 30\\]<\/mathml><\/td><td><mathml>\\[\\Sigma x_i\\ y_i\\ =\\ 250\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(2)\\ becomes\\ a\\ (10)\\ +\\ 5b\\ =\\ 99\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 19.8\\ &#8212;&#8212;&#8211;\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(3)\\ becomes\\ a\\ (30)\\ +\\ b\\ (10)\\ =\\ 250\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 25\\ &#8212;&#8212;&#8211;\\ (5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Solving\\ (4)\\ and\\ (5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 19.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 25\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ =\\ 5.2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ a\\ =\\ 5.2\\ in\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(5.2)\\ +\\ b\\ =\\ 19.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.4\\ +\\ b\\ =\\ 19.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 19.8\\ -\\ 10.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Eqn\\ (1)\\ becomes\\ y\\ =\\ 5.2\\ x\\ +\\ 9.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 22\\ .}\\ Fit\\ a\\ straight\\ line\\ to\\ the\\ following\\ data:\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>x<\/td><td>0<\/td><td>1<\/td><td>2<\/td><td>3<\/td><td>4<\/td><\/tr><tr><td>y<\/td><td>10<\/td><td>14<\/td><td>19<\/td><td>26<\/td><td>31<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2024,\\ Supp (June)\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ the\\ line\\ be\\ y\\ =\\ a\\ x\\ +\\ b\\ &#8212;&#8212;&#8211;\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Normal\\ equations\\ are\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i\\ +\\ nb\\ =\\ \\Sigma y_i\\ &#8212;&#8212;&#8211;\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i^2\\ +\\ b\\ \\Sigma\\ x_i\\ =\\ \\Sigma x_i\\ y_i\\ &#8212;&#8212;&#8211;\\ (3)\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>x<sub>i<\/sub><\/td><td>y<sub>i <\/sub><\/td><td>x<sub>i<\/sub><sup> 2<\/sup><\/td><td>x<sub>i<\/sub> y<sub>i<\/sub><\/td><\/tr><tr><td>0<\/td><td>10<\/td><td>0<\/td><td>0<\/td><\/tr><tr><td>1<\/td><td>14<\/td><td>1<\/td><td>14<\/td><\/tr><tr><td>2<\/td><td>19<\/td><td>4<\/td><td>38<\/td><\/tr><tr><td>3<\/td><td>26<\/td><td>9<\/td><td>78<\/td><\/tr><tr><td>4<\/td><td>31<\/td><td>16<\/td><td>124<\/td><\/tr><tr><td><mathml>\\[\\Sigma x_i\\ =\\ 10\\]<\/mathml><\/td><td><mathml><span class=\"MathJax_Preview\" style=\"color: inherit;\"><span class=\"MJXp-math MJXp-display\" id=\"MJXp-Span-5245\"><span class=\"MJXp-mi\" id=\"MJXp-Span-5246\">\u03a3<span class=\"MJXp-mi MJXp-italic\" id=\"MJXp-Span-5248\" style=\"margin-right: 0.05em;\">y<\/span><span class=\"MJXp-mi MJXp-italic MJXp-script\" id=\"MJXp-Span-5249\" style=\"vertical-align: -0.4em;\">i<\/span>&nbsp;<\/span><span class=\"MJXp-mo\" id=\"MJXp-Span-5251\" style=\"margin-left: 0.333em; margin-right: 0.333em;\">=<\/span><span class=\"MJXp-mtext\" id=\"MJXp-Span-5252\">100&nbsp;<\/span><\/span><\/span><span class=\"mjx-chtml MJXc-display MJXc-processing\">11<span id=\"MathJax-Element-241-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\"><\/span><\/span><script>\\Sigma y_i\\ =\\ 99<\/script><\/mathml><\/td><td><mathml>\\[\\Sigma {x_i}^2\\ =\\ 30\\]<\/mathml><\/td><td><mathml>\\[\\Sigma x_i\\ y_i\\ =\\ 254\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(2)\\ becomes\\ a\\ (10)\\ +\\ 5b\\ =\\ 100\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 20\\ &#8212;&#8212;&#8211;\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(3)\\ becomes\\ a\\ (30)\\ +\\ b\\ (10)\\ =\\ 254\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 25.4\\ &#8212;&#8212;&#8211;\\ (5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Solving\\ (4)\\ and\\ (5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 20\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 25.4\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ =\\ 5.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ a\\ =\\ 5.4\\ in\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(5.4)\\ +\\ b\\ =\\ 20\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.8\\ +\\ b\\ =\\ 20\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 20\\ -\\ 10.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 9.2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Eqn\\ (1)\\ becomes\\ y\\ =\\ 5.4\\ x\\ +\\ 9.2\\]<\/div>\n","protected":false},"excerpt":{"rendered":"<p>SYLLABUS:&nbsp; Statistical data \u2013 Ungrouped data \u2013 Grouped data \u2013 Discrete data \u2013 Continuous data \u2013 Arithmetic mean \u2013 Variance \u2013 Standard deviation \u2013 Fitting a straight line using the method of least squares \u2013 Simple problems \u2013 Engineering applications (not for examinations). Arithmetic Mean: Data Data is a collection of information assembled by observations, [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":52224,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788674],"tags":[],"class_list":["post-43617","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-basic-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>UNIT \u2013 IV STATISTICS (NEW SCHEME \u2013 2023) - YANAMTAKSHASHILA<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/yanamtakshashila.com\/?p=43617\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"UNIT \u2013 IV STATISTICS (NEW SCHEME \u2013 2023) - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"SYLLABUS:&nbsp; Statistical data \u2013 Ungrouped data \u2013 Grouped data \u2013 Discrete data \u2013 Continuous data \u2013 Arithmetic mean \u2013 Variance \u2013 Standard deviation \u2013 Fitting a straight line using the method of least squares \u2013 Simple problems \u2013 Engineering applications (not for examinations). 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