{"id":57538,"date":"2025-03-29T12:01:24","date_gmt":"2025-03-29T06:31:24","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=57538"},"modified":"2026-03-03T15:51:38","modified_gmt":"2026-03-03T10:21:38","slug":"fitting-a-normal-curve","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=57538","title":{"rendered":"FITTING A NORMAL CURVE"},"content":{"rendered":"\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-89bfda0d449bddcb20c4924e685a3000\">Aim:<\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\text{To find the mean \u03bc for the following data of size}\\ \\hspace{16cm}\\\\ \\text{43,45,47,44,41,46,48,47,45,48,45,42,41,49,42,46,44,49,47,}\\ \\hspace{15cm}\\\\ \\text{49,41,50,45,45,49,42,46,49,47,46,45,45,49,41,45,49,50,43,}\\ \\hspace{10cm}\\\\ \\text{45,47,40,45,48,43,42,46,43,45,46,42}\\ \\hspace{10cm}\\]<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\">\\[2.\\ \\text{To find the variance}\\ \u03c3^2\\ \\text{and}\\ \\hspace{20cm}\\\\ \\text{standard deviation \u03c3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\text{To fit the normal curve}\\ \\hspace{20cm}\\\\ f(x)\\ =\\ N(\\mu, \\sigma^2)\\ =\\ f(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}} e^{-\\frac{(x &#8211; \\mu)^2}{2\\sigma^2}},\\ -\\ \\infty\\ \\lt\\ x\\ \\lt\\ \\infty\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\text{To calculate the probability}\\ p = P(44 \\lt X \\lt 47)\\ \\hspace{18cm}\\\\ \\text{using the formula}\\ \\int_{X_1}^{X_2} f(x)\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\text{To calculate the probability}\\ p_1\\ =\\ P(44 \\lt X \\lt 47)\\ \\hspace{18cm}\\\\ \\text{using the the probability calculator}\\ \\hspace{15cm}\\\\ \\text{and verify that}\\ p_1\\ =\\ p\\ approximately\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\text{To calculate the number of data points in the interval}\\ \\hspace{17cm}\\\\ \\text{(44,47) using the the formula}\\  n\\ =\\ N_p\\ \\hspace{15cm}\\\\ \\text{and verify it with given data}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-caaadd60b535c837691a2eb0c3ab170e\">Procedure:<\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 1:}\\ \\text{Open Geogebra classic 5 (by double clicking on the icon)}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 2:}\\ \\text{Open spread sheet view}\\ \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 4:}\\ \\text{type the following data one by one in cells A1 to A50}\\ \\hspace{15cm}\\\\ \\text{of column A of the spread sheet}\\ \\hspace{13cm}\\\\ \\text{43,45,47,44,41,46,48,47,45,48,45,42,41,49,42,46,44,49,47,}\\ \\hspace{10cm}\\\\ \\text{49,41,50,45,45,49,42,46,49,47,46,45,45,49,41,45,49,50,43,}\\ \\hspace{10cm}\\\\ \\text{45,47,40,45,48,43,42,46,43,45,46,42}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 5:}\\ \\text{To Calculate the mean of the data by using the}\\ \\hspace{15cm}\\\\ \\text{input command}\\ \\mu\\ =\\  mean (A1:A50)\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 6:}\\ \\text{To Calculate the variance of the data by using the}\\ \\hspace{15cm}\\\\ \\text{input command  \u03c32 = variance (A1:A50)}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 7:}\\ \\text{To Calculate the standard deviation  of the data by using the}\\ \\hspace{15cm}\\\\ \\text{input command  \u03c3 = sqrt(\u03c32)}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 8:}\\ \\text{To fit the normal curve of the given data}\\ \\hspace{15cm}\\\\ f(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}} e^{-\\frac{(x &#8211; \\mu)^2}{2\\sigma^2}}\\ \\hspace{14cm}\\\\ \\text{by using the input bar to type}\\ \\hspace{15cm}\\\\ \\text{f(x) = ((1)\/ (\u03c3 sqrt(2 pi))) e^(-((1)\/(2))(((x &#8211; \u03bc)\/(\u03c3)))^(2))}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 9:}\\ \\text{To place the cursor near x-axis and right click}\\ \\hspace{15cm}\\\\ \\text{then choose the option &#8220;Zoom to fit&#8221;}\\ \\hspace{16cm}\\\\ \\text{do the same for y-axis}\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 10:}\\ \\text{To draw the line x = \u03bc by using the input bar}\\ \\hspace{18cm}\\\\ \\text{to type x = \u03bc}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 11:}\\  \\hspace{25cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i.\\ \\hspace{1cm}\\ \\text{To mark the point of intersection of the line x = \u03bc}\\ \\hspace{15cm}\\\\ \\text{and the normal curve by using the intersection tool}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ \\hspace{1cm}\\ \\text{To mark the point of intersection of the line x = \u03bc}\\ \\hspace{15cm}\\\\ \\text{with the x-axis by using the intersection tool}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Note:}\\ \\text{The points of intersection are the points A and B}\\ \\hspace{18cm}\\\\ \\text{are appearing in the graphics view}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 12:}\\ \\text{Deselect the line x = \u03bc in the algebraic view}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 13:}\\ \\text{Join the points A and B by using the line segment tool}\\ \\hspace{15cm}\\\\ \\text{by using input command  Mean = segment (A,B)}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 14:}\\ \\text{Create sliders for}\\ X_1\\ and\\ X_2\\ \\hspace{15cm}\\\\ \\text{and set the values}\\ X_1\\ =\\ 44\\ and\\ x_2\\ =\\ 47\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 15:}\\ \\text{Calculate the probability}\\ p\\ =\\ \\int_{X_1}^{X_2} f(x)\\ dx\\ \\hspace{15cm}\\\\ \\text{by using the input bar to type p = Integral(f, X_1,X_2)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 16:}\\ \\text{Create a slider for N}\\ \\hspace{18cm}\\\\ \\text{and set the value N=50(the data size)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 17:}\\ \\text{Calculate the theoretical value of number of data}\\ \\hspace{15cm}\\\\ \\text{between 44 and 47 using the command}\\ \\hspace{14cm}\\\\ \\text{n=N p and round &#8211; off it to the nearest integer value.}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Note:}\\ \\text{Count the number of data &#8211; points}\\ x_i\\ such\\ that\\ 44\\ \\leq\\ x_i\\ \\leq\\ 47,\\ \\hspace{15cm}\\\\ \\text{from the given data and compare it with n}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 18:}\\ \\text{Go to view and open Probability calculator}\\ \\hspace{15cm}\\\\ \\text{Choose the Normal under the distribution}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 18:}\\ \\text{Enter the values of \u03bc and \u03c3 (which are 45.34 and 2.69,}\\ \\hspace{15cm}\\\\ \\text{respectively, for the above data}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 19:}\\ \\text{Choose closed interval option []. Enter 44 in the left blank}\\ \\hspace{15cm}\\\\ \\text{and 47 in the right blank and calculate the probability.}\\ \\hspace{14cm}\\\\ \\text{Verify that this probability is approximately equal to}\\ \\hspace{13cm}\\\\ \\text{the probability calculated in the step 15}\\ \\hspace{11cm}\\]<\/div>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-883bedab9864fd79dff46cd9aa765019\">Output:<\/h5>\n\n\n\n<p class=\"has-text-align-center\">Output for Normal curve fitting and data estimation<\/p>\n\n\n\n<table id=\"tablepress-3\" class=\"tablepress tablepress-id-3 tbody-has-connected-cells\">\n<thead>\n<tr class=\"row-1\">\n\t<th colspan=\"2\" class=\"column-1\">Description<\/th><th class=\"column-3\">Value \/ Expression<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td colspan=\"2\" class=\"column-1\">Number of Data Point (N)<\/td><td class=\"column-3\">N = 50<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td colspan=\"2\" class=\"column-1\">Mean (\u03bc)<\/td><td class=\"column-3\">45.34<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td colspan=\"2\" class=\"column-1\">Variance(\u03c3^2)<\/td><td class=\"column-3\">7.22<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td colspan=\"2\" class=\"column-1\">Standard deviation<\/td><td class=\"column-3\">2.69<\/td>\n<\/tr>\n<tr class=\"row-6\">\n\t<td class=\"column-1\">Equation of the Normal curve<\/td><td class=\"column-2\"><\/td><td class=\"column-3\">X=45.34<\/td>\n<\/tr>\n<tr class=\"row-7\">\n\t<td rowspan=\"2\" class=\"column-1\">Probability<br \/>\np = P(44<=X<=47)<\/td><td class=\"column-2\">From the Equation of the Normal curve<\/td><td class=\"column-3\">p = 0.42<\/td>\n<\/tr>\n<tr class=\"row-8\">\n\t<td class=\"column-2\">From the probability calculator<\/td><td class=\"column-3\">p = 0.4222<\/td>\n<\/tr>\n<tr class=\"row-9\">\n\t<td rowspan=\"2\" class=\"column-1\">Number of Data points in the interval 44<=X<=47<\/td><td class=\"column-2\">Theoretical Result from the Fitted Normal curve <\/td><td class=\"column-3\">n=21<\/td>\n<\/tr>\n<tr class=\"row-10\">\n\t<td class=\"column-2\">From the Given Data<\/td><td class=\"column-3\">21<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n<p><iframe width=\"968\" height=\"605\" src=\"https:\/\/www.youtube.com\/embed\/ilW4r85EwiA?list=PLQIom4Rz29vyp5ZiCGuc0-zCEqs8Pvslf\" title=\"Normal Distribution Curve - 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