{"id":1935,"date":"2023-10-02T07:27:34","date_gmt":"2023-10-02T07:27:34","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=1935"},"modified":"2023-10-02T07:28:26","modified_gmt":"2023-10-02T07:28:26","slug":"week-10-ss1-first-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-10-ss1-first-term-further-mathematics-notes\/","title":{"rendered":"Week 10 &#8211; SS1 First Term Further Mathematics Notes"},"content":{"rendered":"<p>\u00a0<br \/>\n\u00a0<strong>WEEK TEN<br \/>\n<\/strong><strong>TOPIC: STATISTICS: MEASURES OF CENTRAL TENDENCY<br \/>\n<\/strong><strong>MEASUREMENT OF CENTRAL TENDENCY<br \/>\n<\/strong><\/p>\n<ul>\n<li>Mean, Median and Mode of ungrouped<strong><br \/>\n\t\t\t\t<\/strong><\/li>\n<li>Mean, Median and Mode of grouped data<strong><br \/>\n\t\t\t\t<\/strong><\/li>\n<\/ul>\n<p>Measures of central tendency: This is a measure of how the data are centrally placed. The three commonest measures of position, depending on the information required are the arithmetic mean, median and the mode.<br \/>\n<strong>MEAN: <\/strong>It is most widely used measure and sometimes called the arithmetical averages. The mean of the number x<sub>1<\/sub>, x<sub>2<\/sub>, x<sub>3<\/sub>, x<sub>4<\/sub> \u2026\u2026\u2026\u2026\u2026\u2026\u2026x<sub>n<\/sub> is given by:<br \/>\n   X = \u2211x   where \u2211x is the sum of all items.     n = number of items<br \/>\n           n<br \/>\nWhen the data involves frequency; mean = \u2211fx\/\u2211f<br \/>\nExamples:<br \/>\n1. Calculate the mean of the numbers 15, 17, 19, 21, 23, 25, 27, 29.<br \/>\n<strong>Solution:<br \/>\n<\/strong>Mean (x) = 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29    = 176  = 22<br \/>\n                                                 8                                  8<br \/>\n2. The table shows the number of suitcases possessed by a group of travelers.<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>No of suitcases\u00a0<\/td>\n<td>          0\u00a0<\/td>\n<td>            1\u00a0<\/td>\n<td>             2\u00a0<\/td>\n<td>            3\u00a0<\/td>\n<td>            4\u00a0<\/td>\n<td>            5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>Travelers\u00a0<\/td>\n<td>          2\u00a0<\/td>\n<td>            7\u00a0<\/td>\n<td>             7\u00a0<\/td>\n<td>            2\u00a0<\/td>\n<td>            3\u00a0<\/td>\n<td>           9\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Calculate the mean to the nearest whole number.<\/p>\n<p>\u00a0<strong>Solution:<br \/>\n<\/strong><strong><br \/>\n\t\t\t<\/strong>Mean ( x ) = \u2211 fx\/\u2211 f    =  84\/30 =   2.8  = 3                                                                                                                                                                                                                            <strong><br \/>\n\t\t\t<\/strong><\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td><strong>X<\/strong>\u00a0<\/td>\n<td><strong>F<\/strong>\u00a0<\/td>\n<td><strong>FX<\/strong>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>       0\u00a0<\/td>\n<td>2\u00a0<\/td>\n<td>0\u00a0<\/td>\n<\/tr>\n<tr>\n<td>1\u00a0<\/td>\n<td>7\u00a0<\/td>\n<td>7\u00a0<\/td>\n<\/tr>\n<tr>\n<td>2\u00a0<\/td>\n<td>7\u00a0<\/td>\n<td>14\u00a0<\/td>\n<\/tr>\n<tr>\n<td>3\u00a0<\/td>\n<td>2\u00a0<\/td>\n<td>6\u00a0<\/td>\n<\/tr>\n<tr>\n<td>4\u00a0<\/td>\n<td>3\u00a0<\/td>\n<td>12\u00a0<\/td>\n<\/tr>\n<tr>\n<td>5\u00a0<\/td>\n<td>9\u00a0<\/td>\n<td>45\u00a0<\/td>\n<\/tr>\n<tr>\n<td><strong>Total<\/strong>\u00a0<\/td>\n<td><strong>30<\/strong>\u00a0<\/td>\n<td><strong>84<\/strong>\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<strong>EVALUATION<em><br \/>\n\t\t\t\t<\/em><\/strong>1. Calculate the mean of the numbers 37.5, 25.5, 30.5, 41.5, 52.5, 28.5.<br \/>\n2. Calculate the mean score of the scores represented in the table below. <\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Scores\u00a0<\/td>\n<td>           10\u00a0<\/td>\n<td>              12\u00a0<\/td>\n<td>             14\u00a0<\/td>\n<td>             16\u00a0<\/td>\n<td>             18\u00a0<\/td>\n<\/tr>\n<tr>\n<td>No of Students\u00a0<\/td>\n<td>            5\u00a0<\/td>\n<td>              2\u00a0<\/td>\n<td>              3\u00a0<\/td>\n<td>              4\u00a0<\/td>\n<td>              4\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<br \/>\n\u00a0<strong>Mode:<br \/>\n<\/strong>The mode of a distribution is the value of the variable which occurs most often in the distribution. It is also possible for a distribution to have more than one mode, if there were more than one item having the highest frequency.<br \/>\n<strong>Example:<\/strong><\/p>\n<ol>\n<li>Find the mode of the data 5, 4, 8, 9, 6, 8, 9, 3, 8. The mode is 8 (it appears 3 times more than others)\n<\/li>\n<\/ol>\n<p><strong>Median:<br \/>\n<\/strong>This is the middle value of a set of data, when arranged in ascending or descending order.<br \/>\n<strong>Example:<br \/>\n<\/strong>Find the median of these numbers: (1). 35, 28, 42, 28, 56, 70, 35      (2) 18, 20, 25, 30, 22, 25, 28, 15<br \/>\n<em>Solution:<br \/>\n<\/em><\/p>\n<ol>\n<li>Re \u2013 arranging the numbers: 70, 56, 42, [35] 35, 28, 28. The median is 35\n<\/li>\n<li>\n<div>15, 18, 20, [22, 25], 25, 28, 30.   Median = 22 + 25 =   47    = 23.5\n<\/div>\n<p>                                                                         2              2\n<\/li>\n<\/ol>\n<p><strong>General Example<\/strong>:<br \/>\nThe table below is the distribution of the test scored in a class:<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Scores\u00a0<\/td>\n<td>     1\u00a0<\/td>\n<td>   2\u00a0<\/td>\n<td>    3\u00a0<\/td>\n<td>    4\u00a0<\/td>\n<td>     5\u00a0<\/td>\n<td>     6\u00a0<\/td>\n<td>     7\u00a0<\/td>\n<td>     8\u00a0<\/td>\n<td>     9\u00a0<\/td>\n<td>   10\u00a0<\/td>\n<\/tr>\n<tr>\n<td>Frequency\u00a0<\/td>\n<td>     1\u00a0<\/td>\n<td>   1\u00a0<\/td>\n<td>    5\u00a0<\/td>\n<td>    3\u00a0<\/td>\n<td>     X<\/td>\n<td>     0\u00a0<\/td>\n<td>     6\u00a0<\/td>\n<td>     2\u00a0<\/td>\n<td>     3\u00a0<\/td>\n<td>    4\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>If the mean score of the class is 6, find the (i) value of x   (ii) median score     (iii) modal score.<br \/>\nSolution<em>:<br \/>\n<\/em><\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>X\u00a0<\/td>\n<td>F\u00a0<\/td>\n<td>FX\u00a0<\/td>\n<\/tr>\n<tr>\n<td> 1\u00a0<\/td>\n<td>1\u00a0<\/td>\n<td>1\u00a0<\/td>\n<\/tr>\n<tr>\n<td>2\u00a0<\/td>\n<td>1\u00a0<\/td>\n<td>2\u00a0<\/td>\n<\/tr>\n<tr>\n<td>3\u00a0<\/td>\n<td>5\u00a0<\/td>\n<td>15\u00a0<\/td>\n<\/tr>\n<tr>\n<td>4\u00a0<\/td>\n<td>3\u00a0<\/td>\n<td>12\u00a0<\/td>\n<\/tr>\n<tr>\n<td>5\u00a0<\/td>\n<td>X\u00a0<\/td>\n<td>5x\u00a0<\/td>\n<\/tr>\n<tr>\n<td>6\u00a0<\/td>\n<td>0\u00a0<\/td>\n<td>0\u00a0<\/td>\n<\/tr>\n<tr>\n<td>7\u00a0<\/td>\n<td>6\u00a0<\/td>\n<td>42\u00a0<\/td>\n<\/tr>\n<tr>\n<td>8\u00a0<\/td>\n<td>2\u00a0<\/td>\n<td>16\u00a0<\/td>\n<\/tr>\n<tr>\n<td>9\u00a0<\/td>\n<td>3\u00a0<\/td>\n<td>27\u00a0<\/td>\n<\/tr>\n<tr>\n<td>10\u00a0<\/td>\n<td>4\u00a0<\/td>\n<td>40\u00a0<\/td>\n<\/tr>\n<tr>\n<td>Total\u00a0<\/td>\n<td>25 + x\u00a0<\/td>\n<td>155 + 5x\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<ol>\n<li>\n<div>Mean = \u2211fx\/\u2211x\n\t\t\t\t\t<\/div>\n<p>6 =    155 + 5x\n<\/li>\n<\/ol>\n<p>                            25 + x<br \/>\nCross multiplying:  6(25 + x) = 155 + 5x<br \/>\n                                    150 + 6x = 155 + 5x<br \/>\n                                      6x \u2013 5x = 155 \u2013 150<br \/>\n                                                x  = 5.<br \/>\n(ii)Median score: the median score lies between the 15<sup>th<\/sup> and 16<sup>th<\/sup> scores, hence: median = (5 + 7)\/2 = 6.<br \/>\n(iii)Mode: 7<\/p>\n<p>\u00a0<strong>Evaluation:<br \/>\n<\/strong>Calculate the mode and median of the scores below; 2, 2, 1, 1, 0, 3, 3, 4, 4, 4, 5, 1, 2, 2.<\/p>\n<p>\u00a0<strong>MEAN, MEDIAN AND MODE OF GROUPED DATA<br \/>\n<\/strong><strong>Mean: <\/strong>The arithmetic mean of grouped frequency distribution can be obtained using:<br \/>\n Class Mark Method:<br \/>\n            X  =      where x is the midpoint of the class interval.<br \/>\nAssumed Mean Method: It is also called working mean method.    X  =  A + (\u2211 fd\/\u2211f)<br \/>\nWhere, d = x \u2013 A,   x = class mark and A = assumed mean.<br \/>\n<strong>Example<em>:<\/em><\/strong> The numbers of matches in 100 boxes are counted and the results are shown in the table below: <\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Number of matches\u00a0<\/td>\n<td>       25    &#8211;   28\u00a0<\/td>\n<td>       29    &#8211;    32<\/td>\n<td>   33   &#8211;    36\u00a0<\/td>\n<td>  37      &#8211;     40\u00a0<\/td>\n<\/tr>\n<tr>\n<td>Number of boxes\u00a0<\/td>\n<td>              18\u00a0<\/td>\n<td>                34\u00a0<\/td>\n<td>          37\u00a0<\/td>\n<td>             11\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Calculate the mean (i) using class mark    (ii) assumed mean method given that the assumed mean is 30.5.<\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0Solution<em>:<\/em><\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td><strong>Class interval<\/strong><\/td>\n<td><strong>F<\/strong>\u00a0<\/td>\n<td><strong>X<\/strong>\u00a0<\/td>\n<td><strong>FX<\/strong>\u00a0<\/td>\n<td><strong>d = x &#8211; A<\/strong>\u00a0<\/td>\n<td><strong>Fd<\/strong>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>25     &#8211;     28\u00a0<\/td>\n<td>18\u00a0<\/td>\n<td>26.5\u00a0<\/td>\n<td>477\u00a0<\/td>\n<td>\n<ul>\n<li>4<\/li>\n<\/ul>\n<\/td>\n<td>\n<ul>\n<li>72<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td>  29     &#8211;     32\u00a0<\/td>\n<td>34\u00a0<\/td>\n<td>30.5\u00a0<\/td>\n<td>1037\u00a0<\/td>\n<td>0\u00a0<\/td>\n<td>0\u00a0<\/td>\n<\/tr>\n<tr>\n<td>  33     &#8211;     36\u00a0<\/td>\n<td>37\u00a0<\/td>\n<td>34.5\u00a0<\/td>\n<td>1276.5\u00a0<\/td>\n<td>4\u00a0<\/td>\n<td>148\u00a0<\/td>\n<\/tr>\n<tr>\n<td>  37     &#8211;     40\u00a0<\/td>\n<td>11\u00a0<\/td>\n<td>38.5\u00a0<\/td>\n<td>423.5\u00a0<\/td>\n<td>8\u00a0<\/td>\n<td> 88\u00a0<\/td>\n<\/tr>\n<tr>\n<td>Total\u00a0<\/td>\n<td>100\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>3214\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>164\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<\/p>\n<ol>\n<li>Class Mark Method: X  =     =  3214\/100   = 32. 14 = 32 matches per box (nearest whole no)\n<\/li>\n<li>\n<div>Assumed Mean Method: X  =  A + (\u2211 fd\/\u2211f)\n<\/div>\n<p>                                                  = 30. 5 + (164\/100) =30.5 + 1.64<br \/>\n                                                  = 32.14 = 32 matches per box (nearest whole number)\n<\/li>\n<\/ol>\n<p><strong>Evaluation<em>:<br \/>\n<\/em><\/strong>Calculate the mean shoe sizes of the number of shoes represented in the table below using (i) class mark   (ii) assumed mean method given that the assumed mean is 42.<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Shoe sizes\u00a0<\/td>\n<td>30    &#8211;      34\u00a0<\/td>\n<td>35      &#8211;    39\u00a0<\/td>\n<td>40     &#8211;     44\u00a0<\/td>\n<td>45      &#8211;    49\u00a0<\/td>\n<td>50      &#8211;   54\u00a0<\/td>\n<\/tr>\n<tr>\n<td>No of Men\u00a0<\/td>\n<td>10\u00a0<\/td>\n<td>12\u00a0<\/td>\n<td>8\u00a0<\/td>\n<td>15\u00a0<\/td>\n<td>5\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<strong>Mode<br \/>\n<\/strong>The mode of a grouped frequency distribution can be determined <strong><em>geometrically<\/em><\/strong> and by <strong><em>interpolation method.<br \/>\n<\/em><\/strong>Mode from Histogram: The highest bar is the modal class and the mode can be determined by drawing a straight line from the right top corner of the bar to the right top corner of the adjacent bar on the left. Draw another line from the left top corner to the bar of the modal class to the left top corner of the adjacent bar on the right.<\/p>\n<p>\u00a0<strong><em>Example:<br \/>\n<\/em><\/strong>The table gives the distribution of ages of students in an institution.<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Ages(year)\u00a0<\/td>\n<td>16    &#8211;      18\u00a0<\/td>\n<td>19      &#8211;    21\u00a0<\/td>\n<td>22     &#8211;     24\u00a0<\/td>\n<td>25      &#8211;    27\u00a0<\/td>\n<td>28      &#8211;   30\u00a0<\/td>\n<\/tr>\n<tr>\n<td>No of Students\u00a0<\/td>\n<td>18\u00a0<\/td>\n<td>30\u00a0<\/td>\n<td>35\u00a0<\/td>\n<td>24\u00a0<\/td>\n<td>13<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Draw a histogram and use your histogram to estimate the mode to the nearest whole number.<br \/>\n<em>Solution:<br \/>\n<\/em><\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Class Interval (Ages)\u00a0<\/td>\n<td>F\u00a0<\/td>\n<td>Class Boundary\u00a0<\/td>\n<\/tr>\n<tr>\n<td>16      &#8211;     18\u00a0<\/td>\n<td>18\u00a0<\/td>\n<td>15.5    &#8211;   18. 5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>19     &#8211;      21\u00a0<\/td>\n<td>30\u00a0<\/td>\n<td>18.5    &#8211;   21.5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>22     &#8211;      24\u00a0<\/td>\n<td>35\u00a0<\/td>\n<td>21.5    &#8211;   24.5<\/td>\n<\/tr>\n<tr>\n<td>25     &#8211;      27\u00a0<\/td>\n<td>24\u00a0<\/td>\n<td>24.5    &#8211;  27.5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>28     &#8211;      30\u00a0<\/td>\n<td>13\u00a0<\/td>\n<td>27.5   &#8211;   30.5\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0Histogram<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F1.png\" alt=\"\"\/><br \/>\n\t\t<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F2.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F3.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F4.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F5.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F6.png\" alt=\"\"\/>   35<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F7.png\" alt=\"\"\/><br \/>\n\t\t<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F8.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F9.png\" alt=\"\"\/>    30<\/p>\n<p>\u00a0<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F10.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F11.png\" alt=\"\"\/>    25<\/p>\n<p>\u00a0    20<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F12.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F13.png\" alt=\"\"\/><br \/>\n\t\t    15<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F14.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F15.png\" alt=\"\"\/><br \/>\n\t\t    10<\/p>\n<p>\u00a0      5<\/p>\n<p>\u00a0<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F16.png\" alt=\"\"\/>      0<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Masses(kg)\u00a0<\/td>\n<td>Frequency\u00a0<\/td>\n<td>Cumulative Frequency\u00a0<\/td>\n<td>Upper Class Boundary\u00a0<\/td>\n<\/tr>\n<tr>\n<td>10 \u2013 14<\/td>\n<td>3\u00a0<\/td>\n<td>3\u00a0<\/td>\n<td>&lt; 14.5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>15 \u2013 19\u00a0<\/td>\n<td>7\u00a0<\/td>\n<td>10\u00a0<\/td>\n<td>&lt;19.5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>20 \u2013 24\u00a0<\/td>\n<td>9\u00a0<\/td>\n<td>19\u00a0<\/td>\n<td>&lt;24.5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>25 \u2013 29\u00a0<\/td>\n<td>5\u00a0<\/td>\n<td>24\u00a0<\/td>\n<td>&lt; 29.5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>30 \u2013 34\u00a0<\/td>\n<td>11\u00a0<\/td>\n<td>35\u00a0<\/td>\n<td>&lt; 34.5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>35 \u2013 39\u00a0<\/td>\n<td>6\u00a0<\/td>\n<td>41\u00a0<\/td>\n<td>&lt; 39.5\u00a0<\/td>\n<\/tr>\n<tr>\n<td>40 \u2013 44\u00a0<\/td>\n<td>9\u00a0<\/td>\n<td>50\u00a0<\/td>\n<td>&lt; 44.5\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>              15.5     18.5    21.5     24.5    27.5    30.5                                  <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0Modal class = 22   &#8211;    24<br \/>\nMode = 21.5 + 0.9 = 22.4, approximately 22 yrs.<\/p>\n<p>\u00a0<strong>MODE FROM INTERPOLATION: <\/strong>The mode can be obtained using the formula.<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F17.png\" alt=\"\"\/>                                                     Mode =  L<sub>m <\/sub> +        \u2206<sub>1              <\/sub>C<sub><br \/>\n\t\t\t<\/sub><br \/>\n\t\t<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F18.png\" alt=\"\"\/>                                                                                  \u2206<sub>1 <\/sub>+ \u2206<sub>2<br \/>\n<\/sub>Where L<sub>m<\/sub> = lower class boundary of the modal class.<br \/>\n             \u2206<sub>1  <\/sub>= difference between the frequency of the modal class and the class before it.<br \/>\n             \u2206<sub>2 <\/sub> = difference between the frequency of the modal class and the class after it.<br \/>\n             C   = class width of the modal class.<br \/>\nExample: Using the table given in the example above:<br \/>\n  Modal class = 22 \u2013 24,   \u2206<sub>1 <\/sub>= 35 \u2013 30 = 5,  \u2206<sub>2 <\/sub>= 35 \u2013 24 = 11,   C = 3,   L<sub>m<\/sub> = 21.5<\/p>\n<p>\u00a0<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F19.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F20.png\" alt=\"\"\/>                                      Mode = 21.5   +       5        x3<br \/>\n                                                                      5 + 11<br \/>\n                                                 = 21.5 + (15\/16)   = 21.5 + 0.9375<br \/>\n                                                 = 22.44, approximately 22 yrs.<br \/>\n<strong>MEDIAN FROM INTERPOLATION FORMULA<br \/>\n<\/strong><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F21.png\" alt=\"\"\/>Median = L<sub>1<\/sub> +   N\/2 \u2013 cfm C<br \/>\n                               fm<br \/>\nWhere, L<sub>1<\/sub> = lower class boundary of the median class.<br \/>\n            Cfm = cumulative frequency of the class before the median class.<br \/>\n            Fm = frequency of the median class.<br \/>\n            C   = class width of the median class and N   = Total frequency<\/p>\n<p>\u00a0The median class: 30 \u2013 34, L<sub>1<\/sub> = 29.5, cfm = 24,   fm = 11,   C = 5<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F22.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week10SS1F23.png\" alt=\"\"\/>              Median = 29.5 +  25 &#8211; 24   x 5<br \/>\n                                               11<br \/>\n                            = 29.5 + 5     = 30kg<br \/>\n<strong>Evaluation<\/strong>: Calculate the modal shoe sizes and median of the number of shoes represented in the table below using interpolation method.<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Shoe sizes\u00a0<\/td>\n<td>30    &#8211;      34<\/td>\n<td>35      &#8211;    39\u00a0<\/td>\n<td>40     &#8211;     44\u00a0<\/td>\n<td>45      &#8211;    49\u00a0<\/td>\n<td>50      &#8211;   54\u00a0<\/td>\n<\/tr>\n<tr>\n<td>No of Men\u00a0<\/td>\n<td>10\u00a0<\/td>\n<td>12\u00a0<\/td>\n<td>8\u00a0<\/td>\n<td>15\u00a0<\/td>\n<td>5\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<strong>General evaluation<\/strong>:<br \/>\n1. The table below gives the distribution of masses (kg) of 40 people<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Masses(kg)<\/td>\n<td>1 \u2013 5<\/td>\n<td>6 &#8211; 10\u00a0<\/td>\n<td>11 -15\u00a0<\/td>\n<td>16    &#8211;  20<\/td>\n<td>21     &#8211; 25\u00a0<\/td>\n<td>26    &#8211; 30\u00a0<\/td>\n<td>31     &#8211; 35<\/td>\n<td>36    &#8211;  40<\/td>\n<\/tr>\n<tr>\n<td>Frequency\u00a0<\/td>\n<td>9\u00a0<\/td>\n<td>20\u00a0<\/td>\n<td>32\u00a0<\/td>\n<td>42\u00a0<\/td>\n<td>35\u00a0<\/td>\n<td>22\u00a0<\/td>\n<td>15\u00a0<\/td>\n<td>5\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<ol>\n<li>State the modal class of the distribution and find the mode.\n<\/li>\n<li>Calculate the mean of the distribution.\n<\/li>\n<\/ol>\n<p>\u00a02. The following table shows the distribution of marks obtained by a class.<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Marks\u00a0<\/td>\n<td>  0       \u00a0<\/td>\n<td>      1\u00a0<\/td>\n<td>      2\u00a0<\/td>\n<td>      3\u00a0<\/td>\n<td>      4\u00a0<\/td>\n<td>      5\u00a0<\/td>\n<td>      6\u00a0<\/td>\n<td>      7\u00a0<\/td>\n<td>      8\u00a0<\/td>\n<td>     9\u00a0<\/td>\n<\/tr>\n<tr>\n<td>No of students \u00a0<\/td>\n<td>  1\u00a0<\/td>\n<td>      1\u00a0<\/td>\n<td>     3\u00a0<\/td>\n<td>      4\u00a0<\/td>\n<td>      4\u00a0<\/td>\n<td>    12\u00a0<\/td>\n<td>      7\u00a0<\/td>\n<td>      3\u00a0<\/td>\n<td>      3\u00a0<\/td>\n<td>     2\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Using this table, find the (1) median mark     (2) modal mark      (3) mean of the distribution.<\/p>\n<p>\u00a0<strong>Reading Assignment<\/strong>: Further Mathematics Project Book 1(New third edition), pg 328, Exercise18, No 15 -20<\/p>\n<p>\u00a0<strong>Weekend Assignment<\/strong>\u00a0\u00a0\u00a0\u00a0<strong><br \/>\n\t\t\t<\/strong><\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Marks\u00a0<\/td>\n<td>  3\u00a0<\/td>\n<td> 4\u00a0<\/td>\n<td>   5\u00a0<\/td>\n<td>6\u00a0<\/td>\n<td>7\u00a0<\/td>\n<td> 8\u00a0<\/td>\n<\/tr>\n<tr>\n<td>Frequency\u00a0<\/td>\n<td>   5 \u00a0<\/td>\n<td> x \u2013 1\u00a0<\/td>\n<td>   x\u00a0<\/td>\n<td> 9  \u00a0<\/td>\n<td>4\u00a0<\/td>\n<td>1\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>If the mean is 5, calculate the (a) value of x     (b) mode      (c) median of the distribution.<\/p>\n<p>\u00a02.\u00a0\u00a0\u00a0\u00a0The table gives the frequency distribution of a random sample of 250 steel bolts according to their head diameter, measured to the nearest 0.01mm.<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>Diameter (mm)\u00a0<\/td>\n<td>23.06 \u201323.10<\/td>\n<td>23.11 \u2013 23.15\u00a0<\/td>\n<td>23.16 \u201323.20<\/td>\n<td>23.21 \u201323.25<\/td>\n<td>23.26-23.30<\/td>\n<td>23.31 \u201323.35<\/td>\n<td>23.36-23.40\u00a0<\/td>\n<td>23.41-23.45\u00a0<\/td>\n<td>23.46-23.50\u00a0<\/td>\n<\/tr>\n<tr>\n<td>No of bolts\u00a0<\/td>\n<td>10\u00a0<\/td>\n<td>20\u00a0<\/td>\n<td>28\u00a0<\/td>\n<td>36\u00a0<\/td>\n<td>52\u00a0<\/td>\n<td>38\u00a0<\/td>\n<td>32\u00a0<\/td>\n<td>       21\u00a0<\/td>\n<td>13\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<ol>\n<li>State the median class and calculate the median using interpolation method.\n<\/li>\n<li>Draw the histogram and use it to estimate the mode.\n<\/li>\n<li>Calculate the mean value using a working mean of 23.28mm.\n<\/li>\n<li>The table gives the frequency distribution of marks obtained by a group of students in a test.\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0 \u00a0WEEK TEN TOPIC: STATISTICS: MEASURES OF CENTRAL TENDENCY MEASUREMENT OF CENTRAL TENDENCY Mean, Median&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,173],"tags":[],"class_list":["post-1935","post","type-post","status-publish","format-standard","hentry","category-posts","category-first-term-ss1-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1935","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=1935"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1935\/revisions"}],"predecessor-version":[{"id":1936,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1935\/revisions\/1936"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=1935"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=1935"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=1935"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}