document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191524>Question 1191524</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3.	Marks of 75 students are summarized in the following frequency distribution:       <BR>\n" );
document.write( "Marks 	40  -  44 	45  -  49 	50  -  54 	55  -  59 	60  -  64 	65  -  69 	70  -  74 <BR>\n" );
document.write( "No. of students 	7	10	22	F4	F5	6	3<BR>\n" );
document.write( "If 20% of the students have marks between 55 and 59 <BR>\n" );
document.write( "i.	Find the missing frequencies f4 and f5. <BR>\n" );
document.write( "ii.	Find the mean, median and mode<BR>\n" );
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document.write( "Part (i)<BR>\n" );
document.write( "Original Table:<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Marks </TD><TD >40  -  44 </TD><TD >45  -  49 </TD><TD >50  -  54 </TD><TD >55  -  59 </TD><TD >60  -  64 </TD><TD >65  -  69 </TD><TD >70  -  74 </TD></tr><tr><TD >No. of students </TD><TD >7</TD><TD >10</TD><TD >22</TD><TD >F4</TD><TD >F5</TD><TD >6</TD><TD >3</TD></tr>\n" );
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document.write( "20% of all 75 students have marks between 55 and 59.<BR>\n" );
document.write( "20% of 75 = 0.20*75 = 15<BR>\n" );
document.write( "The number 15 replaces F4\r<BR>\n" );
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document.write( "Let x be the value of F5<BR>\n" );
document.write( "The items in the frequency column must add to 75<BR>\n" );
document.write( "7 + 10 + 22 + 15 + x + 6 + 3 = 75<BR>\n" );
document.write( "x+63 = 75<BR>\n" );
document.write( "x = 75-63<BR>\n" );
document.write( "x = 12<BR>\n" );
document.write( "The number 12 replaces F5\r<BR>\n" );
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document.write( "Updated Table:<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Marks </TD><TD >40  -  44 </TD><TD >45  -  49 </TD><TD >50  -  54 </TD><TD >55  -  59 </TD><TD >60  -  64 </TD><TD >65  -  69 </TD><TD >70  -  74 </TD></tr><tr><TD >No. of students </TD><TD >7</TD><TD >10</TD><TD >22</TD><TD >15</TD><TD >12</TD><TD >6</TD><TD >3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "Part (ii)\r<BR>\n" );
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document.write( "The class interval 40 to 44 has the midpoint (A+B)/2 = (40+44)/2 = 84/2 = 42<BR>\n" );
document.write( "Through similar calculations, the midpoint of the interval 45 to 49 is 47.<BR>\n" );
document.write( "This process is carried out until you address each interval.\r<BR>\n" );
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document.write( "The midpoints in order from left to right are: 42, 47, 52, 57, 62, 67, 72<BR>\n" );
document.write( "The midpoints are increasing by 5 each time, which is exactly the gap from each left endpoint of adjacent classes (same goes for the right endpoints as well).\r<BR>\n" );
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document.write( "I'll add the row of midpoints which I'll refer to as M<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Marks </TD><TD >40  -  44 </TD><TD >45  -  49 </TD><TD >50  -  54 </TD><TD >55  -  59 </TD><TD >60  -  64 </TD><TD >65  -  69 </TD><TD >70  -  74 </TD></tr><tr><TD >No. of students </TD><TD >7</TD><TD >10</TD><TD >22</TD><TD >15</TD><TD >12</TD><TD >6</TD><TD >3</TD></tr><tr><TD >Midpoint (M)</TD><TD >42</TD><TD >47</TD><TD >52</TD><TD >57</TD><TD >62</TD><TD >67</TD><TD >72</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "For each column, we'll multiply the frequency value f with its corresponding midpoint M<BR>\n" );
document.write( "Eg: the first column has 7*42 = 294<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Marks </TD><TD >40  -  44 </TD><TD >45  -  49 </TD><TD >50  -  54 </TD><TD >55  -  59 </TD><TD >60  -  64 </TD><TD >65  -  69 </TD><TD >70  -  74 </TD></tr><tr><TD >No. of students (f) </TD><TD >7</TD><TD >10</TD><TD >22</TD><TD >15</TD><TD >12</TD><TD >6</TD><TD >3</TD></tr><tr><TD >Midpoint (M)</TD><TD >42</TD><TD >47</TD><TD >52</TD><TD >57</TD><TD >62</TD><TD >67</TD><TD >72</TD></tr><tr><TD >f*M</TD><TD >294</TD><TD >470</TD><TD >1144</TD><TD >855</TD><TD >744</TD><TD >402</TD><TD >216</TD></tr>\n" );
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document.write( "Adding up the values in that bottom row gets us<BR>\n" );
document.write( "294 + 470 + 1144 + 855 + 744 + 402 + 216 = 4125\r<BR>\n" );
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document.write( "Divide that over the total number of students (75)<BR>\n" );
document.write( "4125/75 = 55\r<BR>\n" );
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document.write( "The mean is 55.\r<BR>\n" );
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document.write( "There are an odd number of students (75)<BR>\n" );
document.write( "75/2 = 37.5 this rounds to 38<BR>\n" );
document.write( "The median is in slot 38. The first 37 items are below it, the upper 37 items are above the median.<BR>\n" );
document.write( "That gives 37+1+37 = 75 scores total.\r<BR>\n" );
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document.write( "Clearly the median is not in class 40-44 since we only have 7 students here.<BR>\n" );
document.write( "The median is also not in the 45-49 class because 7+10 = 17 is too small also<BR>\n" );
document.write( "7+10+22 = 39 is now too large since we want to land on 38.\r<BR>\n" );
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document.write( "The median mark is somewhere between 50 and 54.<BR>\n" );
document.write( "We don't have enough info to determine its exact value because of the grouped nature of the marks.\r<BR>\n" );
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document.write( "The mode is the most frequent score.<BR>\n" );
document.write( "The largest frequency is 22 and it occurs for the 50-54 class.<BR>\n" );
document.write( "Like with the median, we can't pinpoint its exact value.<BR>\n" );
document.write( "It's possible that the median = mode, but it's not a guarantee.\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "Mean = 55<BR>\n" );
document.write( "Median = somewhere between 50 and 54.<BR>\n" );
document.write( "Mode = somewhere between 50 and 54.<BR>\n" );
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