document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202358>Question 1202358</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The following are the annual salaries of 22 chief executive officers of major companies (The salaries are written ir <BR>\n" );
document.write( "thousands of dollars)<BR>\n" );
document.write( "785, 152, 176, 89, 80, 537, 514, 333, 581, 628, 671, 738, 447, 423, 210, 230, 695, 490, 1103, 604, 345, 540.<BR>\n" );
document.write( "Find the 25th and 70th percentile of these salaries. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837134 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=837134\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Unfortunately percentiles do not have a standard agreed upon definition.<BR>\n" );
document.write( "Sources:<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statisticshowto.com/probability-and-statistics/percentiles-rank-range/#:~:text=but%20there%20is%20no%20universal%20definition%20for%20it\">https://www.statisticshowto.com/probability-and-statistics/percentiles-rank-range/#:~:text=but%20there%20is%20no%20universal%20definition%20for%20it</a><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Introductory_Statistics_(Lane)/01%3A_Introduction_to_Statistics/1.07%3A_Percentiles#:~:text=There%20is%20no%20universally%20accepted%20definition%20of%20a%20percentile\">https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Introductory_Statistics_(Lane)/01%3A_Introduction_to_Statistics/1.07%3A_Percentiles#:~:text=There%20is%20no%20universally%20accepted%20definition%20of%20a%20percentile</a>\r<BR>\n" );
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document.write( "Despite this glaring flaw, I'll be using the formulas mentioned on this page<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.calculatorsoup.com/calculators/statistics/percentile-calculator.php\">https://www.calculatorsoup.com/calculators/statistics/percentile-calculator.php</a><BR>\n" );
document.write( "That page also provides a calculator to verify the answer.\r<BR>\n" );
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document.write( "Original data set = {785, 152, 176, 89, 80, 537, 514, 333, 581, 628, 671, 738, 447, 423, 210, 230, 695, 490, 1103, 604, 345, 540}\r<BR>\n" );
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document.write( "Sorted data set = {80, 89, 152, 176, 210, 230, 333, 345, 423, 447, 490, 514, 537, 540, 581, 604, 628, 671, 695, 738, 785, 1103}\r<BR>\n" );
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document.write( "n = number of items in the set<BR>\n" );
document.write( "n = 22\r<BR>\n" );
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document.write( "Let's list each value with its associated rank.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Score</TD><TD >Rank</TD></tr><tr><TD >80</TD><TD >1</TD></tr><tr><TD >89</TD><TD >2</TD></tr><tr><TD >152</TD><TD >3</TD></tr><tr><TD >176</TD><TD >4</TD></tr><tr><TD >210</TD><TD >5</TD></tr><tr><TD >230</TD><TD >6</TD></tr><tr><TD >333</TD><TD >7</TD></tr><tr><TD >345</TD><TD >8</TD></tr><tr><TD >423</TD><TD >9</TD></tr><tr><TD >447</TD><TD >10</TD></tr><tr><TD >490</TD><TD >11</TD></tr><tr><TD >514</TD><TD >12</TD></tr><tr><TD >537</TD><TD >13</TD></tr><tr><TD >540</TD><TD >14</TD></tr><tr><TD >581</TD><TD >15</TD></tr><tr><TD >604</TD><TD >16</TD></tr><tr><TD >628</TD><TD >17</TD></tr><tr><TD >671</TD><TD >18</TD></tr><tr><TD >695</TD><TD >19</TD></tr><tr><TD >738</TD><TD >20</TD></tr><tr><TD >785</TD><TD >21</TD></tr><tr><TD >1103</TD><TD >22</TD></tr>\n" );
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document.write( "Then,<BR>\n" );
document.write( "rank = (percentile/100)*(n - 1) + 1<BR>\n" );
document.write( "rank = (25/100)*(22 - 1) + 1<BR>\n" );
document.write( "rank = 6.25\r<BR>\n" );
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document.write( "integer part = x = 6<BR>\n" );
document.write( "fractional part = y = 0.25 \r<BR>\n" );
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document.write( "The scores at ranks 6 and 7 are 230 and 333 respectively. Refer to the table above. Let's call these scores P and Q.\r<BR>\n" );
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document.write( "And finally,<BR>\n" );
document.write( "P + y*(Q - P)<BR>\n" );
document.write( "230 + 0.25*(333 - 230)<BR>\n" );
document.write( "255.75<BR>\n" );
document.write( "is the 25th percentile.\r<BR>\n" );
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document.write( "Again, there is no agreed upon universal definition of percentile. <BR>\n" );
document.write( "If your stats textbook uses another definition, then be sure to use that.\r<BR>\n" );
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document.write( "Now let's find the 70th percentile<BR>\n" );
document.write( "rank = (percentile/100)*(n - 1) + 1<BR>\n" );
document.write( "rank = (70/100)*(22 - 1) + 1<BR>\n" );
document.write( "rank = 15.7\r<BR>\n" );
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document.write( "x = integer part = 15<BR>\n" );
document.write( "y = fractional part = 0.7\r<BR>\n" );
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document.write( "The values at ranks 15 and 16 are 581 and 604 in that order.<BR>\n" );
document.write( "P = 581<BR>\n" );
document.write( "Q = 604\r<BR>\n" );
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document.write( "P + y*(Q - P)<BR>\n" );
document.write( "581 + 0.7*(604 - 581)<BR>\n" );
document.write( "597.1<BR>\n" );
document.write( "That represents the 70th percentile.<BR>\n" );
document.write( "About 70% of the scores are below this cutoff point.<BR>\n" );
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