Question 1202358
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Unfortunately percentiles do not have a standard agreed upon definition.
Sources:
<a 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>
and
<a 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>


Despite this glaring flaw, I'll be using the formulas mentioned on this page
<a href="https://www.calculatorsoup.com/calculators/statistics/percentile-calculator.php">https://www.calculatorsoup.com/calculators/statistics/percentile-calculator.php</a>
That page also provides a calculator to verify the answer.




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}


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}


n = number of items in the set
n = 22


Let's list each value with its associated rank.
<table border = "1" cellpadding = "5"><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></table>


Then,
rank = (percentile/100)*(n - 1) + 1
rank = (25/100)*(22 - 1) + 1
rank = 6.25


integer part = x = 6
fractional part = y = 0.25 


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.


And finally,
P + y*(Q - P)
230 + 0.25*(333 - 230)
255.75
is the 25th percentile.


Again, there is no agreed upon universal definition of percentile. 
If your stats textbook uses another definition, then be sure to use that.


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Now let's find the 70th percentile
rank = (percentile/100)*(n - 1) + 1
rank = (70/100)*(22 - 1) + 1
rank = 15.7


x = integer part = 15
y = fractional part = 0.7


The values at ranks 15 and 16 are 581 and 604 in that order.
P = 581
Q = 604


P + y*(Q - P)
581 + 0.7*(604 - 581)
597.1
That represents the 70th percentile.
About 70% of the scores are below this cutoff point.
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