Question 1202358: The following are the annual salaries of 22 chief executive officers of major companies (The salaries are written ir
thousands of dollars)
785, 152, 176, 89, 80, 537, 514, 333, 581, 628, 671, 738, 447, 423, 210, 230, 695, 490, 1103, 604, 345, 540.
Find the 25th and 70th percentile of these salaries.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Unfortunately percentiles do not have a standard agreed upon definition.
Sources:
https://www.statisticshowto.com/probability-and-statistics/percentiles-rank-range/#:~:text=but%20there%20is%20no%20universal%20definition%20for%20it
and
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
Despite this glaring flaw, I'll be using the formulas mentioned on this page
https://www.calculatorsoup.com/calculators/statistics/percentile-calculator.php
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.
| Score | Rank | | 80 | 1 | | 89 | 2 | | 152 | 3 | | 176 | 4 | | 210 | 5 | | 230 | 6 | | 333 | 7 | | 345 | 8 | | 423 | 9 | | 447 | 10 | | 490 | 11 | | 514 | 12 | | 537 | 13 | | 540 | 14 | | 581 | 15 | | 604 | 16 | | 628 | 17 | | 671 | 18 | | 695 | 19 | | 738 | 20 | | 785 | 21 | | 1103 | 22 |
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.
------------------------------------
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.
|
|
|
