Percentile

KWF

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Have I calculated the percentile correctly from the following?

A small business employees five people with the following salaries: $35,000, $30,000, $25,000, $22,000 and $20,000.

The employee making $30,000 wants to know the percentile that he is in regarding the other employee's salaries.

$30,000/$132,000 = 0.277 = 22.7%

Is he in the 22.7% percentile?

I thank you for your reply.
 
Have I calculated the percentile correctly from the following?

A small business employees five people with the following salaries: $35,000, $30,000, $25,000, $22,000 and $20,000.

The employee making $30,000 wants to know the percentile that he is in regarding the other employee's salaries.

$30,000/$132,000 = 0.277 = 22.7%

Is he in the 22.7% percentile?

That percentile is not a reasonable one. That employee's salary is the second highest.
 
That percentile is not a reasonable one. That employee's salary is the second highest.


How can the percentile become a "reasonable one"? In other words, how can the situation be changed to make a reasonable percentile?



Thanks!



 
Can you offer a proper calcualtion? I thought mine was correct.
 
I'm not willing to do the exercise for you. I want you to study the lesson at the link which I provided, followed by returning here and telling us about the part(s) that confuse you.

You do not understand the meaning of a percentile; hence, you are not yet ready for this exercise.

(If you'd prefer doing something other than studying or asking specific questions, then please speak with your instructor.)

Cheers :cool:
 
Thanks, but where is the link?

If you look at the response which mmm4bot gave to your original post, you'll see some text in BLUE. That is a link. Click on it, and you'll be taken to the lesson which explains percentiles.
 
Okay, I found it hiding at "Have you seen any formulas for doing this?"

I saw the calculation, but the person who wrote it did not explain why "0.5" is used for one of the percentile calculations. Thanks mmm4444bot for the link and thanks Mrspi for pointing it out to me!

-----------------------

Is this now correct to my earlier posting. See above:

$20000, $22,000, $25,000, $30,000, $35,000

Number of salaries below $30,000, 3

{3+(0.5 X 0)/5} = 3/5 = 0.6 X 100 or 60th percentile (?) Is this correct or perhaps close?
 
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Okay, I found it hiding at "Have you seen any formulas for doing this?"

I saw the calculation, but the person who wrote it did not explain why "0.5" is used for one of the percentile calculations. Thanks mmm4444bot for the link and thanks Mrspi for pointing it out to me!

-----------------------

Is this now correct to my earlier posting. See above:

$20000, $22,000, $25,000, $30,000, $35,000

Number of salaries below $30,000, 3

{3+(0.5 X 0)/5} X 100 = 3/5 = 0.6 or 60th percentile (?) Is this correct or perhaps close?
You got an acceptable answer, but not the answer that is consistent with the formula that you used.

Let T = the total number of scores = 5.
Let B = the number of scores below the target score = 3.
Let E = the number of scores equal to the target score = 1.
Let P = the percentile rank of the target score.

Formula 1: \(\displaystyle P = \dfrac{B + (0.5 * E)}{T} * 100 = \dfrac{3 + (1 * 0.5)}{5} *100 = \dfrac{3 + 0.5}{5} *100 = \dfrac{3.5}{5} *100 = 0.7 * 100 = 70.\)

Formula 2: \(\displaystyle P = \dfrac{B}{T} * 100 = \dfrac{3}{5} * 100 = 0.6 * 100 = 60.\)
 
Thanks JeffM for the reply and explanation!

Do you know why the 0.5 is used for the first formula?
Yes. It does not make much sense when talking about a small list of numbers, but then breaking up an ordered list of numbers into hundredths does not make much sense unless you have more than 100 numbers to begin with.

Score 1, count 20
Score 2, count 100
Score 3, count 70
Score 4, count 10

Percentile of score 3 according to formula 2 is \(\displaystyle \dfrac{20 + 100}{20 + 100 + 70 + 10} * 100 = \dfrac{120 * 100}{200} = 60.\)

That seems to imply that a score of 3 is not very good; perhaps as many as 39% of the scores were better than 3.

Percentile of score 3 according to formula 1 is \(\displaystyle \dfrac{120 + (70 * 0.5)}{200} * 100 = \dfrac{155.5 * 100}{200} = 77.75.\)

Now it is clear that at most 22% of the scores were better than 3.

The factor of 0.5 in formula 1 reduces the distortion caused by "lumpiness" in the data.

Make sense?

Now personally, I do not find using percentiles computed under either formula very revealing when dealing with data that is very lumpy. So I personally would use percentiles as a descriptive device only when the number of cases was large and the data were not particularly lumpy, and then I would use formula 1.
 
You got an acceptable answer, but not the answer that is consistent with the formula that you used.

Let T = the total number of scores = 5.
Let B = the number of scores below the target score = 3.
Let E = the number of scores equal to the target score = 1.
Let P = the percentile rank of the target score.

Formula 1: \(\displaystyle P = \dfrac{B + (0.5 * E)}{T} * 100 = \dfrac{3 + (1 * 0.5)}{5} *100 = \dfrac{3 + 0.5}{5} *100 = \dfrac{3.5}{5} *100 = 0.7 * 100 = 70.\)

Formula 2: \(\displaystyle P = \dfrac{B}{T} * 100 = \dfrac{3}{5} * 100 = 0.6 * 100 = 60.\)

Thanks JeffM for the reply and examples! Do you know why 0.5 is used in the first formula?
 
Thanks JeffM for the reply and examples! Do you know why 0.5 is used in the first formula?
Please see my immediately preceding post for a brief explanation of the reasoning behind formula 1.
 
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Yes. It does not make much sense when talking about a small list of numbers, but then breaking up an ordered list of numbers into hundredths does not make much sense unless you have more than 100 numbers to begin with.

Score 1, count 20
Score 2, count 100
Score 3, count 70
Score 4, count 10

Percentile of score 3 according to formula 2 is \(\displaystyle \dfrac{20 + 100}{20 + 100 + 70 + 10} * 100 = \dfrac{120 * 100}{200} = 60.\)

That seems to imply that a score of 3 is not very good; perhaps as many as 39% of the scores were better than 3.

Percentile of score 3 according to formula 1 is \(\displaystyle \dfrac{120 + (70 * 0.5)}{200} * 100 = \dfrac{155.5 * 100}{200} = 77.75.\)

Now it is clear that at most 22% of the scores were better than 3.

The factor of 0.5 in formula 1 reduces the distortion caused by "lumpiness" in the data.

Make sense?

Now personally, I do not find using percentiles computed under either formula very revealing when dealing with data that is very lumpy. So I personally would use percentiles as a descriptive device only when the number of cases was large and the data were not particularly lumpy, and then I would use formula 1.

JeffM-
I want to thank you for the reply and examples. They are indeed helpful. I do not understand what you mean by "lumpiness" or "lumpy."
And also, why wouldn't 0.3 or 0.75 be used instead of 0.5?

I apoloigize for my lack of knowledge regarding percentiles, and for any difficult questions I send to you.

Thanks again!
 
First, read the article linked to by denis, which gives yet a third and more sophisticated formula for percentile, which is one type of descriptive statistic.

Second, the purpose of descriptive statistics is to reduce a large and apparently confusing mass of numbers to a few easily comprehended numbers. These simplifying numbers virtually always lose information. The hope is that the information lost is worth less than the gain in understanding. The statistician's duty is to pick the type or types of simplified number that provide the maximum degree of understanding.

Third, the "better behaved" the data are, the less information is lost by relying on descriptive statisitics.

Fourth, by "well behaved" data, we mean that a graph of their frequency results in a relatively simple curve. The intuitive idea is that a few simple numbers are sufficient to describe a simple curve. By "lumpy," I meant that the data are heavily concentrated in a small number of values rather than being spread across a large number of values.

The various formulas discussed attempt to improve the reliability of the percentile as an informative summarizing number when the data are few, not well behaved, or quite lumpy. When the data are ample, well behaved, and not lumpy, all three formulas give virtually the same result. Because mathematicians have not agreed on them, it should be obvious that none of them is conclusively "right." Consequently, I at least cannot give you an answer to your question about 0.5 except that it is easy to work with and takes a neutral position between counting all of the instances of one score and none of the instances. It is a reasonable adjustment.

In my opinion, percentiles simply are not very good summarizing numbers whenever the number of data points is small, the data are not well behaved, or the data are lumpy. I'd presume when someone reputable gives data in percentiles that the data are numerous, well behaved, and not lumpy, and the question of which formula was used has virtually no effect on the numbers given. It approximately gives the percentage of total scores that are below the indicated score, and 99 minus the percentile approximately gives the percentage of total scores that are above the indicated score. So a score in the 70th percentile means approximately 70% of the scores were lower, and approximately 29% were higher. If those statements are quite misleasing, then someone reputable will probably not use percentiles as a descriptor.
 
First, read the article linked to by denis, which gives yet a third and more sophisticated formula for percentile, which is one type of descriptive statistic.

Second, the purpose of descriptive statistics is to reduce a large and apparently confusing mass of numbers to a few easily comprehended numbers. These simplifying numbers virtually always lose information. The hope is that the information lost is worth less than the gain in understanding. The statistician's duty is to pick the type or types of simplified number that provide the maximum degree of understanding.

Third, the "better behaved" the data are, the less information is lost by relying on descriptive statisitics.

Fourth, by "well behaved" data, we mean that a graph of their frequency results in a relatively simple curve. The intuitive idea is that a few simple numbers are sufficient to describe a simple curve. By "lumpy," I meant that the data are heavily concentrated in a small number of values rather than being spread across a large number of values.

The various formulas discussed attempt to improve the reliability of the percentile as an informative summarizing number when the data are few, not well behaved, or quite lumpy. When the data are ample, well behaved, and not lumpy, all three formulas give virtually the same result. Because mathematicians have not agreed on them, it should be obvious that none of them is conclusively "right." Consequently, I at least cannot give you an answer to your question about 0.5 except that it is easy to work with and takes a neutral position between counting all of the instances of one score and none of the instances. It is a reasonable adjustment.

In my opinion, percentiles simply are not very good summarizing numbers whenever the number of data points is small, the data are not well behaved, or the data are lumpy. I'd presume when someone reputable gives data in percentiles that the data are numerous, well behaved, and not lumpy, and the question of which formula was used has virtually no effect on the numbers given. It approximately gives the percentage of total scores that are below the indicated score, and 99 minus the percentile approximately gives the percentage of total scores that are above the indicated score. So a score in the 70th percentile means approximately 70% of the scores were lower, and approximately 29% were higher. If those statements are quite misleasing, then someone reputable will probably not use percentiles as a descriptor.



Hello JeffM:


I cannot thank you enough for your explanations. I underdand "lumpy" or "lumpiness" better in regard to your examples.

I do have one last question, I promise. In one of your replies, you determined the percentile for the following:

Score 1, count 20
Score 2, count 100
Score 3, count 70
Score 4, count 10

Percentile of score 3 according to formula 2 is: (20 + 10)/(20 + 100+ 70 + 10) X 100 = 120/200 X 100 = 60


The information at the site MMM4444bot recommended me to read indicated that the test scores should be listed from lowest to highest. In your example you did not list the scores as 10, 20, 70, 100 and then use (10 + 20 /200) X 100. Can you explain why?

I think the test scores at the regentsprep.org/site were from 20 different students. In your example, were you making up test scores from one person and then find his percentile from the last two tests?

Once again, I thank you for helping me understand percentiles!
 
Hello JeffM:


I cannot thank you enough for your explanations. I underdand "lumpy" or "lumpiness" better in regard to your examples.

I do have one last question, I promise. In one of your replies, you determined the percentile for the following:

Score 1, count 20
Score 2, count 100
Score 3, count 70
Score 4, count 10

Percentile of score 3 according to formula 2 is: (20 + 10)/(20 + 100+ 70 + 10) X 100 = 120/200 X 100 = 60


The information at the site MMM4444bot recommended me to read indicated that the test scores should be listed from lowest to highest. In your example you did not list the scores as 10, 20, 70, 100 and then use (10 + 20 /200) X 100. Can you explain why?

I think the test scores at the regentsprep.org/site were from 20 different students. In your example, were you making up test scores from one person and then find his percentile from the last two tests?

Once again, I thank you for helping me understand percentiles!
Percentiles summarize values usually called scores such as scores on a test. It is the scores that are to be ranked. For my example, I was showing the number of students who got a particular score as count. It is not the counts that are ranked from lowest to highest, but the scores.

I now think that I have given far more information than was required.

A score's percentile tells approximately what percentage of the total number of scores were lower than the given score, and the difference between 99 and the score's percentile tells approximately what percentage of the total number of scores were higher than the given score.
 
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