This is a two-part question from the sublime to the ridiculous:
a) In simple, practical, everyday terms, how would you explain percentiles to someone?
b) Now, beyond the basics, Wiki offers this math definition (as the most general of equations) "The p-th percentile of N ordered values is obtained by first calculating the rank n [where] n = N/100*p +1/2
But, if I knew “p” I wouldn’t need the bloody formula! It’s like, you have to find rank (n) before you can solve for percentile § , but I’d need to know percentile § before I can find the order (n). …And, what the heck is an ordered value (N), anyhow? How is one going to find that? I know, it’s a statistics thing…you [nor anyone else] wouldn’t understand, right? Don’t you just love math talk? - Jinx
The nth percentile is the 100th part of a group having n/100 of the group below it and (100-n)/100 above it.
If you are described as being in the nth percentile of the entire population for some property (height, wealth, age, number of freckles), then you would expect that n% of that population would have less of that property than you, and (100-n)% of that population would have more, with the simplifying assumption here that the property is infinitely finely divided so that nobody has exactly the same amount as you.
I admit, it’s a bit unclear, but if I understand correctly, they’re telling you how to find which value, out of a set of values, corresponds to the pth percentile. Like, if you wanted to know what score on a test corresponded to the 90th percentile: the p in that case would be 90. N would be how many test scores there are altogether. The formula, then, would give you n, the position of the score you’re looking for. For example, if you used this formula to find what score corresponded to the 90th percentile, and it gave you n=47, the score corresponding to the 90th percentile would be the 47th from the bottom (if they’re arranged in ascending numerical order).
Suppose you take a test that has 25 questions. You get 15 questions right. What would your score be? As each question is worth 4 points (25 x 4 = 100) your score would be 60%. Now suppose that out of all the students in the class, you grade was in the middle. You would be at the 49th percentile, which means that you did better than half of the students in your class (note that it doesn’t matter whether there are 50, 500, or 50,000 students.) On the other hand, if you did the best, you would be at the 99th percentile, because you’ve done better than 99% of the other people taking that exam. (Of course these are simplified - it could actually be 99.99% etc.) So your grade of 60% gives information about much material you answered correctly, but the percentile ranking (of 50th, 99th, etc.) gives information about how you did compared to others.
Suppose you take a test that has 25 questions. You get 15 questions right. What would your score be? As each question is worth 4 points (25 x 4 = 100) your score would be 60%. Now suppose that out of all the students in the class, you grade was in the middle. You would be at the 49th percentile, which means that you did better than half of the students in your class (note that it doesn’t matter whether there are 50, 500, or 50,000 students.) On the other hand, if you did the best, you would be at the 99th percentile, because you’ve done better than 99% of the other people taking that exam. (Of course these are simplified - it could actually be 99.99% etc.) So your grade of 60% gives information about much material you answered correctly, but the percentile ranking (of 50th, 99th, etc.) gives information about how you did compared to others.
A percentile is a group of people or things that together comprise a particular single ordered 1/100th of a statistical universe when it is arrayed according to a parameter that the percentile is a rank under. For example, if your universe is 500 students who took a given math test, and you’re using an ascending ranking of their scores on that test, the 99th percentile is the students who scored 6th to 10th best of all 500 students. (The 100th percentile would be the top five; the 1st percentile, the five students who did the absolute worst.)
Percentages are used generally where it is useful to have a convenient way to express what proportion of a given statistical universe has a particular characteristic. Whether your sample size is 200 or 2,000,000, approximately 2.5% of a randomized population will have Type B blood. (The figure is done from memory as an example and is not intended to be an accurate statement.)
In short, a percentile is an ordinal figure showing where on a given scale a particular grouping falls. If you’re in the 95th percentile, you rank above 94% of the populace in that particular measurement, and below 5% of it. A percentage, on the other hand, is a cardinal number representing a group of X out of a universe of Y as a quantity of N 1/100ths, or N%. E.g., if you have 3,828 widget customers and 531 of them bought the deluxe edition of the widget, you can work with those awkward numbers, but it’s handy to know that just under 14% of your customers bought the deluxe version of the widget.
Percentages are 100-based, and are typically used with discrete variables. So one could classify the severity of incoming ER casualties on a scale from Mild, to Moderate, to Severe and to Dead on Arrival. One would then summarize a sample of ER admissions as "x% of arriving ER cases presented as Mild, "y% of arriving ER cases presented as Moderate,"z% of arriving ER cases presented as Severe and “w% of arriving ER cases presented as Dead on Arrival.”
Percentages can refer to variable values taken on by a population, or by a sample.
Percentiles are also 100-based, but are used with continuous variables, or with ordinal discrete variables with lots and lots of values. One can define a percentile for any value between 0 and 100. The 0th Percentile is the minimum value observed in a sample, or observable in a population. The 100th Percentile is the maximum value observed in a sample, or observable in a population.The median is the 50th Percentile, and is the “middle value” in a sample, or a population.
As an example, one could consider age at death(lifespan) in a random sample of adults. If 50 were the 20th percentile, that would mean that approximately 20% of the sampled adults died at age 50 or younger.
Percentiles can be used to describe samples. A common use of percentiles is in a reference application. Percentiles are compiled for a population of interest, and individual values can be compared to these reference values.
One’s SAT or ACT or GRE or IQ scores are frequently reported as both an absolute score or scores, and also in terms of referent percentiles. There are also referent percentile charts in medical usage, and by checking a patient’s score against reference charts a problem can be identified.