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Let's just say we know 1st to 99th percentiles of the height of men in country X.

Could I use this data to answer the question:

What is the probability of a male in this country being greater than 6ft?

If I have the following percentile information:

enter image description here

We can see that 6ft is the 60th Percentile.

So the probability of a male being greater than 6ft is 1-(60/100) = 0.4 or 40%

This makes sense to me, however it has been many years since stats class.

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  • $\begingroup$ As a general warning, just from your data you can draw the conclusion you asked about as explained in the answer but you cannot make draw any conclusions on the average or variance just from the data table. Just from the table it could be possible that there are say 0.5% of people who are 200 feet tall which would totally throw off any averages computed from the numbers you have. Only if you also include context and use the fact that your data represents height of real human beings you can exclude such scenarios. $\endgroup$
    – quarague
    Aug 9, 2023 at 7:15

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Yes, your interpretation is exactly correct. The table tells us that 60% of men are 6 foot or under. Therefore, the probability of someone being taller than that is one minus 60%, or 40%. (I am skipping over some nitpickery concerning "less than or equal" as opposed to "less than", which really does not make a difference.)

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    $\begingroup$ Thank you. My googling could not really find anything to state definitively that percentiles could be used in this specific way. Could a (approximate) probability density function also be created from these percentiles, which would allow us to get the area under the curve and therefore probability that someone is greater than 6ft? $\endgroup$
    – SCool
    Aug 8, 2023 at 11:31
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    $\begingroup$ Absolutely. What you would do is to first plot an empirical cumulative distribution function to your data, which you can simply read off your table: the ECDF starts at 0, jumps to 0.01 at 4, to 0.02 at 4.1, and so on until it jumps to 0.99 at 8.2. (In principle, it should jump to 1 somewhere beyond that.) The PDF is the derivative of that, you would need to approximate a bit with your step function. Then the probability you are looking for is the integral of the PDF from 6.0 to infinity, which... drum roll... is just one minus the value of the CDF at 6.0, i.e., exactly what is in the table. $\endgroup$ Aug 8, 2023 at 11:39
  • $\begingroup$ I'm perplexed with "does not make a difference". I would think the proper technical answer for percent of people over 6' would be (1 - 0.61) = 39%, as 6'1" or above is greater than 6' (1% of people are 6'). Indeed if we're talking exact values to absolute precision, basically no one will be EXACTLY 6', and your idea fits. But in reality in society we tend to measure heights in stratified values (as alluded to by the table increments). It's a nitpick, but I'd think it reasonable on like a homework or test to be graded on how you treat that nitpick nonetheless; it does make a (small) difference $\endgroup$ Aug 9, 2023 at 6:40
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    $\begingroup$ @JeopardyTempest: of course you are completely right. However, when we are discussing how to get from percentile tables to probabilities, I would say that a discussion of these details is simply too technical. Perhaps I should have written "makes a small difference, and a discussion of this would confuse the main point". $\endgroup$ Aug 9, 2023 at 6:43
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    $\begingroup$ @JeopardyTempest: ah, but 6' is above average, so one would expect slightly less than half the people listed at 6' to be above 6', and slightly more to be below! $\endgroup$ Aug 9, 2023 at 6:47

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