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I know that exists the Student's t-test for comparing mean and sample mean.

My question is: there exists a similar test to compare the nth percentile?

Or Student's t-test can be used to compare percentiles instead of means?

My hypothesis would ideally be something like: "the 75th percentile of this sample is greater than x"

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    $\begingroup$ Hypotheses are about populations, not samples. You can observe the 75th percentile of a sample, so there's no need to hypothesize about it. $\endgroup$
    – Glen_b
    Nov 3, 2017 at 10:17
  • $\begingroup$ If you have a small amount of data, you could bootstrap the 75th percentile. In fact, you can use the bootstrap as an alternative to the t test itself $\endgroup$
    – call-in-co
    Nov 4, 2017 at 4:26

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One sample binomial proportions tests can be used when making inferences about population quantiles.

For example, if your hypothesis is that the 75th percentile of a continuous variable in some population is 80, then under the null and assuming random sampling the number of values in the sample $\leq 80$ will be binomial$(n,0.75)$.

It sounds like you have a one-tailed alternative; there's no difficulty there.

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    $\begingroup$ The choice of the percentile to compare would need to be "magic", i.e., non-arbitrary. If you have trouble choosing you might just compare the entire distribution, using a Kolmogorov-Smirnov test. $\endgroup$ Nov 3, 2017 at 11:53
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    $\begingroup$ +1 Certainly, if by magic you mean something along the lines of "determined by theory" or at least "determined before you see data", then yes, that's the case, as it generally is with choosing any other population quantity to test. One should not look at the data and then choose a quantile to test using a test like this one, because this doesn't consider the effect of such a choice. (In a similar vein I'd also mention that the very common practice of choosing which of two possible tests to use based on seeing the data is also problematic, for much the same reason.). ... ctd $\endgroup$
    – Glen_b
    Nov 3, 2017 at 23:30
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    $\begingroup$ ctd... The suggestion of using a goodness of fit test (such as the Kolmogorov-Smirnov) - comparing all quantiles if one is unsure of a quantile to compare - is a good thing to mention. Indeed, the Kolmogorov-Smirnov is particularly relevant to this situation, since it would be exactly the right test to use if you were tempted to choose the quantile that was most discrepant (the quantile with the largest size of difference between sample proportion and population proportion below or above it), since that's exactly the test statistic for the Kolmogorov-Smirnov (I guess that's why you raised it) $\endgroup$
    – Glen_b
    Nov 3, 2017 at 23:35
  • $\begingroup$ what if I want to test diff of median or percentiles? $\endgroup$
    – yabchexu
    May 1, 2020 at 2:26
  • $\begingroup$ Perhaps best posted as a new question, though there's a number of relevant posts t be found that might already cover what you need. $\endgroup$
    – Glen_b
    May 1, 2020 at 3:30

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