Editing Quantile (section)

=== Relationship to the mean ===
For any population probability distribution on finitely many values, and generally for any probability distribution with a mean and variance, it is the case that
<math display="block">\mu - \sigma\cdot\sqrt{\frac{1-p}{p}} \le Q(p) \le \mu + \sigma\cdot\sqrt{\frac{p}{1-p}}\,,</math>
where {{mvar|Q(p)}} is the value of the {{mvar|p}}-quantile for {{math|0 < ''p'' < 1}} (or equivalently is the  {{mvar|k}}-th {{mvar|q}}-quantile for {{math|1=''p'' = ''k''/''q''}}), where {{mvar|μ}} is the distribution's [[arithmetic mean]], and where {{mvar|σ}} is the distribution's [[standard deviation]].<ref>{{Cite journal |last1=Bagui |first1=S. |last2=Bhaumik |first2=D. |date=2004 |title=Glimpses of inequalities in probability and statistics |journal=International Journal of Statistical Sciences |volume=3 |pages=9–15 |issn=1683-5603 |url=http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/01/P3.V3s.pdf |access-date=2021-08-12 |archive-date=2021-08-12 |archive-url=https://web.archive.org/web/20210812115620/http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/01/P3.V3s.pdf |url-status=dead }}</ref>  In particular, the median {{math|1=(''p'' = ''k''/''q'' = 1/2)}} is never more than one standard deviation from the mean.

The above formula can be used to bound the value {{math|''μ'' + ''zσ''}} in terms of quantiles.
When {{math|''z'' ≥ 0}}, the value that is [[standard score|{{math|''z''}} standard deviations above the mean]] has a lower bound
<math display="block">\mu + z \sigma \ge Q\left(\frac{z^2}{1+z^2}\right)\,,\mathrm{~for~} z \ge 0.</math>
For example, the value that is {{math|1=''z'' = 1}} standard deviation above the mean is always greater than or equal to {{math|1=''Q''(''p'' = 0.5)}}, the median, and the value that is {{math|1=''z'' = 2}} standard deviations above the mean is always greater than or equal to {{math|1=''Q''(''p'' = 0.8)}}, the fourth quintile.

When {{math|''z'' ≤ 0}}, there is instead an upper bound
<math display="block">\mu + z \sigma \le Q\left(\frac{1}{1+z^2}\right)\,,\mathrm{~for~} z \le 0.</math>
For example, the value {{math|''μ'' + ''zσ''}} for {{math|1=''z'' = −3}} will never exceed {{math|1=''Q''(''p'' = 0.1)}}, the first decile.