सामान्यीकृत माध्य

यदि p एक अशून्य वास्तविक संख्या है, तथा धनात्मक वास्तविक संख्याएँ हैं, तो इन संख्याओं का सामान्यीकृत माध्य (generalized mean) या p घात वाला घात माध्य (power mean) ' निम्नलिखित है- विशिष्ट स्थितियाँ thumb|A visual depiction of some of the specified cases for n 2 with a x1 M+∞, b x2 M−∞, magenta harmonic mean, H M−1(a, b), blue geometric mean, G M0(a, b) red arithmetic mean, A M1(a, b), and lime quadratic mean, Q M2(a, b). निम्निष्ट हरात्मक माध्य (harmonic mean) गुणोत्तर माध्य (geometric mean) समान्तर माध्य (arithmetic mean) वर्ग माध्य (quadratic mean]])Quadratic घन माध्य (cubic mean) उचिष्ट (maximum) {| class="toccolours collapsible collapsed" width="90%" style="text-align:left" !Proof of (geometric mean) |- | We can rewrite the definition of Mp using the exponential function In the limit p'' → 0, we can apply L'Hôpital's rule to the argument of the exponential function. Differentiating the numerator and denominator with respect to p, we have By the continuity of the exponential function, we can substitute back into the above relation to obtain as desired. |} {| class="toccolours collapsible collapsed" width="90%" style="text-align:left" !Proof of and |- | Assume (possibly after relabeling and combining terms together) that . Then The formula for follows from |} इन्हें भी देखें समान्तर माध्य गुणोत्तर माध्य हरात्मक माध्य वर्ग माध्य मूल श्रेणी: माध्य