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Weighted Arithmetic Mean in Ancient India ̊І ˨Ͻ˨Ξϑі˨ ͬϞͻ˨Ξ ̙Ϟϑϑ˨

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Weighted Arithmetic Mean in Ancient India ̊І ˨Ͻ˨Ξϑі˨ ͬϞͻ˨Ξ ̙Ϟϑϑ˨ ͖͑χϑΎξ͖̐˨ͱ Weighted Arithmetic Mean in Ancient India ̊І ˨ͻ˨ξϑІ˨ ͬϟͻ˨ξ ̙ϟϑϑ˨ ߝ vȘɾɟȩƇʕżɾǞȩȘ Òׯ»× ŁÁÎ »ÒÜÎł Á¨ ×­¯Ò Ŋ¼×ε ×¼¼ìŌ ŁµÒÁ µµ ŊåΩŌł Ò³Ò ×Á ¯¼×¯¨ì ×­× ×ì̯µ ÁÎ ¼×ε åµÜ Á¨ ×­ ¯Ò×ίÜׯÁ¼ æ­¯­ æÁܵį ¯¼ ÒÁ» Ò¼Òį ¯×Ò ůŬ ūŬŬŷ ŹŬŮŨŹū ×­× VĮĮ E­µ¼Á¯Ò ­ ¨ÁÎ ×­ ËÐ ÇÅÇ˶Ð}ЪÜĮ a­ žÎ¯×­»×¯ E¼į E¯¼ ¼ EÁ »×­»×¯µ ¼ Ò¯¼×¯¨¯ ¯¼×µµ× Á¨ ¼¯¼× Î ×­Î Ò׼Π»ÒÜÎÒ Á¨ ¼×ε ×¼¼ì ÜÒ ¯¼ 2¼¯ ¯Ò ëÌÎÒÒ ¯¼ ×­ åÎì ¼» Z}¶­¨ã ×­× ­ Ò×ׯÒׯÒĮ a­ÜÒį ×­ ÌÎ¯Ò Î¯×­»×¯ ŁÎ×­Îį µ©Î¯ł a­ÁÒ ¨ÁÎ ×­ ±ÁÜμµ Á¼ Ò×ׯÒ×¯Ò ×­× ­ ¨Áܼ ¯¼ ĂĊĄĄĮ 2¼ ¨Áλܵ ŁĂł ¯Ò å¯æ ¯¼ Ò×ׯÒ×¯Ò Ò ¼ Òׯ»× Á¨ ×­ ¼×ε ×¼¼ì Á¨ ¯Ò×ίÜׯÁ¼Į ×­ ¯¼Ü©Üε ¯×Áίµ Á¨ Z}¶­¨ã [1]į Ò µÒÁ ¯¼ ­¯Ò µÎ× Îׯµ Ŋp­ì \×ׯÒׯÒĵŌ [2]į E­µ¼Á¯Ò ­¯©­µ¯©­×Ò ÒÁ» Á¨ p­¼ ×­ n ÍÜ¼×¯×¯Ò x1,x2,...,xn Î ÒÒ¯©¼ ×­ ¼¯¼× 2¼¯¼ ¯Ò Á¼ »¯¼¯Ò×Îׯå Ò×ׯÒׯÒį ¼ 毩­×Ò w1,w2,...,wn ÎÒÌׯåµìį ×­¼ ×­ »ÁÎ ©¼Îµ Î»Î³Ò ×­× ×­ ŮÇШ}ÌËÐÇ} Á¨ <Üׯ¯µì ŊÁ¼×¯¼Ò ׯµ Á¼Ì× Ŋ毩­× ί׭»×¯ »¼Ō x ¯Ò ¨¯¼ Ò ÒίÌׯÁ¼ ¨ÁÎ ×­ Á¼Ü× Á¨ ©Î¯Üµ×Üεį ÌÁÌܵׯÁ¼į ¼ w x + + w x Á¼Á»¯ ¼ÒÜÒÒ ¯¼ 寵µ©Ò Ò æµµ Ò ¯¼ ¯×¯Ò ¼ ×Áæ¼Ò Á¼ x = 1 1 ··· n n . UkV w + + w Òµ æ­¯­ ¯Ò ÎÎ ¯¼ ¼ì Áܼ×Îì å¼ × ×­ ÌÎÒ¼× ×¯»Į 1 ··· n a­ ׯµ ÒίÌׯÁ¼ Á¨ Á¼×»ÌÁÎÎì ¯¼ÜÒ×ίµ ¼ Á»»Î¯µ ÌÎׯ ÌÁ¯¼×Ò ×Á ­¯©­µì åµÁÌ Ò×ׯÒׯµ ߝࡏߞ ÷Ǖƚ ÚȩɔʕȀŏɟǞɾʿ ŏȘƇ vȒɔȩɟɾŏȘżƚ ȩǀ ɾǕƚ ÒìÒ×»ĮŌ Ł[2]į ÌĮ ĂĊćł ɟǞɾǕȒƚɾǞż ƚŏȘ žÒ ×ίÜ× ×Á VĮĮ E­µ¼Á¯Ò Á¼ ­¯Ò ĂăĆ×­ ¯Î×­ ¼¼¯åÎÒÎì Ł­ æÒ Áμ Á¼ ăĊ 4ܼ ĂĉĊĄłį æ »¼×¯Á¼ a­ žÎ¯×­»×¯ E¼ ŁžĮEĮ ¯¼ Ò­ÁÎ׳ ¯Ò ¨ÎÍܼ׵ì ÜÒ ¼Á× ¨æ Á¨ ×­ Òåε ¨ÁλܵׯÁ¼Ò ¼ Ì̵¯×¯Á¼Ò Á¨ ×­ Á¼µì ¯¼ Ò×ׯÒ×¯Ò ¼ »×­»×¯Òį Ü× µÒÁ ¯¼ ëÌί»¼×µ žÎ¯×­»×¯ E¼ ×­× ÁÜÎ ¯¼ ×­ æÁÎ³Ò Á¨ ¼¯¼× 2¼¯¼ Ò¯¼į Á¼Á»¯Òį ÒÁ¯ÁµÁ©ìį ¼ Á×­Î ¯åÎÒ »¯ »×­»×¯¯¼ÒĮ ¯Ò¯Ìµ¯¼ÒĮ +Á»×ίµµìį ×­ ί׭»×¯ »¼ ĿÍÜׯÁ¼ p ¨¯ÎÒ× »³ ¨æ ¯¼×ÎÁÜ×ÁÎì Î»Î³Ò Á¼ ×­ Á¼Ì× ŁĂłŀ Ì×ÜÎÒ ×­ Á¼Ì× Á¨ ×­ Ŋ¼×ÎŌ Á¨ Á¼¨¯©ÜÎׯÁ¼ Ń Á¨ ×­ žÎ¯×­»×¯ E¼ ¼ ¯×Ò ­¯Ò×ÁÎìĮ ×­ ÁÁί¼×Ò Á¨ ×­ Ŋ¼×ÎÁ¯Ō Á¨ ×ί¼©µ ŁÁÎ ¼ì Á×­Î ¨¯©ÜÎ Áܼ ì µ¯¼ Ò©»¼×Òł ¯Ò ×­ žÎ¯×­»×¯ E¼ Á¨ ߝࡏߝ ɟǞɾǕȒƚɾǞż ƚŏȘ ǞȘ ŏɾǕƚȒŏɾǞżɯ ŏȘƇ ×­ ÁÁί¼×Ò Á¨ ×­ åÎׯÒĮ 2¼ Ì­ìÒ¯Òį ×­ 毩­× žĮEĮ ëɾŏɾǞɯɾǞżɯ ĿÍÜׯÁ¼ Łăłŀ ÌÌÎÒ ¯¼ ×­ ¨Áλ Á¨ ×­ Ŋ¼×Î Á¨ »ÒÒŌĮ *ÁÎ Ò× Á¨ n ÌÁ¯¼×Ò æ¯×­ ÁÁί¼×Ò `B ¼ »ÒÒÒ miį 2¼ »×­»×¯Òį ×­ žÎ¯×­»×¯ E¼ x Á¨ n ¼Ü»ÎÒ ×­ ÁÁί¼×Ò Á¨ ×­ ¼×Î Á¨ »ÒÒ ¯Ò ×­ 毩­× žĮEĮ x1,x2,...,xn ¯Ò ¨Áλµµì ¨¯¼ ×Á ×­ ¼Ü»Î (m ` + + m ` )/(m + + m )Į 1 R ··· n M 1 ··· n a­ ¨¯¼¯×¯Á¼ Á¨ ×­ žÎ¯×­»×¯ E¼ ĿÍÜׯÁ¼ ŁĂłŀ ¯Ò x + x + + x x = 1 2 ··· n . URV µ©¼×į Òì ×Á ܼÎÒ×¼į ¼ ¯× ¯Ò Ò×ί©­×¨ÁÎæÎ ×Á æÎ¯× n Á»ÌÜ×Î ÌÎÁ©Î»» ¨ÁÎ Á»ÌÜׯ¼© ×­ žÎ¯×­»×¯ E¼Į 2× ¯Ò ¼ ë× »×­»×¯µ Á¼Ì×Į Ü× ¯× ¯Ò 毵ì Ì̵¯ 2× ¯Ò ׭ΨÁÎ ¼Á× ÒÜÎÌίү¼© ×­× ×­ žÎ¯×­»×¯ E¼ ­Ò ¯¼ ×­ Ò×Üì Á¨ ÌÌÎÁ믻ׯÁ¼ÒĮ *ÁÎį ×­ Á¼Ì× ÌÌÎÒ ¼ ×­ »ÁÒ× Á»»Á¼µì ÜÒ »ÒÜÎ Á¨ ¼×ε ×¼¼ì ¯¼ ×­ ëÌί»¼×µ Ò¯¼Ò ¯¼ ¯×Ò ¨ÁµµÁ毼© å×Îĭ ×­ ¼ ×­ æÁÎ }ÜÇ}£ ­Ò ÌÎׯµµì Á» Òì¼Á¼ì» žÎ¯×­»×¯ E¼ Á¨ »ÒÜλ¼×Ò ¯¼ ÎÁÎ × ¯Ò ¨ÁÎ ×­ žÎ¯×­»×¯ E¼Į žÌÎ× ¨ÎÁ» ¯¼© »ÒÜÎ Á¨ ×­ ÒÜ» Á¨ µµ ×­ »ÒÜλ¼×Ò ¯å¯ ì ×­ ¼Ü»Î ¼×ε ×¼¼ìį ×­ žĮEĮ ¯Ò µÒÁ ÜҨܵ Ìλ×Î ¨ÁÎ Á¨ »ÒÜλ¼×ÒĮ ž¼ Ì­ìÒ¯µ »ÒÜλ¼× ¯Ò ÒÁ»×­¯¼© ¯¼Ò¯©­×Ò ¯¼×Á Á×­Î ¨×ÜÎÒ Á¨ ©¯å¼ × µ¯³ ×­ åί¼ ¯¼×ί¼Ò¯µµì ÌÌÎÁ믻×Į ÁÎ ×­ Ò׼Πå¯×¯Á¼ æ­¯­ »ÒÜÎ ×­ ×¼¼ì Á¨ a­Î ¯Ò ¼Á×­Î æì Á¨ µÁÁ³¯¼© × ×­ ܵ ÎÁµ Á¨ ¯ÒÌÎÒ¯Á¼ ¯¼ ×­ ×Į \Á» Á¨ ×­ ÌÎ Á¼Ì×ܵ ÒÌ×Ò ×­ žÎ¯×­»×¯ E¼ Ò ŊÌÜÎŌ Ü» ŊÌ̵¯Ō Á¼Ì×Į Á¨ ×­ žÎ¯×­»×¯ E¼ Î »¼×¯Á¼ ¯¼ ÒׯÁ¼ ăĮĆĮ VÎׯµ ëÌί¼ Ò­ÁæÒ ×­× ¼Ü»ÎÒ ¯¼ Ò×ׯÒׯµ × 2¼ Ò̯µ µ×ÜÎ × ×­ 2¼¯¼ \×ׯÒׯµ 2¼Òׯ×Ü×į ÜÒܵµì ×¼ ×Á µÜÒ×Î ÎÁܼ ÒÁ» ¼×ε åµÜĮ ž¼ì <Áµ³×į Á¼ ×­ *¯ÎÒ× pÁε \×ׯÒ×¯Ò ì ŁăāĮĂāĮăāĂāłį \ĮEĮ Ȓŏɟɾʿŏ ʕȒŏɟ 4ʕɾɾŏ ࡫K`ivXk3!;KBHX+QK࡬ Ǟɯ ŏɾ ɾǕƚ ëɾŏɾࡷŏɾǕ ĆȘǞɾ ȩǀ ɾǕƚ vȘƇǞŏȘ ëɾŏɾǞɯɾǞżŏȀ vȘɯɾǞɾʕɾƚࡈ ȩȀǺŏɾŏࡏ ÷ǕǞɯ ŏɟɾǞżȀƚ Ǟɯ ŏ ɯȀǞǃǕɾȀʿ ŏųɟǞƇǃƚƇ ŏȘƇ ȒȩƇǞ˙ƚƇ ʲƚɟɯǞȩȘ ȩǀ [7]ࡏ ̊͑́ϻ˨;́ Ύ̐ϑ ߞߜߝߣ Ͼ̥͖͇͑ϑ̥̙ ˨ࡏͻࡏ ͖; ˨;͖̥̐;ϑ ͖;̙͖˨ ृߝ · ͖͑χϑΎξ͖̐˨ͱ \ׯ©µÎ Ò¯¼©µ ÁÜ× ×­ žÎ¯×­»×¯ E¼ Ò ×­ ¨¯ÎÒ× Á¨ Á¨ ݪ£¨Ð ŮǪШµÐª C}¶ ĿÍÜׯÁ¼ ŁăłŀĮ ž¼ ¯¼×ܯׯå ×­ Ŋ*¯å 2Ò ×­× ­¼© \×ׯÒ×¯Ò ¼ Á¼×¯¼Ü ×Á æμÒÒ Á¨ ×­ µæ Á¨ µÎ© ¼Ü»ÎÒ ¨ÁÎ žĮEĮ µÒÁ Á»Ò ­¼© ×­ pì p a­¯¼³ žÁÜ× ×­ pÁεŌĮ a­ Ò¯ ÁÜ× ¯¼ Á»»¼×Îì ì ×­ »×­»×¯¯¼ +¼ŻÒŻ ŁĂĆąĆ ¯ × ×­ ­Î× Á¨ ×­ žÎ¯×­»×¯ E¼ ¯Ò ×Á Á»¯¼ ŪŬłŃÒ ÒׯÁ¼ ĄĮćĮ ÁÒÎåׯÁ¼ÒŃл Çů} ËÜÇ}¯ ¶ÓµÇË ã } ˪¶£¯ ¶ÓµÇĮ 2¼ ÒׯÁ¼ Ąį æ Ò­µµ ÍÜÁ× Ò××»¼×Ò Á¼ žĮEĮ a­¯¼³ÎÒ Á¼ Ò×ׯÒ×¯Ò µ¯³ \ׯ©µÎ [24] ­å ÌÁ¯¼× ÁÜ× ×­× ¨ÎÁ» ­Ì×ÎÒ ¯¼ 2¼¯¼ ί׭»×¯ µµ ­¨Ð}ĵÜã}Ü}¨Ç} ×­¯Ò ¯ ¯Ò Áܼ×ί¼×ܯׯåĭ ¨ÁÎ ¯× Ò³Òį ÌÎÁ믵µìį ×Á Ł»×­»×¯µ ÌÎÁÒÒÒ ÌÎׯ¼¯¼© ×Á ëåׯÁ¼ÒłĮ a­Ò ©¯¼ ¯¼¨ÁλׯÁ¼ ÁÜ× ×­ × ì ¯Òί¼© ¯¼¨ÁλׯÁ¼į ­Ì×ÎÒ Òί ­Áæ ×Á Á»ÌÜ× ×­ ŁåΩł Ì×­į ¼»µìį ×­ ¯¼¯å¯Üµ¯×ì Á¨ ×­ ÁÒÎåׯÁ¼ÒĮ Ò¯Òį ¯¼ ×­ æ¯×­ ÁÎ µ¼©×­ Á¨ ¼ ¯ÎΩܵÎŅÒ­Ì ÌÁÁµ Á¨ æ×Î ¼ æÁÎÒ Á¨ \ׯ©µÎ Ł[24]į ÌĮ Ăąłĭ ×­Îì Òׯ»× ¯×Ò åÁµÜ»Į 2× ¯Ò ¯¼ ×­Ò ­Ì×ÎÒ ×­× ×­ In ancient and even modern times, too much žÎ¯×­»×¯ E¼ ¯Ò ¨¯¼ ¼ ÜÒ ¯¼ Ò×ׯÒׯµ Ò¼Òį familiarity with the circumstances of each ¯ĮĮį Ò ×­ Ò× ÎÌÎÒ¼×ׯå åµÜ ¨ÁÎ Ò× Á¨ ÁÒÎåׯÁ¼ÒĮ observation could undermine intentions to έ»©ÜÌ× ¯Ò Á¼ Á¨ ×­ ε¯Ò× 2¼¯¼ »×­»×¯¯¼Ò ¯¼ combine them. The strong temptation is, and has æ­ÁÒ ×ë× ×­ Á¼Ì× Á¨ žÎ¯×­»×¯ E¼ Ł¯¼ ¨×į 毩­× always been, to select one observation thought to žĮEĮł ÁÜÎÒ ë̵¯¯×µì ¯¼ ×­¯Ò Ò×ׯÒׯµ Ò¼ÒĮ 0 ÜÒÒ ¯× ×Á be the best, rather than to corrupt it by averaging ÎÌÎÒ¼× ×­ Ì×­ Á¨ ¯×­į æ­¼ ×­ Ì×­ ¯Ò ¯¨¨Î¼× ¯¼ with others of suspected lesser value. ¯¨¨Î¼× ÌÁÎׯÁ¼Ò Á¨ ×­ ¯×­Į Ü× ÌέÌÒ ×­ ¼Ü»ÎÁÜÒ ¼Áåµ ¯Ò ¯¼ ×­ æÁγ Á¨ ÁµÁÒÒÜÒ µ¯³ έ»©ÜÌ× a­ÜÒį µ×­ÁÜ©­ ×­ žÎ¯×­»×¯ E¼ Łµ¯³ ×­ ¯»µ ­å ÁåÎÒ­Áæ ×­¯Ò ÁÜÎμ Á¨ ×­ žÎ¯×­»×¯ E¼ ÒìÒ×»ł ¼Áæ Ò»Ò ¼×Üε Á¼Ì× ×Á ÜÒ æ­Á ­å ©ÎÁæ¼ ¯¼ ­¯Ò æÁγĮÞ oÎì μ׵ìį έ»©ÜÌ×ōÒ æÁγ ­Ò ¼ ÜÌ æ¯×­ ¯×į ¯×Ò ¯¼×ÎÁÜׯÁ¼ »ÜÒ× ­å ¼ Á¼Ì×ܵ ³¼Á浩 ì \ׯ©µÎ ¯¼ [24] ŁÌĮ ĄāłĮ ÒÜ×µ×ìĮ a­ ¯ Á¨ žĮEĮ Ò Ò×ׯÒׯµ Òׯ»× ÌÌÎÒ ¯¼ žÒ »¼×¯Á¼ ε¯Îį ×­ žÎ¯×­»×¯ E¼ ŊÁ»¯¼ÒŌ Ò¯¼×¯¨¯ ×ÎׯÒÒ Á¨ ÜÎÁÌ ÍÜ¯× µ×Į Òåε ¼Ü»ÎÒ ¯¼×Á Ò¯¼©µ ¼Ü»ÎĮ ž¼¯¼× 2¼¯¼ »×­»×¯Òį Ò̯µµì ×­ »×­»×¯Ò Á¨ έ»©ÜÌ×į ߝࡏߟ ȘɾǞɛʕǞɾʿ ȩǀ ɾǕƚ ɟǞɾǕȒƚɾǞż ƚŏȘ ÌÌÎÒ ×Á ÎÌµ× æ¯×­ åίÁÜÒ ¯Ò Á¨ ŊÁ»¯¼×¯Á¼ŌĮ 2¼ ×­ ¯¼Ü©Üε ¯ÒÒÜ Á¨ ¨Ü}¶į æ ­ Ò¼ ×­× \åε Ò­ÁµÎÒ Á¼ ­¯Ò×ÁÎì Á¨ Ò×ׯÒ×¯Ò ­å ¼ ÒÒÎׯ¼© έ»©ÜÌ× ­ ¯¼×ÎÁÜ Ìί¼¯Ìµ Á¨ Á»ÌÁүׯÁ¼ ×­× ¯× ¯Ò Á¼µì ¨ÎÁ» ×­ ĂĈ×­ ¼×ÜÎì ŪŬ ×­× Á¼ Á»Ò ÎÁÒÒ æ­¯­ »µª¶Ë ×æÁ ÒÁµÜׯÁ¼Ò Á¨ Îׯ¼ ÍÜÎׯ ÍÜׯÁ¼ ×­ ÜÒ Á¨ ×­ žÎ¯×­»×¯ E¼ Ò ÎÌÎÒ¼×ׯå åµÜ ¨ÁÎ ¯¼ ×­Î åίµÒ ×Á ÌÎÁÜ ¼Á×­Î ÒÁµÜׯÁ¼ Á¨ ×­ ¼ì × Á»Ìίү¼© »ÁÎ ×­¼ ×æÁ ÁÒÎå åµÜÒĮ a­ì ÍÜׯÁ¼į Ìί¼¯Ìµ æ¯×­ »Á»¼×ÁÜÒ Á¼ÒÍÜ¼Ò ¯¼ Ò ¯¼ æÁγ × ĂćĄĆ Á¨ ×­ ¼©µ¯Ò­ Ò×ÎÁ¼Á»Î 0¼Îì »×­»×¯Ò [6]Į 2¼ ¨×į ×­ ¼» Á¨ ×­¯Ò Ìܵ¯×¯Á¼ ¯Ò +µµ¯Î¼ ×­ ε¯Ò× Ü¼»¯©ÜÁÜÒ ÜÒ Á¨ ×­ žÎ¯×­»×¯ ¯¼Ò̯Πì ×­ ×λ ¨Ü}¶ ÜÒ ¯¼ 2¼¯¼ »×­»×¯Ò E¼ ¯¼ Ò×ׯÒׯµ Ò¼ÒĮ 2¼ ­¯Ò ÌÎÒ¯¼×¯µ ÎÒÒ × ×­ ¨ÁΠέ»©ÜÌ×ōÒ µæ Á¨ Á»ÌÁүׯÁ¼Į \¯¼ ŊŁ¨¯¼¯¼© ìł ž»Î¯¼ \×ׯÒׯµ žÒÒÁ¯×¯Á¼ ¯¼ ĂĊĈĂį ­Üέ¯µµ ¯Ò¼­Î× Á»¯¼×¯Á¼Ō ¯Ò Á¼ Á¨ ×­ »¼¯¼©Ò Á¨ ×­ \¼Ò³Î¯× æÁÎ [8] Î»Î³Ò Ł į ÌĮ Ăąłĭ ¨Ü}¶ Ł[6]į ÌĮ Ăąłį Á¼ ¼ å¯æ ×­ žÎ¯×­»×¯ E¼ ×ÁÁ Ò ×­ ÁÜ×Á» Á¨ ¨Ü}¶ı …I fully expected that I would find some good examples of mean taking in ancient astronomy; 2¼ ÒׯÁ¼ ąį æ Ò­µµ ÍÜÁ× Ò××»¼×Ò Á¼ žĮEĮ ¨ÎÁ» and, perhaps, also in ancient physics. I have not ­Ì×ÎÒ ¯¼ 2¼¯¼ ί׭»×¯ µµ µªÌÇ}­}ĵÜã}Ü}¨Ç} found any.And I now believe that no such examples ŁÁ»ÌÜ×ׯÁ¼Ò ÌÎׯ¼¯¼© ×Á »¯ë×ÜÎÒł æ­¯­ ÎÒÒ ×­ will be found in ancient science. ÌÎÁµ» Á¨ Á»ÌÜׯ¼© ×­ ÌÎÁÌÁÎׯÁ¼ Á¨ ÌÜÎ ©Áµ ¯¼ ¼ µµÁì ¨Áλ ì µ¼¯¼© Á¨ Òåε Ì¯Ò Á¨ ©Áµ Á¨ ¯Ò¼­Î× ÁÒÎåÒ ×­× ¼¯¼× Ò¯¼×¯Ò×Ò ¯ ¼Á× ¯¨¨Î¼× 毩­×Ò ¼ ÌÜίׯÒĮ 0Îį 毩­× žÎ¯×­»×¯ ­å ¼ì ÌÎ¯Ò »×­Á ¨ÁÎ ­ÁÁÒ¯¼© Ò× Òׯ»× E¼ ÌÌÎÒ Ò ¼ ë× »×­»×¯µ Á¼Ì× Î×­Î Á¼ ×­ Ò¯Ò Á¨ Òåε ÁÒÎåׯÁ¼ÒŃ×­ì ε¯ Á¼ ×­¼ Ò ÎÌÎÒ¼×ׯå ÁÎ ¼ Òׯ»× ¯¼ Ò×ׯÒׯµ ÎÁ¯×µì ­ÁÒ¼ ÁÒÎåׯÁ¼ÒĮ Ü×į Ò ¯¼ ×­ Ò Á¨ Ò¼ÒĮ 0ÁæåÎį å¼ ¨ÁÎ Ò­ÁµÎÒ ¯¼×ÎÒ× Ìί»Î¯µì ¯¼ ¼Ü»ÎÁÜÒ Á×­Î Ò¯¼×¯¨¯ Á¼Ì×Òį ÒÜ­ ¼ ÁÜ¼× Á¨ ×­ ­¯Ò×ÁÎì Á¨ ×­ Ò×ׯÒׯµ žÎ¯×­»×¯ E¼į ×­ ÌÒÒ©Ò ­¯Ò×ÁÎì Á»Ìµ×µì ÁåεÁÁ³Ò ×­ µÎ ¼ ÌÎ¯Ò ÜÒ Á¨ Á¼ µªÌÇ}­}ĵÜã}Ü}¨Ç} »ì ©¯å ÌÎÒÌׯå Ωί¼© ×­ ×­ žÎ¯×­»×¯ E¼ ¯¼ ×­ ×ÎׯÒÒ Á¨ ¼¯¼× 2¼¯¼ »×­»×¯µ ¼å¯ÎÁ¼»¼× æ­¯­ ¨¯µ¯×× ×­ »Î©¼ »×­»×¯¯¼Ò µ¯³ έ»©ÜÌ× Łćăĉ ŪŬłį ]ÎǢ­ÎÎì Á¨ ×­¯Ò Ò×ׯÒׯµ ¯Į žÌÎ× ¨ÎÁ» ÌÎÁµ»Ò Á¼ Á»¯¼¯¼© ŁĮ ĈĆā ŪŬłį E­åǢÎÎì ŁĉĆā ŪŬłį Vί׭â³Òå»Ǣ Łĉćą ©Áµ ̯Òį ×­Î Î Á×­Î Á¼×ë×Òį µ¯³ ÌÎÁµ»Ò Á¨ ŪŬłį ­Ò³ÎÎì ŁĂĂĆā ŪŬł ¼ Á×­ÎÒĮ 2¼ ¨×į Ò æ Ò­µµ Á»¯¼¯¼© ¯¨¨Î¼× ¯¼åÒ×»¼×Ò Łæ¯×­ Ò¯»Ìµ ¯¼×ÎÒ׳ ¯¼×Á ¯µµÜÒ×Î× ¯¼ ÒׯÁ¼Ò Ą ¼ ąį ×­Ò 2¼¯¼ »×­»×¯¯¼Ò ¼ ÍÜ¯åµ¼× Ò¯¼©µ ¯¼åÒ×»¼×į æ­¯­ ­å µÒÁ µ ×Á ¨¯¼ ¼ Ì̵ì ×­ »ÁÎ ©¼Îµ ¼ ÒÁÌ­¯Ò×¯× Á¼Ì× µÜµ×¯Á¼Ò Á¨ 毩­× žÎ¯×­»×¯ E¼Į a­ì 毵µ ¼Á× \×Ì­¼ EĮ \ׯ©µÎ ŁĮ ĂĊąĂłį Ò×Ü¼× Á¨ >ܯ¼ > »į ¯Ò ¯Òׯ¼©Ü¯Ò­ Ò×ׯÒׯ¯¼ × ×­ f¼¯åÎÒ¯×ì Á¨ ­¯©Áį ³¼Áæ¼ ¨ÁÎ ­¯Ò æÁγ Á¼ ×­ ­¯Ò×ÁÎì Á¨ Ò×ׯÒׯÒĮ Þ 2¼ ­¯Ò ÌÌÎ ŊVÎÁ¯µ¯×ì ¯¼ ž¼¯¼× 2¼¯Ō Ł¯¼ 0}¶»»­ »¢ Ш T¨ª¯»Ë»Å¨ã »¢ Zª¶Į ŁÒĮ VĮ\Į ¼ìÁÌ­ìì ¼ EĮXĮ *ÁÎÒ×ÎłĮ µÒå¯ÎĮ ăāĂāĮ Ĉĭ ĂĂĈĂńĂĂĊăłį Į<Į X±Ü ÁÒ »¼×¯Á¼ ×­ ÁÜÎμ Á¨ 毩­× žĮEĮ ¯¼ ×­ æÁÎ³Ò Á¨ ]ÎǢ­ÎÎì ¼ Á×­ÎÒ ¯¼ ×­ Á¼×ë× Á¨ »×­»×¯µ Á»ÌÜ×ׯÁ¼Ò Á¼ ×­ ¼Ò¯×ì Á¨ ©ÁµĹ Ü× å¼ ­¯Ò ÁÜ¼× ÁåεÁÁ³Ò ×­ »Ü­ ε¯Î ËÐ}ЪËЪ}¯ ÜÒ Á¨ ×­ 毩­× žĮEĮ ì έ»©ÜÌ×Į ߞ ̊͑́ϻ˨;́ Ύ̐ϑ ߞߜߝߣ · ͖͑χϑΎξ͖̐˨ͱ ¯ÒÜÒÒ ¯¼ ×­ ÌÎÒ¼× ÎׯµĮ 2¼ ×­¯Î ̵¼×Îì »ÁµÒį 0ÁæåÎį ¼Á× ×­× ×­ Vì×­©Áμ ¨¯¼¯×¯Á¼ ¼å¯Ò©Ò ¼¯¼× 2¼¯¼ Ò×ÎÁ¼Á»ÎÒ ÎÁÜׯ¼µì ¯ÒÜÒÒ ×­ Ŋ»¼ ×­ žÎ¯×­»×¯ E¼ Ò ©Á»×ίµ Á¼Ì× Ł×­ »¯ÌÁ¯¼× »ÁׯÁ¼Ō Á¨ ̵¼× æ­¯­ ×ÁÁ 毵µ ¼Á× ¯ÒÜÒÒ ¯¼ ×­¯Ò Á¨ µ¯¼ Ò©»¼×ł ¼ ¼Á× Ò ×­ Ò×ׯÒׯµ Á¼Ì× Á¨ ÎׯµĮ ŊåΩŌ ÁÎ Ò Ŋ×­ Ò× ÎÌÎÒ¼×ׯåŌ »Á¼© ŁÁÎ Ò 2¼ 2¼¯¼ ×ÎׯÒÒ Á¼ ί׭»×¯į ×­ ×Á̯ µªÌÇ}­}ĵ ÒÜÒׯ×Ü× ¨ÁÎł Òåε ¼Ü»ÎÒĮ 2× æÒ ÜÒ ¯¼ ×­ Á¼×ë× Üã}Ü}¨Ç} ¯Ò ¯ÒÜÒÒ »Ü­ ¨ÁÎ ×­ ×Á̯ ­¨Ð}ĵ Á¨ »ÜÒ¯ ¼ ÌÜÎ »×­»×¯Òį ¼ ¼Á× ¯¼ ¼µìÒ¯Ò Á¨ ×Į Üã}Ü}¨Ç}Į p ­å ­ÁæåÎ ­¯©­µ¯©­× ×­ åÎÒÒ ¨ÎÁ» 2¼ ×­¯Î ÁÜ¼× Á¼ ×­ ­¯Ò×ÁÎì Á¨ ×­ žÎ¯×­»×¯ E¼ ­¨Ð}ĵÜã}Ü}¨Ç} ¨¯ÎÒ× Ł¯ĮĮį ¯¼ ÒׯÁ¼ Ął ÜÒ Á¨ ×­¯Î ¼ ×­ E¯¼į ³³Î ¼ +Î廯±Î æÎ¯× Ł[3]į ÌĮ ĂĆąłĭ ¯»»¯× Îµå¼ ¨ÁÎ ×­ ­¯Ò×ÁÎì Á¨ ×­ žÎ¯×­»×¯ E¼ ¯¼ ¯×Ò ËÐ}ЪËЪ}¯ Ò¼ÒĮ Not until the sixteenth century was it recognized that the arithmetic mean could be generalized to more than two cases: a =(x1 +x2 + +xn)/n.
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