❈❤÷ì♥❣ ■❱ ❚â♠ t➢t ✈➔ tr➻♥❤ ❜➔② ❞ú ❧✐➺✉ ❜➡♥❣ ❝→❝ ✤↕✐ ❧÷đ♥❣ t❤è♥❣ ❦➯ ♠ỉ t↔ ❇ë ♠ỉ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✻✼ ✴ ✻✻✹ ❈❤÷ì♥❣ ■❱ ✶✵ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✶ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠ỉ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✷ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥ ✶✸ ❙û ❞ư♥❣ ❦➳t ❤ñ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥ ✶✹ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✻✽ ✴ ✻✻✹ ❈❤÷ì♥❣ ■❱ ✶✵ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✶ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠æ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✷ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥ ✶✸ ❙û ❞ư♥❣ ❦➳t ❤đ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥ ✶✹ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✻✽ ✴ ✻✻✹ ❈❤÷ì♥❣ ■❱ ✶✵ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✶ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠ỉ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✷ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥ ✶✸ ❙û ❞ư♥❣ ❦➳t ❤đ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥ ✶✹ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✻✽ ✴ ✻✻✹ ❈❤÷ì♥❣ ■❱ ✶✵ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✶ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠ỉ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✷ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥ ✶✸ ❙û ❞ư♥❣ ❦➳t ❤đ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥ ✶✹ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✻✽ ✴ ✻✻✹ ❈❤÷ì♥❣ ■❱ ✶✵ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✶ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠ỉ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✶✷ ❈→❝ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥ ✶✸ ❙û ❞ư♥❣ ❦➳t ❤đ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥ ✶✹ ❈→❝ ✤↕✐ ❧÷đ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✻✽ ✴ ✻✻✹ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ✶✳ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ✤ì♥ ❣✐↔♥ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ✤ì♥ ❣✐↔♥ ✤÷đ❝ t➼♥❤ ❜➡♥❣ ❝→❝❤ ❝ë♥❣ t➜t ❝↔ ❝→❝ tr q st t ỳ rỗ ❝❤♦ sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✤â✳ ❈æ♥❣ t❤ù❝ t➼♥❤✿ ① ✏ ①✶   ①✷  ♥   ①♥ , tr♦♥❣ ✤â✱ ① ❧➔ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ✤ì♥ ❣✐↔♥✱ ①✐ ❧➔ ❣✐→ trà q✉❛♥ s→t t❤ù ✐✱ ♥ ❧➔ sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉✳ ❱➼ ❞ư✿ ●✐↔ sû sè t✐➲♥ ✭✤ì♥ ✈à ♥❣❤➻♥ ❱◆❉✮ ❞ò♥❣ ❝❤♦ ❝❤✐ t✐➯✉ t❤ü❝ ♣❤➞♠ tr♦♥❣ ♠ët t✉➛♥ ❧➔✿ ✶✷✵✱ ✶✺✵✱ ✶✷✺✱ ✶✵✵✱ ✶✽✵✱ ✶✹✵✱ ✷✵✵✳ ❑❤✐ ✤â sè t✐➲♥ tr✉♥❣ ❜➻♥❤ ♠ët ♥❣➔② tr♦♥❣ t✉➛♥ ❞➔♥❤ ❝❤♦ ❝❤✐ t✐➯✉ t❤ü❝ ♣❤➞♠ ❧➔✿ ① ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮   ✶✽✵   ✶✹✵   ✷✵✵ ✏ ✶✹✺ ✏ ✶✷✵   ✶✺✵   ✶✷✺   ✶✵✵ ✼ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✻✾ ✴ ✻✻✹ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ✷✳ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝â trå♥❣ sè ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ✤ì♥ ❣✐↔♥ ✤÷đ❝ t➼♥❤ ❜➡♥❣ ❝→❝❤ ❝ë♥❣ ❝→❝ t➼❝❤ ❝õ❛ ❣✐→ trà q✉❛♥ s→t t ỳ ợ trồ số tữỡ ự rỗ ❝❤✐❛ ❝❤♦ tê♥❣ ❝→❝ trå♥❣ sè ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✤â✳ ❈æ♥❣ t❤ù❝ t➼♥❤✿ ①✇ ✏ ①✶ ✇✇✶   ①✷✇✇✷   .  ✇①❦ ✇❦ , ✶ ✷ ❦ tr♦♥❣ ✤â✱ ① ✇ ❧➔ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝â trå♥❣ sè✱ ①✐ , ✇✐ t÷ì♥❣ ù♥❣ ❧➔ ❣✐→ trà ✈➔ trå♥❣ sè ❝õ❛ q✉❛♥ s→t t❤ù ✐✳ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✼✵ ✴ ✻✻✹ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❱➼ ❞ö✿ ✣✐➸♠ t❤✐ ❦➳t t❤ó❝ ❤å❝ ❦➻ ❝õ❛ ♠ët s✐♥❤ ✈✐➯♥ ✤÷đ❝ tr ữợ ổ ố ❤å❝ tr➻♥❤ ❳❙❚❑ ✈➔ Ù♥❣ ❞ö♥❣ ✼✳✵ ✺ ◆❣✉②➯♥ ❧➼ ❦➳ t♦→♥ ✺✳✵ ✸ ❚♦→♥ ◗✉↔♥ ❧➼ ✽✳✵ ✸ ▼æ ❤➻♥❤ ❑✐♥❤ t➳ ✹✳✵ ✸ ❚❤à tr÷í♥❣ ❝❤ù♥❣ ❦❤♦→♥ ✻✳✵ ✸ ❑❤✐ ✤â ✤✐➸♠ tr✉♥❣ ❜➻♥❤ ❝→❝ ♠æ♥ ❤å❝ ❝õ❛ s✐♥❤ ✈✐➯♥ tr➯♥ tr♦♥❣ ❦➻ ♥➔② s➩ ❧➔✿ ①✇ ✏ ✼ ✂ ✺   ✺ ✂✺  ✸  ✸  ✽ ✂✸  ✸  ✸  ✹ ✂✸ ✸   ✻ ✂ ✸ ✏ ✻.✶✷ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✼✶ ✴ ✻✻✹ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ×✉ ✤✐➸♠✿ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ởt số ữợ t õ t→❝ ✤ë♥❣ ✈➔♦ ♠å✐ ♣❤➛♥ tû ❝õ❛ t➟♣ ❞ú ❧✐➺✉❀ ▼å✐ t➟♣ ❞ú ❧✐➺✉ ✤➲✉ ❝â ❞✉② ♥❤➜t ♠ët tr✉♥❣ ❜➻♥❤ ❝ë♥❣❀ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ t♦→♥ ❤å❝ q✉❡♥ t❤✉ë❝❀ ❤ì♥ ♥ú❛ ♥â ❝â ♥❤✐➲✉ t➼♥❤ ❝❤➜t t♦→♥ ❤å❝ ❣✐ó♣ ❝❤♦ t❛ ❝â t❤➸ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ❝→❝ s✉② ❞✐➵♥ tr♦♥❣ t❤è♥❣ ❦➯✳ ◆❤÷đ❝ ✤✐➸♠✿ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❜à ↔♥❤ ❤÷ð♥❣ ❜ð✐ ❝→❝ ❣✐→ trà ♥❣♦↕✐ ❜✐➯♥❀ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝❤➾ ❤❛② ❞ò♥❣ ❝❤♦ ❞ú ❧✐➺✉ ✤♦ ❜➡♥❣ t❤❛♥❣ ✤♦ t➾ ❧➺✳ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✼✷ ✴ ✻✻✹ ✣ë tr↔✐ ❣✐ú❛ ✣ë tr↔✐ ❣✐ú❛ ❧➔ ❤✐➺✉ ✤ë ❝❤➯♥❤ ❧➺❝❤ ❣✐ú❛ tù ♣❤➙♥ ✈à t❤ù ❜❛ ✈➔ tù ♣❤➙♥ ✈à t❤ù ♥❤➜t ❝õ❛ t➟♣ ❞ú ❧✐➺✉✳ ❈→❝❤ t➼♥❤ ✤ë tr↔✐ ❣✐ú❛ ❘◗ ✿ ❘◗ ✏ ◗✸ ✁ ◗✶ ✱ tr♦♥❣ ✤â ◗✸ , ◗✶ ❧➔ tù ♣❤➙♥ ✈à t❤ù ❜❛ ✈➔ t❤ù ♥❤➜t ❝õ❛ t➟♣ ❞ú ❧✐➺✉✳ ❱➼ ❞ö✿ ❚➟♣ ❞ú ❧✐➺✉ ✶✵✱ ✶✸✱ ✶✺✱ ✷✺✱ ✸✺✱ ✹✵✱ ✻✵✱ ✼✵ ❝â ◗ ✤â ❝â ✤ë tr↔✐ ❣✐ú❛ ❘◗ ✶ ✏ ◗✸ ✁ ◗✶ ✏ ✺✺ ✁ ✶✸.✺ ✏ ✹✶.✺ ✏ ✶✸.✺ ✈➔ ◗✸ ✏ ✺✺✱ tø ✣ë tr↔✐ ❣✐ú❛ ❦❤➢❝ ♣❤ư❝ ✤÷đ❝ ♥❤÷đ❝ ✤✐➸♠ ❝õ❛ ✤ë tr↔✐ ❣✐ú❛ ♥❤÷♥❣ ♥â ❦❤ỉ♥❣ ①❡♠ ①➨t ✤÷đ❝ ✤➳♥ ❝→❝❤ t❤ù❝ ♣❤➙♥ ❜è ❝õ❛ t➜t ❝↔ ❝→❝ q✉❛♥ s→t tr♦♥❣ t➟♣ ❞ú ❧✐➺✉✳ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✼✾ ✴ ✻✻✹ ✣ë ❧➺❝❤ t✉②➺t ✤è✐ tr✉♥❣ ❜➻♥❤ ✣ë ❧➺❝❤ t✉②➺t ✤è✐ tr✉♥❣ ❜➻♥❤ ❝õ❛ ởt t ỳ ỗ q st õ trà ✶ , ①✷ , , ①♥ ợ tr ữủ ổ tự ➦♥ ✐ ✏✶ ⑤①✐ ✁ ① ⑤ ♥ ✣ë ❧➺❝❤ t✉②➺t ✤è✐ tr✉♥❣ ❜➻♥❤ ❝➠♥ ❝ù ✈➔♦ ♠å✐ ✤✐➸♠ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✈➔ ❝❤➾ r❛ ♠ët ❝→❝❤ tr✉♥❣ ❜➻♥❤ ♠é✐ ✤✐➸♠ ❞ú ❧✐➺✉ ♥➡♠ ❝→❝❤ ①❛ tr✉♥❣ ❜➻♥❤ ❜❛♦ ♥❤✐➯✉✳ ❚✉② ♥❤✐➯♥✱ ✈➻ ❝æ♥❣ t❤ù❝ t➼♥❤ ✤ë ❧➺❝❤ t✉②➺t ✤è✐ tr✉♥❣ ❜➻♥❤ ❞ò♥❣ ✤➳♥ ❣✐→ trà t✉②➺t ✤è✐ ♥➯♥ ❦❤â t❤ü❝ ❤✐➺♥ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ t♦→♥ ❤å❝✳ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✵ ✴ ✻✻✹ P❤÷ì♥❣ s❛✐ ✈➔ ✣ë ❧➺❝❤ ❝❤✉➞♥ P❤÷ì♥❣ s❛✐ ❝õ❛ ♠ët t➟♣ ❞ú ❧✐➺✉ tê♥❣ t❤➸✱ ❦➼ ❤✐➺✉ ❧➔ ❝æ♥❣ t❤ù❝✿ ✏ σ✷ ➦◆ ✐ ✏✶ ♣①✐ ✁ µq✷ ◆ ữủ ❧➔ tr✉♥❣ ❜➻♥❤ ❝õ❛ tê♥❣ t❤➸ ✈➔ ◆ ❧➔ sè q✉❛♥ s→t tr♦♥❣ tê♥❣ t❤➸✳ ✷ P❤÷ì♥❣ s❛✐ ❝õ❛ ♠ët t➟♣ ❞ú ❧✐➺✉ ♠➝✉✱ ❦➼ ❤✐➺✉ ❧➔ s ✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝✿ s ✷ ✏ ➦♥ ✐ ✏✶ ♣①✐ ♥ tr♦♥❣ ♠➝✉✳ ✁ ① q✷ ✁✶ ✱ ð ✤➙② ① ❧➔ tr✉♥❣ ❜➻♥❤ ❝õ❛ ♠➝✉ ✈➔ ♥ ❧➔ sè q✉❛♥ s→t ✣ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ tê♥❣ t❤➸ ✭♠➝✉✮✱ ❦➼ ❤✐➺✉ ❧➔ σ ♣s q✱ ❧➔ ❝➠♥ ❜➟❝ ❤❛✐ ❝õ❛ ♣❤÷ì♥❣ s❛✐ ❝õ❛ tê♥❣ t❤➸ ✭♠➝✉✮✿ ❞ σ ✏ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❞ ➦◆ ✷ ✐ ✏✶ ♣①✐ ✁ µq ◆ s ✏ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ➦♥ ✐ ✏✶ ♣①✐ ✁ ① q ♥ ✁✶ ✷ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✶ ✴ ✻✻✹ ❱➼ ❞ư t➼♥❤ ♣❤÷ì♥❣ s❛✐ ✈➔ ởt tờ t ỗ tû ❝â ❣✐→ trà ❧➔✿ ✶✵✱ ✶✺✱ ✸✷✱ ✶✽✱ ✷✺✱ ✻✺✱ ✸✵✱ ✸✽✳ ❚➼♥❤ t♦→♥ ❝❤♦ t❛ tr✉♥❣ ❜➻♥❤ σ✷ µ ✏ ✷✾.✶✷✺✱ tø ✤â ♣❤÷ì♥❣ s❛✐ ❝õ❛ tê♥❣ t❤➸ ❧➔✿ ✷ ✷ ✏ ♣✶✵ ✁ ✷✾.✶✷✺q   ✽   ♣✸✽ ✁ ✷✾.✶✷✺q ✏ ✷✻✵.✶✵✾✹, ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ tê♥❣ t❤➸ ❧➔✿ σ ✏ ✶✻.✶✷✼✾✶✳ ◆➳✉ ❝→❝ ❣✐→ trà ❧➔ sè ✤♦ ❝õ❛ ❝→❝ ✤ì♥ ✈à ♠➝✉ ✤÷đ❝ ❧➜② tø ♠ët tê♥❣ t❤➸ t❤➻ t❛ ❝â ♣❤÷ì♥❣ s❛✐ ♠➝✉ ❧➔✿ ✷ ✷ ✏ ♣✶✵ ✁ ✷✾.✶✷✺q  ✽.✁ .✶  ♣✸✽ ✁ ✷✾.✶✷✺q ✏ ✷✾✼.✷✻✼✾, ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ ♠➝✉ ❧➔✿ s ✏ ✶✼.✷✹✶✹✻✳ s ✷ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✷ ✴ ✻✻✹ ❚r✉♥❣ ❜➻♥❤✱ P❤÷ì♥❣ s❛✐ ✈➔ ✣ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ t❤✉ ❣å♥ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝õ❛ ❞ú ❧✐➺✉ t❤✉ ữợ t số ữủ ❝æ♥❣ t❤ù❝✿ ① ➦❦ ✏ ✐ ✏✶ ① ✐ ❢ ✐ , ✐ ✏✶ ❢ ✐ ➦❦ tr♦♥❣ ✤â✱ ①✐ ❧➔ ❣✐→ trà q✉❛♥ s→t t❤ù ✐ ❤♦➦❝ ❣✐→ trà ✤↕✐ ❞✐➺♥ ❝õ❛ tê t❤ù ✐ ✱ ❢✐ ❧➔ t➛♥ sè ❝õ❛ q✉❛♥ s→t ❤♦➦❝ tê t❤ù ✐ ✱ ➦❦ ✐ ✏✶ ❢✐ ✏ ♥✳ P❤÷ì♥❣ s❛✐ ❝õ❛ ♠ët t➟♣ ❞ú ❧✐➺✉ tê♥❣ t❤➸ t❤✉ ❣å♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝✿ σ✷ ✏ ➦❦ ✐ ✏✶➦♣①✐ ✁ µq ❢✐ , ❦ ✐ ✏✶ ❢✐ ✷ ð ✤➙②✱ ①✐ ❧➔ ❣✐→ trà q✉❛♥ s→t t❤ù ✐ ❤♦➦❝ ❣✐→ trà ✤↕✐ ❞✐➺♥ ❝õ❛ tê t❤ù ✐ ✱ ❢✐ ❧➔ t➛♥ ➦ sè ❝õ❛ q✉❛♥ s→t ❤♦➦❝ tê t❤ù ✐ ✱ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❦ ✐ ✏✶ ❢✐ ✏ ◆✳ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✸ ✴ ✻✻✹ ❚r✉♥❣ ❜➻♥❤✱ P❤÷ì♥❣ s❛✐ ✈➔ ✣ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ t❤✉ ❣å♥ P❤÷ì♥❣ s❛✐ ❝õ❛ ♠ët t➟♣ ❞ú ❧✐➺✉ ♠➝✉ t❤✉ ❣å♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝✿ s ✷ ✏ ➦❦ ✐ ✏✶ ♣①✐ ✁ ① q ❢✐ , ✐ ✏✶ ❢ ✐ ✁ ✶ ✷ ➦❦ ð ✤➙②✱ ①✐ ❧➔ ❣✐→ trà q✉❛♥ s→t t❤ù ✐ ❤♦➦❝ ❣✐→ trà ✤↕✐ ❞✐➺♥ ❝õ❛ tê t❤ù ✐ ✱ ❢✐ ❧➔ t➛♥ ➦ sè ❝õ❛ q✉❛♥ s→t ❤♦➦❝ tê t❤ù ✐ ✱ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❦ ✐ ✏✶ ❢✐ ✏ ♥✳ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✹ ✴ ✻✻✹ ◗✉✐ t➢❝ t❤ü❝ ♥❣❤✐➺♠ ◗✉✐ t➢❝ t❤ü❝ ♥❣❤✐➺♠ ❝❤♦ t❛ ❜✐➳t ♣❤➛♥ tr➠♠ ❝→❝ ❣✐→ trà ♥➡♠ tr♦♥❣ ✈á♥❣ ♠ët sè ❧➛♥ ✤ë ❧➺❝❤ ❝❤✉➞♥ t➼♥❤ tø tr✉♥❣ ❜➻♥❤ ♥➳✉ t➟♣ ❞ú ❧✐➺✉ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳ ❈ö t❤➸ t❛ ❝â ❝→❝ ❦➳t ❧✉➟♥ s❛✉ ❝❤♦ t➟♣ ❞ú ❧✐➺✉ t✉➙♥ t❤❡♦ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✿ ❈â ❦❤♦↔♥❣ ✻✽% sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ t➟♣ tr✉♥❣ tr♦♥❣ ❦❤♦↔♥❣ ♣µ ✁ σ, µ   σq✳ ❈â ❦❤♦↔♥❣ ✾✺% sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ t➟♣ tr✉♥❣ tr♦♥❣ ❦❤♦↔♥❣ ♣µ ✁ ✷σ, µ   ✷σq✳ ❈â ❦❤♦↔♥❣ ✾✾.✼% sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ t➟♣ tr✉♥❣ tr♦♥❣ ❦❤♦↔♥❣ ♣µ ✁ ✸σ, µ   ✸σq✳ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✺ ✴ ✻✻✹ ◗✉✐ t➢❝ t❤ü❝ ♥❣❤✐➺♠ Empirical Rule for one standard Deviation Empirical Rule for two standard Deviations Empirical Rule for three standard Deviations 68% 95% 99.7% − 1σ σ µ 1σ σ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ − 2σ σ µ 2σ σ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ư♥❣ − 3σ σ µ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ 3σ σ ✽✻ ✴ ✻✻✹ ✣à♥❤ ❧➼ ❚❝❤❡❜②s❤❡✈ ✣à♥❤ ❧➼ ❚❝❤❡❜②s❤❡✈ ♣❤→t ❜✐➸✉ r➡♥❣ ✈ỵ✐ t➟♣ ❞ú ❧✐➺✉ ❝â ố tũ ỵ t t t ✷ , ❦ → ✶ ❣✐→ trà ❝õ❛ t➟♣ ❞ú ❧✐➺✉ s➩ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ ✟❦ ❧➛♥ ✤ë ❧➺❝❤ ❝❤✉➞♥ s♦ ✈ỵ✐ tr✉♥❣ ❜➻♥❤✳ ❈ư t❤➸ t❛ ❝â ♠ët sè ❦➳t q✉↔ s❛✉✿ ❈â ❦❤♦↔♥❣ ✼✺% sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ t➟♣ tr✉♥❣ tr♦♥❣ ❦❤♦↔♥❣ ♣µ ✁ ✷σ, µ   ✷σq✳ ❈â ❦❤♦↔♥❣ ✽✹% sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ t➟♣ tr✉♥❣ tr♦♥❣ ❦❤♦↔♥❣ ♣µ ✁ ✷.✺σ, µ   ✷.✺σq✳ ❈â ❦❤♦↔♥❣ ✽✾% sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ t➟♣ tr✉♥❣ tr♦♥❣ ❦❤♦↔♥❣ ♣µ ✁ ✸σ, µ   ✸σq✳ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✼ ✴ ✻✻✹ ✣à♥❤ ❧➼ ❚❝❤❡❜②s❤❡✈ Application Chebyshev's Theorem for Two Standard Deviations 75% 2σ σ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ µ − 2σ σ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✽ ✴ ✻✻✹ ❍➺ sè ❜✐➳♥ t❤✐➯♥ ❍➺ sè ❜✐➳♥ t❤✐➯♥ ❧➔ t➾ sè ✭t➼♥❤ ❜➡♥❣ ♣❤➛♥ tr➠♠✮ ❝õ❛ ✤ë ❧➺❝❤ ❝❤✉➞♥ ✈➔ tr✉♥❣ ❜➻♥❤✳ ❈æ♥❣ t❤ù❝ t➼♥❤ ❤➺ sè ❜✐➳♥ t❤✐➯♥ ❈❱✿ ❍➺ sè ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❞ú ❧✐➺✉ tê♥❣ t❤➸✿ ❈❱ ✏ σµ ✶✵✵%❀ ❍➺ sè ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❞ú ❧✐➺✉ ♠➝✉✿ ❈❱ ✏ ①s ✶✵✵%❀ ❍➺ sè ❜✐➳♥ t❤✐➯♥ ❞ò♥❣ ✤➸ s♦ s→♥❤ ✤ë ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤❛✐ t➟♣ ❞ú ❧✐➺✉ ❦❤✐ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝❤ó♥❣ ❦❤→❝ ♥❤❛✉✳ ❱➼ ❞ö✿ ●✐↔ sû tr♦♥❣ ✺ t✉➛♥✱ ❣✐→ ❝õ❛ ❝ê ♣❤✐➳✉ ❆ ❧➔✿ ✷✷✱ ✷✽✱ ✷✵✱ ✷✼✱ ✷✺ ✈➔ ❣✐→ ❝õ❛ ❝ê ♣❤✐➳✉ ❇ ❧➔✿ ✹✵✱ ✺✵✱ ✹✽✱ ✺✺✱ ✹✺✳ t t ữủ . . tø ✤â ❈❱❆ ✸.✸✻ ✏ ✷✹ ✶✵✵% ✏ ✶✸.✼✼%; ✹ ✺.✻✵ ✹✼.✻ µ❆ ✏ ✷✹.✹✱ σ❆ ✏ ✸.✸✻ ✈➔ ✶✵✵% ✏ ✶✶.✼✼% ◆❤÷ ✈➟② ❣✐→ ❝õ❛ ❝ê ♣❤✐➳✉ ❆ ❜✐➳♥ t❤✐➯♥ ♥❤✐➲✉ ❤ì♥✳ ❇ë ♠ỉ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✽✾ ✴ ✻✻✹ ❙❦❡✇♥❡ss ❙❦❡✇♥❡ss ❧➔ ✤↕✐ ❧÷đ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë ❧➺❝❤ ❝õ❛ ♣❤➙♥ ♣❤è✐✱ ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ sè ❜➜t ✤è✐ ①ù♥❣✳ ❙❦❡✇♥❡ss ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ✈ỵ✐ tr✉♥❣ ❜➻♥❤✱ tr✉♥❣ ✈à ✈➔ ♠❡❞✐❛♥✳ ◆➳✉ ♣❤➙♥ ♣❤è✐ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❧➔ ✤è✐ ①ù♥❣ t❤➻ ❦❤æ♥❣ ❝â ❤➺ sè s❦❡✇♥❡ss ✈➔ ❦❤✐ ✤â tr✉♥❣ ❜➻♥❤✱ tr✉♥❣ ✈à ✈➔ ♠♦❞❡ trò♥❣ ♥❤❛✉❀ ◆➳✉ ♣❤➙♥ ♣❤è✐ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ♥❣❤✐➯♥❣ ✈➲ ❜➯♥ tr→✐ t❤➻ ❤➺ sè s❦❡✇♥❡ss ➙♠ ✈➔ tr✉♥❣ ❜➻♥❤ ❁ tr✉♥❣ ✈à ❁ ♠♦❞❡❀ ◆➳✉ ♣❤➙♥ ♣❤è✐ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ♥❣❤✐➯♥❣ ✈➲ ❜➯♥ ♣❤↔✐ t❤➻ ❤➺ sè s❦❡✇♥❡ss ❞÷ì♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ❃ tr✉♥❣ ✈à ❃ ♠♦❞❡✳ ▼æ t↔ ❤➻♥❤ ❤å❝ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✾✵ ✴ ✻✻✹ ❙❦❡✇♥❡ss Symmetric Distribution Mean Median Distribution Skewed Left (Negatively Skewed) Mode Mean Median Mode ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ Distribution Skewed Right (Positively Skewed) Mean Mode Median ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✾✶ ✴ ✻✻✹ ❑✉rt♦s✐s ❑✉rt♦s✐s ❧➔ ✤↕✐ ❧÷đ♥❣ ✤♦ ♠ù❝ ✤ë t➟♣ tr✉♥❣ t÷ì♥❣ ✤è✐ ❝õ❛ ❝→❝ q✉❛♥ s→t q✉❛♥❤ tr✉♥❣ t➙♠ ❝õ❛ ♥â tr♦♥❣ ♠è✐ q✉❛♥ ❤➺ s♦ s→♥❤ ✈ỵ✐ ❤❛✐ ✤✉æ✐✳ ❍➺ sè ❑✉rt♦s✐s ✈➔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉✿ ❑❤✐ ♣❤➙♥ ♣❤è✐ t➟♣ tr✉♥❣ ð ♠ù❝ ✤ë ❜➻♥❤ t❤÷í♥❣ t❤➻ ❤➺ sè ❑✉rt♦s✐s ❂ ✸❀ ❑❤✐ ♣❤➙♥ ♣❤è✐ t➟♣ tr✉♥❣ ❤ì♥ ♠ù❝ ❜➻♥❤ t❤÷í♥❣ ✭❤➻♥❤ ❞→♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❝❛♦ ✈➔ ♥❤å♥ ✈ỵ✐ ❤❛✐ ✤✉ỉ✐ ❤➭♣✮ t❤➻ ❑✉rt♦s✐s ❃ ✸❀ ❑❤✐ ♣❤➙♥ ♣❤è✐ ❦❤ỉ♥❣ t➟♣ tr✉♥❣ ♥❤÷ ♠ù❝ ❜➻♥❤ t❤÷í♥❣ ✭❤➻♥❤ ❞→♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♣❤➥♥❣ ✈➔ tr↔✐ ❞➔✐✮ t❤➻ ❑✉rt♦s✐s ❁ ✸✳ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ◆❣➔② ✶✹ t❤→♥❣ ỗ r ỗ r ởt ỗ t❤➸ ❤✐➺♥ ✤÷đ❝ ❝→❝ t❤ỉ♥❣ t✐♥✿ ❣✐→ trà ❝ü❝ ✤↕✐✱ ❣✐→ trà ❝ü❝ t✐➸✉✱ ❜❛ tù ♣❤➙♥ ✈à ✈➔ ❝→❝ q st ỗ r ① ¯ ♦ ◗✶ ✶✳✺✯■◗❘ ✸✯■◗❘ ❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮ ■◗❘ ❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣ ✯ ◗✸ ✶✳✺✯■◗❘ ✸✯■◗❘ ◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾ ✾✸ ✴ ✻✻✹