Bài giảng xác suất thống kê ứng dụng trong kinh tế xã hội chương 4 tóm tắt và trình bày dữ liệu bằng các đại lượng thống kê mô tả

❈❤÷ì♥❣ ■❱
❚â♠ t➢t ✈➔ tr➻♥❤ ❜➔② ❞ú ❧✐➺✉ ❜➡♥❣ ❝→❝ ✤↕✐ ❧÷ñ♥❣ t❤è♥❣
❦➯ ♠æ t↔

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✻✼ ✴ ✻✻✹


❈❤÷ì♥❣ ■❱
✶✵

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✶

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠æ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✷

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥

✶✸

❙û ❞ö♥❣ ❦➳t ❤ñ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥

✶✹

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✻✽ ✴ ✻✻✹


❈❤÷ì♥❣ ■❱
✶✵

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✶

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠æ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✷

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥

✶✸

❙û ❞ö♥❣ ❦➳t ❤ñ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥

✶✹

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✻✽ ✴ ✻✻✹


❈❤÷ì♥❣ ■❱
✶✵

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✶

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠æ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✷

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥

✶✸

❙û ❞ö♥❣ ❦➳t ❤ñ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥

✶✹

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✻✽ ✴ ✻✻✹


❈❤÷ì♥❣ ■❱
✶✵

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✶

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠æ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✷

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥

✶✸

❙û ❞ö♥❣ ❦➳t ❤ñ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥

✶✹

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✻✽ ✴ ✻✻✹


❈❤÷ì♥❣ ■❱
✶✵

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ♠ù❝ ✤ë t➟♣ tr✉♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✶

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠æ t↔ sü ♣❤➙♥ ❜è ❝õ❛ t➟♣ ❞ú ❧✐➺✉

✶✷

❈→❝ ✤↕✐ ❧÷ñ♥❣ ✤♦ ❧÷í♥❣ ✤ë ♣❤➙♥ t→♥

✶✸

❙û ❞ö♥❣ ❦➳t ❤ñ♣ tr✉♥❣ ❜➻♥❤ ✈➔ ✤ë ❧➺❝❤ ❝❤✉➞♥

✶✹

❈→❝ ✤↕✐ ❧÷ñ♥❣ ♠✐➯✉ t↔ ❤➻♥❤ ❞→♥❣ ❝õ❛ t➟♣ ❞ú ❧✐➺✉

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✻✽ ✴ ✻✻✹


❚r✉♥❣ ❜➻♥❤ ❝ë♥❣
✶✳ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ✤ì♥ ❣✐↔♥
❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ✤ì♥ ❣✐↔♥ ✤÷ñ❝ t➼♥❤ ❜➡♥❣ ❝→❝❤ ❝ë♥❣ t➜t ❝↔ ❝→❝
❣✐→ trà q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ rç✐ ❝❤✐❛ ❝❤♦ 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 ❧✐➺✉✳
❱➼ ❞ö✿ ●✐↔ 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♥ ❣✐↔♥ ✤÷ñ❝ t➼♥❤ ❜➡♥❣ ❝→❝❤ ❝ë♥❣ ❝→❝ t➼❝❤
❝õ❛ ❣✐→ trà q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✈î✐ trå♥❣ sè t÷ì♥❣ ù♥❣ rç✐
❝❤✐❛ ❝❤♦ 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✐♥❤ ✈✐➯♥ ✤÷ñ❝ ❝❤♦ tr♦♥❣ ❜↔♥❣ ❞÷î✐
✤➙②✿

▼æ♥ ❤å❝

✣✐➸♠ ❙è ✤✳✈ ❤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♥❣ ❧➔ ♠ët sè ✤♦ ❤÷î♥❣ t➙♠ ♣❤ê ❜✐➳♥ ✈➻ ♥â 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❝ ❤✐➺♥ ✤÷ñ❝ ❝→❝ s✉② ❞✐➵♥ tr♦♥❣ t❤è♥❣
❦➯✳

◆❤÷ñ❝ ✤✐➸♠✿

❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❜à ↔♥❤ ❤÷ð♥❣ ❜ð✐ ❝→❝ ❣✐→ trà ♥❣♦↕✐ ❜✐➯♥❀
❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝❤➾ ❤❛② ❞ò♥❣ ❝❤♦ ❞ú ❧✐➺✉ ✤♦ ❜➡♥❣ t❤❛♥❣ ✤♦ t➾ ❧➺✳

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✼✷ ✴ ✻✻✹


❚r✉♥❣ ✈à

❚r✉♥❣ ✈à ❧➔ ❣✐→ trà ✤ù♥❣ ❣✐ú❛ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ✤➣ ✤÷ñ❝ s➢♣ t❤ù tü✳ ◆❤÷ ✈➟②✱
❦❤æ♥❣ ❦➸ tr✉♥❣ ✈à s➩ ❝â ✺✵% sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❝â ❣✐→ trà ❧î♥ ❤ì♥
tr✉♥❣ ✈à ✈➔ ✺✵% sè q✉❛♥ s→t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❝â ❣✐→ trà ♥❤ä ❤ì♥ tr✉♥❣ ✈à✳
❚➼♥❤ tr✉♥❣ ✈à ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❝â ♥ q✉❛♥ s→t✿

❙➢♣ ①➳♣ ❧↕✐ t➟♣ ❞ú ❧✐➺✉❀
◆➳✉ sè q✉❛♥ s→t ♥ ❧➔ sè ❧➫ t❤➻ tr✉♥❣ ✈à ❧➔ q✉❛♥ s→t ð ✈à tr➼ t❤ù ♣♥   ✶q④✷✳
◆➳✉ sè q✉❛♥ s→t ♥ ❧➔ sè ❝❤➤♥ t❤➻ tr✉♥❣ ✈à ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝õ❛ ❤❛✐ q✉❛♥
s→t ð ✈à tr➼ ❝❤➼♥❤ ❣✐ú❛ ❝õ❛ t➟♣ ❞ú ❧✐➺✉✱ tù❝ ❧➔ ❤❛✐ q✉❛♥ s→t ð ✈à tr➼ t❤ù ♥④✷ ✈➔
♣♥   ✷q④✷✳

❱➼ ❞ö✿ ❚➻♠ tr✉♥❣ ✈à ❝õ❛ t➟♣ ❞ú ❧✐➺✉✿ ✺, ✾, ✶✷, ✶✵, ✷✵, ✶✺, ✸✵, ✷✺✳ ✣➛✉ t✐➯♥ t❛ s➢♣
①➳♣ ❧↕✐ ❞ú ❧✐➺✉ t❤❡♦ t❤ù tü ✭t➠♥❣ ❞➛♥✮✿ ✺, ✾, ✶✵, ✶✷, ✶✺, ✷✵, ✷✺, ✸✵✳ ❉♦ ❞ú ❧✐➺✉ ❝â
♥

✏ ✽ q✉❛♥ s→t ❧➔ sè ❝❤➤♥ ♥➯♥ t❛ t➻♠ q✉❛♥ s→t ð ✈à tr➼ t❤ù ✹ ✈➔ ✺ ❧➔ ✶✷ ✈➔ ✶✺✳
♣✶✷   ✶✺q④✷ ✏ ✶✸.✺

❑❤✐ ✤â tr✉♥❣ ✈à ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❜➡♥❣

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✼✸ ✴ ✻✻✹


❚r✉♥❣ ✈à

×✉ ✤✐➸♠✿

❚r✉♥❣ ✈à ❦❤æ♥❣ ❜à ↔♥❤ ❤÷ð♥❣ ❜ð✐ ♥❤ú♥❣ ❣✐→ trà ♥❣♦↕✐ ❜✐➯♥❀
▼å✐ t➟♣ ❞ú ❧✐➺✉ ✤➲✉ ❝â ❞✉② ♥❤➜t ♠ët tr✉♥❣ ✈à❀
❈â t❤➸ t➻♠ tr✉♥❣ ✈à ❝❤♦ t➟♣ ❞ú ❧✐➺✉ sû ❞ö♥❣ t❤❛♥❣ ✤♦ t❤ù ❜➟❝✱ ❦❤♦↔♥❣✱ t➾ ❧➺✳

◆❤÷ñ❝ ✤✐➸♠✿

❚r✉♥❣ ✈à ❦❤æ♥❣ t→❝ ✤ë♥❣ ✤➳♥ ♠å✐ ❣✐→ trà ❝õ❛ t➟♣ ❞ú ❧✐➺✉❀
✣➸ t➻♠ tr✉♥❣ ✈à t❛ ♣❤↔✐ s➢♣ ①➳♣ ❧↕✐ t❤ù tü ❞ú ❧✐➺✉ ✈➔ ✈✐➺❝ ♥➔② tè♥ r➜t ♥❤✐➲✉ t❤í✐
❣✐❛♥✳

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✼✹ ✴ ✻✻✹


❙è ♠♦❞❡
❙è ♠♦❞❡ ❧➔ ❣✐→ trà ❣➦♣ ♥❤✐➲✉ ♥❤➜t tr♦♥❣ t➟♣ ❞ú ❧✐➺✉✳
❱➼ ❞ö✿

❈❤♦ t➟♣ ❞ú ❧✐➺✉✿ ✵✱ ✶✱ ✸✱ ✶✱ ✺✱ ✷✱ ✻✱ ✷✱ ✾✱ ✷✳ ❑❤✐ ✤â ♠♦❞❡ ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❧➔ ✷✳
❚➟♣ ❞ú ❧✐➺✉✿ ✵✱ ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻ ❦❤æ♥❣ ❝â ♠♦❞❡✳
❚➟♣ ❞ú ❧✐➺✉✿ ✷✱ ✸✱ ✷✱ ✺✱ ✼✱ ✽✱ ✼✱ ✶✺ ❝â ❤❛✐ ♠♦❞❡ ❧➔ ✷ ✈➔ ✼✳

×✉ ✤✐➸♠ ✈➔ ♥❤÷ñ❝ ✤✐➸♠ ❝õ❛ sè ♠♦❞❡

×✉ ✤✐➸♠✿

❙è ♠♦❞❡ ❦❤æ♥❣ ❜à ↔♥❤ ❤÷ð♥❣ ❜ð✐ ❣✐→ trà ♥❣♦↕✐ ❜✐➯♥❀
❙è ♠♦❞❡ ❧➔ ✤↕✐ ❧÷ñ♥❣ t❤è♥❣ ❦➯ ♠æ t↔ ❞✉② ♥❤➜t ❝â t❤➸ sû ❞ö♥❣ ❝❤♦ t➟♣ ❞ú ❧✐➺✉
✤♦ ❜➡♥❣ t❤❛♥❣ ✤♦ ✤à♥❤ ❞❛♥❤✳

◆❤÷ñ❝ ✤✐➸♠✿

❈â t➟♣ ❞ú ❧✐➺✉ ❦❤æ♥❣ ❝â sè ♠♦❞❡ ✈➔ ❝ô♥❣ ❝â t➟♣ ❞ú ❧✐➺✉ ❝â ♥❤✐➲✉ ♠♦❞❡✳

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✼✺ ✴ ✻✻✹


❚ù ♣❤➙♥ ✈à
❚ù ♣❤➙♥ ✈à ❝❤✐❛ t➟♣ ❞ú ❧✐➺✉ ✤➣ s➢♣ ①➳♣ t❤❡♦ tr➟t tü t➠♥❣ ❞➛♥ t❤➔♥❤ ❜è♥ ♣❤➛♥
❝â sè q✉❛♥ s→t ❜➡♥❣ ♥❤❛✉✳
❈→❝❤ t➼♥❤ tù ♣❤➙♥ ✈à ❝õ❛ t➟♣ ❞ú ❧✐➺✉ ❝â ♥ ♣❤➛♥ tû✿

❚ù ♣❤➙♥ ✈à t❤ù ♥❤➜t ❦➼ ❤✐➺✉ ❧➔ ◗ ❧➔ ❣✐→ trà ❝õ❛ q✉❛♥ s→t t↕✐ ✈à tr➼ ①→❝ ✤à♥❤ ❜ð✐
❝æ♥❣ t❤ù❝ ✷✺%♣♥   ✶q❀
❚ù ♣❤➙♥ ✈à t❤ù ❤❛✐ ❦➼ ❤✐➺✉ ❧➔ ◗ ❝❤➼♥❤ ❧➔ tr✉♥❣ ✈à❀
❚ù ♣❤➙♥ ✈à t❤ù ❜❛ ❦➼ ❤✐➺✉ ❧➔ ◗ ❧➔ ❣✐→ trà ❝õ❛ q✉❛♥ s→t t↕✐ ✈à tr➼ ①→❝ ✤à♥❤ ❜ð✐
❝æ♥❣ t❤ù❝ ✼✺%♣♥   ✶q❀

✶

✷

✸

❱➼ ❞ö✿ ❈❤♦ t➟♣ ❞ú ❧✐➺✉ ❣ç♠ ✽ ♣❤➛♥ tû✿ ✶✵✱ ✶✸✱ ✶✺✱ ✷✺✱ ✸✺✱ ✹✵✱ ✻✵✱ ✼✵✳ ❑❤✐ ✤â
tù ♣❤➙♥ ✈à ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
◗✶
◗✷
◗✸

ð ✈à tr➼ ✷✺%♣✽   ✶q ✏ ✷.✷✺✱ ♥❤÷ ✈➟② ◗ ✏ ✶✸   ✵.✷✺♣✶✺ ✁ ✶✸q ✏ ✶✸.✺❀
ð ✈à tr➼ ✺✵%♣✽   ✶q ✏ ✹.✺✱ ♥❤÷ ✈➟② ◗ ✏ ✷✺   ✵.✺♣✸✺ ✁ ✷✺q ✏ ✸✵❀
ð ✈à tr➼ ✵.✼✺%♣✽   ✶q ✏ ✻.✼✺✱ ♥❤÷ ✈➟② ◗ ✏ ✹✵   ✵.✼✺♣✻✵ ✁ ✹✵q ✏ ✺✺✳

❇ë ♠æ♥ ❚❖⑩◆ ✭✣❍ ❚❍❿◆● ▲❖◆●✮

✶

✷

✸

❳→❝ s✉➜t ❚❤è♥❣ ❦➯ ù♥❣ ❞ö♥❣

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✼✻ ✴ ✻✻✹

P
P tự ừ ởt t ỳ ữủ s tự tỹ tr t ỳ
t ởt ỗ % số q st õ tr ọ ỡ
tự ỏ õ

%q số q st ợ ỡ

tự
t tự ừ t ỳ õ tỷ P tự tr

õ tr ữủ ổ tự






q

ử ợ t ỳ t

tự q st tr


. ữ tự õ


q
tr . q .

ở ổ

st ố ự ử

t




❑❤♦↔♥❣ ❜✐➳♥ t❤✐➯♥
❑❤♦↔♥❣ ❜✐➳♥ t❤✐➯♥ ❧➔ ❤✐➺✉ ❣✐ú❛ ❣✐→ trà ❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ t➟♣ ❞ú
❧✐➺✉✳
❈→❝❤ t➼♥❤ ❦❤♦↔♥❣ ❜✐➳♥ t❤✐➯♥ ❘✿ ❘

✏ ①♠❛① ✁ ①♠✐♥ ✱ tr♦♥❣ ✤â ①♠❛① , ①♠✐♥ ❧➔ ❣✐→ trà

❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ t➟♣ ❞ú ❧✐➺✉✳
❱➼ ❞ö✿

❑❤♦↔♥❣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ t➟♣ ❞ú ❧✐➺✉✿ ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✽ ❧➔ ❘ ✏ ✽ ✁ ✶ ✏ ✼❀
❑❤♦↔♥❣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ t➟♣ ❞ú ❧✐➺✉✿ ✶✱ ✶✱ ✶✱ ✶✱ ✽✱ ✽✱ ✽✱ ✽ ❧➔ ❘ ✏ ✽ ✁ ✶ ✏ ✼❀
❑❤♦↔♥❣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ t➟♣ ❞ú ❧✐➺✉✿ ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✽✱ ✶✵✵ ❧➔ ❘ ✏ ✶✵✵ ✁ ✶ ✏ ✾✾✳

❑❤♦↔♥❣ ❜✐➳♥ t❤✐➯♥ ❧➔ sè ✤♦ ✤ë ♣❤➙♥ t→♥ ✤ì♥ ❣✐↔♥ ✈➔ ❞➵ ❤✐➸✉ ♥❤÷♥❣ ♥â ❝❤➾ ♣❤ö
t❤✉ë❝ ✈➔♦ ❣✐→ trà ❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ t➟♣ ❞ú ❧✐➺✉ r➜t ♥❤↕② ❝↔♠ ✈î✐
❝→❝ ❣✐→ trà ♥❣♦↕✐ ❜✐➯♥ ✈➔ ❜ä q✉❛ ❝→❝❤ ♣❤➙♥ ❜è ♥ë✐ ❜ë ❝õ❛ t➟♣ ❞ú ❧✐➺✉✳

❇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❛ ❦❤➢❝ ♣❤ö❝ ✤÷ñ❝ ♥❤÷ñ❝ ✤✐➸♠ ❝õ❛ ✤ë tr↔✐ ❣✐ú❛ ♥❤÷♥❣ ♥â ❦❤æ♥❣
①❡♠ ①➨t ✤÷ñ❝ ✤➳♥ ❝→❝❤ 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❛ ♠ët t➟♣ ❞ú ❧✐➺✉ ❣ç♠ ♥ q✉❛♥ s→t ❝â ❣✐→ trà

✶ , ①✷ , . . . , ①♥

①

✈î✐ tr✉♥❣ ❜➻♥❤ ① ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝✿

➦♥
✐

✏✶ 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✷

◆

✱ ð ✤➙②

µ

σ✷ ✱

✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐

❧➔ tr✉♥❣ ❜➻♥❤ ❝õ❛ tê♥❣ t❤➸ ✈➔ ◆ ❧➔

sè q✉❛♥ s→t tr♦♥❣ tê♥❣ t❤➸✳

✷

P❤÷ì♥❣ s❛✐ ❝õ❛ ♠ët t➟♣ ❞ú ❧✐➺✉ ♠➝✉✱ ❦➼ ❤✐➺✉ ❧➔ s ✱ ✤÷ñ❝ 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❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✽✶ ✴ ✻✻✹


ử t ữỡ s ở
ởt tờ t ỗ tỷ õ tr
t t tr



à .

tứ õ ữỡ s ừ tờ t



.q . . . .q .,

ở ừ tờ t

.

tr số ừ ỡ ữủ tứ ởt tờ t t t
õ ữỡ s



.q .. . .q .,
ở ừ s .
s



ở ổ

st ố ự ử

t




❚r✉♥❣ ❜➻♥❤✱ P❤÷ì♥❣ s❛✐ ✈➔ ✣ë ❧➺❝❤ ❝❤✉➞♥ ❝õ❛ t➟♣ ❞ú ❧✐➺✉
t❤✉ ❣å♥
❚r✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝õ❛ ❞ú ❧✐➺✉ t❤✉ ❣å♥ ✭❝❤♦ ❞÷î✐ ❞↕♥❣ ❜↔♥❣ t➛♥ sè✮ ✤÷ñ❝ ❝❤♦
❜ð✐ ❝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♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ 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♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ 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♥❣ ❞ö♥❣

− 3σ
σ

µ

◆❣➔② ✶✹ t❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

3σ
σ

✽✻ ✴ ✻✻✹


✣à♥❤ ❧➼ ❚❝❤❡❜②s❤❡✈
✣à♥❤ ❧➼ ❚❝❤❡❜②s❤❡✈ ♣❤→t ❜✐➸✉ r➡♥❣ ✈î✐ t➟♣ ❞ú ❧✐➺✉ ❝â ♣❤➙♥ ♣❤è✐ tò② þ t❤➻ ➼t
♥❤➜t ✶

✁ ✶④❦ ✷ , ❦ → ✶ ❣✐→ trà ❝õ❛ t➟♣ ❞ú ❧✐➺✉ s➩ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ ✟❦ ❧➛♥ ✤ë

❧➺❝❤ ❝❤✉➞♥ s♦ ✈î✐ tr✉♥❣ ❜➻♥❤✳ ❈ö 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❤→♥❣ ✷ ♥➠♠ ✷✵✵✾

✽✼ ✴ ✻✻✹