Cov txheej txheem rau kev txiav txim siab qhov nruab nrab ntawm qhov sib txawv hauv cov ntaub ntawv suav lej
Hauv kev suav lej, kev nkag siab txog "qhov chaw nruab nrab" ntawm cov ntaub ntawv - piv txwv li, los ntawm qhov nruab nrab lossis nruab nrab - feem ntau tsis txaus. Ob pawg ntaub ntawv tuaj yeem muaj qhov nruab nrab zoo ib yam, tab sis sib txawv ntau hauv lawv cov qib ntawm "kev hloov pauv." Yog li ntawd, kev ntsuas ntawm kev sib kis yog qhov tseem ceeb. Ib qho kev ntsuas ntawm kev sib kis uas yooj yim to taub thiab siv yog qhov sib txawv nruab nrab. Tsab xov xwm no tham txog cov txheej txheem rau kev txiav txim siab qhov sib txawv nruab nrab hauv ntau hom ntaub ntawv suav lej, ua tiav nrog cov kauj ruam suav thiab piv txwv.
Nkag Siab Txog Qhov Nruab Nrab Deviation
Qhov kev hloov pauv nruab nrab yog qhov ntsuas uas qhia qhov nruab nrab ntawm txhua qhov ntaub ntawv los ntawm qhov ntsuas ntawm qhov sib txawv hauv nruab nrab, feem ntau yog qhov nruab nrab ntawm lej (mean) lossis median. Qhov kev ncua deb hauv nqe lus nug yog tus nqi tag nrho ntawm qhov sib txawv ntawm cov ntaub ntawv thiab tus nqi hauv nruab nrab, yog li ntawd tsis muaj qhov sib txawv tsis zoo "tshem tawm" qhov sib txawv zoo.
Feem ntau, qhov kev hloov pauv nruab nrab piav qhia txog qhov deb npaum li cas cov ntaub ntawv tau nthuav tawm ntawm nws tus nqi hauv nruab nrab. Qhov kev hloov pauv nruab nrab me dua, cov ntaub ntawv tau sib sau ua ke ntau dua nyob ib puag ncig nruab nrab; qhov loj dua tus nqi, cov ntaub ntawv hloov pauv ntau dua.
Vim li cas ho siv Tus Nqi Tag Nrho?
Yog tias peb xam qhov nruab nrab ntawm qhov sib txawv ntawm cov ntaub ntawv thiab qhov nruab nrab yam tsis muaj tus nqi kiag li, qhov sib ntxiv ntawm qhov sib txawv yuav ib txwm yog xoom (vim tias qhov nruab nrab yog qhov sib npaug). Piv txwv li, yog tias muaj qhov sib txawv ntawm +5 thiab -5, lawv ntxiv rau 0. Yog li ntawd, cov nqi kiag li yog siv kom qhov kev hloov pauv tiag tiag qhia txog qhov deb ntawm cov ntaub ntawv los ntawm qhov chaw.
Tus Qauv Sib Txawv Nruab Nrab rau Cov Ntaub Ntawv Ib Leeg
Rau cov ntaub ntawv ib leeg (tsis yog pawg), qhov nruab nrab ntawm qhov nruab nrab yog tsim los ntawm:
\[
SR = \frac{\sum |x_i – x_i|}{n}
\]
Cov Ntaub Ntawv:
– \( SR \): qhov nruab nrab ntawm qhov sib txawv
– \( x_i \): cov ntaub ntawv thib i
– \( \bar{x} \): lej nruab nrab (nruab nrab)
– \( n \): tus nqi ntawm cov ntaub ntawv
Kev Siv Cov Txheej Txheem Xam Cov Ntaub Ntawv Ib Leeg (Cov Kauj Ruam)
1. Xam tus nqi nruab nrab ntawm tag nrho cov ntaub ntawv.
2. Xam qhov sib txawv ntawm txhua cov ntaub ntawv thiab qhov nruab nrab: \( x_i - \bar{x} \).
3. Siv tus nqi tag nrho ntawm txhua qhov sib txawv: \( |x_i – \bar{x}| \).
4. Muab tag nrho cov nqi ntawm qhov sib txawv ntxiv ua ke.
5. Faib los ntawm tus naj npawb ntawm cov ntaub ntawv \(n \).
Piv txwv ntawm cov ntaub ntawv ib leeg
Cov ntaub ntawv tus nqi: 6, 8, 10, 12, 14
1) Nruab Nrab:
\[
\bar{x}=\frac{6+8+10+12+14}{5}=\frac{50}{5}=10
\]
2) Tus nqi tseeb ntawm qhov sib txawv:
– |6 − 10| = 4
– |8 − 10| = 2
– |10 − 10| = 0
– |12 − 10| = 2
– |14 − 10| = 4
Tag Nrho = 4 + 2 + 0 + 2 + 4 = 12
3) Qhov nruab nrab ntawm kev sib txawv:
\[
SR=\frac{12}{5}=2{,}4
\]
Qhov no txhais tau tias qhov nruab nrab txhua tus nqi hloov pauv los ntawm 2,4 units ntawm qhov nruab nrab (10).
Qhov Nruab Nrab Sib Txawv rau Cov Ntaub Ntawv Nquag (Discrete)
Yog tias cov ntaub ntawv tau nthuav tawm hauv daim ntawv ntawm cov nqi thiab zaus (piv txwv li lub rooj), cov mis yuav dhau los ua:
\[
SR = \frac{\sum f_i | x_i – x_i|}{\sum f_i}
\]
Cov Ntaub Ntawv:
– \( f_i \): zaus ntawm cov ntaub ntawv \( x_i \)
Cov Txheej Txheem Xam Cov Ntaub Ntawv Zaus
1. Xam tus nqi nruab nrab: \(\bar{x}=\frac{\sum f_i x_i}{\sum f_i}\)
2. Xam \( |x_i-\bar{x}| \)
3. Muab tus zaus sib npaug: \( f_i | x_i-\bar{x}| \)
4. Muab tag nrho cov txiaj ntsig ntawm kauj ruam 3 ntxiv rau hauv.
5. Faib los ntawm tag nrho cov zaus
Piv txwv ntawm cov ntaub ntawv sib cais
| Tus nqi (x) | Zaus (f) |
|—|—|
| 5 | 2 |
| 7 | 3 |
| 9 | 1 |
Tag nrho zaus: \(2+3+1=6\)
Nruab nrab:
\[
\bar{x}=\frac{(5)(2)+(7)(3)+(9)(1)}{6}=\frac{10+21+9}{6}=\frac{40}{6}=6{,}67
\]
Xam qhov sib txawv:
– Rau x=5: |5−6,67|=1,67 → zaug f=2 → 3,34
– Rau x=7: |7−6,67|=0,33 → zaug f=3 → 0,99
– Rau x=9: |9−6,67|=2,33 → zaug f=1 → 2,33
Tag Nrho: 3,34 + 0,99 + 2,33 = 6,66
Qhov nruab nrab ntawm kev sib txawv:
\[
SR=\frac{6{,}66}{6}=1{,}11
\]
Qhov Nruab Nrab Sib Txawv rau Cov Ntaub Ntawv Muab Pawg (Chav Kawm Nruab Nrab)
Hauv cov ntaub ntawv uas tau muab faib ua pawg (piv txwv li, kev faib tawm zaus ntawm lub sijhawm), cov nqi ntawm cov ntaub ntawv tsis raug tso tawm ib tus zuj zus, tab sis hauv cov chav kawm. Rau lub hom phiaj no, qhov chaw nruab nrab ntawm chav kawm (xi) yog siv los sawv cev rau cov ntaub ntawv hauv ib chav kawm.
Tus qauv:
\[
SR = \frac{\sum f_i | x_i - x_i| }{\sum f_i}
\]
Txawm li cas los xij, \(x_i\) yog qhov nruab nrab ntawm chav kawm.
Cov Txheej Txheem Xam Cov Ntaub Ntawv Pab Pawg
1. Txheeb xyuas qhov nruab nrab ntawm txhua chav kawm:
\[
x_i = \frac{\text{txwv qis dua + txwv siab dua}}{2}
\]
2. Xam qhov nruab nrab ntawm pawg:
\[
\bar{x}=\frac{\sum f_i x_i}{\sum f_i}
\]
3. Xam \( |x_i-\bar{x}| \)
4. Muab tus zaus (f_i) sib npaug
5. Muab tag nrho cov txiaj ntsig los sib ntxiv, ces faib los ntawm tag nrho cov zaus.
Piv txwv ntawm cov ntaub ntawv pab pawg
| Chav Kawm | f |
|—|—|
| 10–14 | 3 |
| 15–19 | 5 |
| 20–24 | 2 |
Nruab Nrab:
– 10–14 → 12
– 15–19 → 17
– 20–24 → 22
Tag nrho f = 10
Nruab nrab:
\[
\bar{x}=\frac{(12)(3)+(17)(5)+(22)(2)}{10}=\frac{36+85+44}{10}=\frac{165}{10}=16{,}5
\]
Kev Sib Txawv:
– |12−16,5|=4,5 → ×3 = 13,5
– |17−16,5|=0,5 → ×5 = 2,5
– |22−16,5|=5,5 → ×2 = 11
Tag Nrho = 13,5 + 2,5 + 11 = 27
Qhov nruab nrab ntawm kev sib txawv:
\[
SR=\frac{27}{10}=2{,}7
\]
Qhov Sib Txawv Nruab Nrab Los Ntawm Nruab Nrab
Ntxiv rau qhov nruab nrab, qhov nruab nrab ntawm qhov sib txawv kuj tseem tuaj yeem suav los ntawm qhov nruab nrab. Lub hauv paus ntsiab lus zoo ib yam, tsuas yog tus nqi nruab nrab txawv. Qhov no muaj txiaj ntsig thaum cov ntaub ntawv muaj cov outliers vim tias qhov nruab nrab muaj zog dua rau cov nqi siab heev.
Rau cov ntaub ntawv ib leeg:
\[
SR_{Me}=\frac{\sum |x_i-Me|}{n}
\]
Rau cov ntaub ntawv zaus:
\[
SR_{Me}=\frac{\sum f_i|x_i-Me|}{\sum f_i}
\]
Cov Zoo thiab Cov Kev Txwv ntawm Qhov Nruab Nrab Deviation
Tshaj:
1. Yooj yim to taub vim nws siv "qhov nruab nrab ntawm qhov deb" ntawm lub chaw khaws ntaub ntawv.
2. Siv tag nrho cov ntaub ntawv (tsis yog tsuas yog qee cov ntaub ntawv xwb).
3. Muaj peev xwm suav tau rau ntau hom kev nthuav qhia cov ntaub ntawv.
Cov Kev Txwv:
1. Tsis tshua nrov dua li qhov kev hloov pauv tus qauv hauv kev tshuaj xyuas cov ntaub ntawv qib siab.
2. Kev siv cov nqi tag nrho ua rau nws tsis yooj yim rau qee qhov kev ua lej algebraic.
3. Tsis muaj zog npaum li qhov kev hloov pauv tus qauv hauv ntau txoj kev xav.
Kev kaw
Cov txheej txheem rau kev txiav txim siab qhov nruab nrab ntawm cov ntaub ntawv suav lej feem ntau ua raws li tus qauv sib xws: txiav txim siab qhov nruab nrab (qhov nruab nrab lossis nruab nrab), xam qhov deb ntawm txhua qhov chaw ntawm cov ntaub ntawv (lossis chav kawm nruab nrab) ntawm qhov chaw nruab nrab, thiab tom qab ntawd nruab nrab lawv - xav txog qhov zaus yog tias cov ntaub ntawv tau nthuav tawm hauv lub rooj. Qhov nruab nrab ntawm qhov sib txawv yog qhov ntsuas yooj yim ntawm kev faib tawm rau kev piav qhia txog kev hloov pauv ntawm cov ntaub ntawv. Los ntawm kev nkag siab cov kauj ruam, koj tuaj yeem sib piv qhov sib txawv ntawm cov ntaub ntawv teeb tsa thiab ntsuas qhov ruaj khov ntawm cov ntaub ntawv teeb tsa.
Yog tias koj xav tau, kuv tuaj yeem pab koj ua ib qho version ntawm tsab xov xwm no hauv hom ntawv ua haujlwm hauv tsev kawm ntawv (nrog rau kev taw qhia - kev sib tham - xaus) lossis ntxiv cov lus nug xyaum ua ke nrog kev sib tham.
