Cov txheej txheem rau kev suav tus nqi nruab nrab rau cov ntaub ntawv ib leeg thiab cov ntaub ntawv ua pawg
Tus lej nruab nrab yog qhov ntsuas ntawm qhov sib txawv ntawm qhov nruab nrab uas feem ntau siv hauv kev suav lej. Tsis zoo li qhov nruab nrab (mean), uas ntxiv tag nrho cov nqi thiab tom qab ntawd faib los ntawm tus lej ntawm cov nqi, tus lej nruab nrab hais txog "tus nqi nruab nrab" ntawm cov ntaub ntawv teeb tsa. Vim nws tsom mus rau qhov chaw, tus lej nruab nrab yog qhov tsis kam rau cov nqi siab heev (outliers), xws li thaum ib tus nqi loj heev lossis me heev piv rau lwm tus. Qhov no yog vim li cas tus lej nruab nrab thiaj siv dav hauv kev tshuaj xyuas cov ntaub ntawv kev lag luam, kev kawm, kev tshawb fawb hauv zej zog, thiab txawm tias kev ntsuam xyuas cov qhab nia xeem.
Hauv tsab xov xwm no, peb yuav tham txog cov txheej txheem rau kev xam tus nqi nruab nrab rau ob hom ntaub ntawv: cov ntaub ntawv ib leeg (tsis muaj pawg) thiab cov ntaub ntawv sib sau ua ke (nthuav tawm hauv daim ntawv faib zaus). Ntxiv rau cov mis, kev sib tham yuav suav nrog cov kauj ruam ua tau yooj yim rau kev siv.
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1. Lub Tswv Yim Tseem Ceeb ntawm Median
Tus lej nruab nrab yog tus nqi nruab nrab tom qab cov ntaub ntawv raug txheeb los ntawm qhov me tshaj plaws mus rau qhov loj tshaj plaws. Yog tias tus lej ntawm cov ntsiab lus ntaub ntawv yog khib, tus lej nruab nrab yog tus nqi nruab nrab tseeb. Yog tias tus lej ntawm cov ntsiab lus ntaub ntawv yog tus lej txawm, tus lej nruab nrab yog tus nqi nruab nrab ntawm ob tus nqi nruab nrab.
Raws li kev xav, qhov nruab nrab faib cov ntaub ntawv ua ob ntu:
– 50% ntawm cov ntaub ntawv yog qis dua (lossis sib npaug rau) qhov nruab nrab
– 50% ntawm cov ntaub ntawv yog siab dua (lossis sib npaug rau) qhov nruab nrab
Vim tias qhov nruab nrab yog raws li kev txiav txim, thawj kauj ruam uas yuav luag txhua zaus xav tau yog kev txheeb cov ntaub ntawv.
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2. Xam Tus Nruab Nrab rau Cov Ntaub Ntawv Ib Leeg
Cov ntaub ntawv ib leeg yog cov ntaub ntawv uas tau nthuav tawm raws li nws yog (piv txwv li daim ntawv teev cov qib kawm ntawm cov tub ntxhais kawm), tsis tau muab sau ua ke hauv cov chav kawm ib ntus xws li cov ntaub ntawv pab pawg.
A. Cov Kauj Ruam Dav Dav
1. Cais cov ntaub ntawv los ntawm qhov me tshaj plaws mus rau qhov loj tshaj plaws.
2. Txheeb xyuas qhov ntau ntawm cov ntaub ntawv, piv txwv li n.
3. Txheeb xyuas qhov chaw ntawm nruab nrab:
- Yog tias n yog khib, qhov nruab nrab yog nyob rau ntawm qhov chaw \((n+1)/2\).
- Yog tias n yog tus lej sib npaug, tus lej nruab nrab yog tus lej nruab nrab ntawm cov ntaub ntawv ntawm qhov chaw \(n/2\) thiab \((n/2)+1\).
B. Tus Qauv Nruab Nrab rau Cov Ntaub Ntawv Ib Leeg
- Yog tias n yog khib:
\[
Kuv = x_{(n+1)/2}
\]
Qhov no txhais tau hais tias tus nruab nrab yog tus nqi ntawm cov ntaub ntawv nyob rau hauv qhov kev txiav txim \((n+1)/2\)th.
- Yog tias n yog tus lej sib npaug:
\[
Kuv = \frac{x_{n/2} + x_{(n/2)+1}}{2}
\]
C. Piv txwv ntawm Cov Ntaub Ntawv Ib Leeg (n Qis)
Cov ntaub ntawv: 7, 2, 9, 4, 3
1) Xaiv: 2, 3, 4, 7, 9
2) n = 5 (khib)
3) Qhov chaw nruab nrab = \((5+1)/2 = 3\)
Nruab nrab = cov ntaub ntawv thib 3 = 4
Yog li ntawd, qhov nruab nrab ntawm cov ntaub ntawv yog 4.
D. Piv txwv ntawm Cov Ntaub Ntawv Ib Leeg (n Tus Nqi Sib Npaug)
Cov ntaub ntawv: 10, 4, 6, 8
1) Xaiv: 4, 6, 8, 10
2) n = 4 (txawm)
3) Qhov chaw nruab nrab yog cov ntaub ntawv thib 2 thiab thib 3
Nruab Nrab = (6 + 8)/2 = 7)
Yog li ntawd, qhov nruab nrab ntawm cov ntaub ntawv yog 7.
E. Lus Cim Tseem Ceeb: Cov Ntaub Ntawv Uas Muaj Zaus
Qee zaum ib pawg ntaub ntawv tuaj yeem muab ua tus nqi thiab zaus (piv txwv li, 60 tshwm sim ob zaug, 70 tshwm sim tsib zaug). Hauv qhov no, qhov nruab nrab tseem pom raws li "kev txiav txim" ntawm cov ntaub ntawv, tab sis peb tuaj yeem siv qhov zaus sib sau ua ke los txiav txim siab qhov chaw nruab nrab yam tsis tau teev cov ntsiab lus ntaub ntawv ib tus zuj zus. Lub hauv paus ntsiab lus yog tib yam: nrhiav qhov (n + 1) / 2nd txoj haujlwm (khib) lossis qhov (n / 2) thiab (n / 2) + 1st txoj haujlwm (txawm), tom qab ntawd saib cov nqi uas npog qhov chaw ntawd raws li qhov zaus sib sau ua ke.
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3. Xam Tus Nruab Nrab rau Cov Ntaub Ntawv Sib Koom Ua Ib Pawg
Cov ntaub ntawv pawg yog cov ntaub ntawv uas tau muab sau ua pawg sib txawv thiab lawv cov zaus. Piv txwv li: 3 tus neeg uas muaj qhov siab ntawm 150–154 cm, 8 tus neeg uas muaj qhov siab ntawm 155–159 cm, thiab lwm yam. Tsis zoo li cov ntaub ntawv ib leeg, qhov nruab nrab ntawm cov ntaub ntawv pawg feem ntau tsis tau txiav txim siab meej vim peb tsis paub cov nqi ntawm tus kheej hauv lub sijhawm sib txawv. Yog li ntawd, qhov nruab nrab yog xam siv kev kwv yees (kwv yees) siv cov mis nruab nrab rau kev faib tawm pawg.
A. Cov Lus Tseem Ceeb hauv Pawg Neeg Cov Ntaub Ntawv Nruab Nrab
Ua ntej siv cov mis no, peb yuav tsum nkag siab txog ntau yam:
- n = tag nrho cov zaus (tag nrho cov ntaub ntawv)
– n/2 = qhov chaw nruab nrab ntawm tag nrho
– Chav kawm nruab nrab = chav kawm nruab nrab thawj zaug uas tsim cov zaus sib sau ua ke ≥ n/2
- L = ntug qis dua ntawm chav kawm nruab nrab (tsis yog qhov txwv qis dua, tab sis ntug chav kawm; rau cov ntaub ntawv txuas ntxiv feem ntau siv 0,5 kev hloov kho yog tias cov ntaub ntawv yog cov lej)
- F = zaus sib sau ua ke ua ntej chav kawm nruab nrab
– f = zaus ntawm chav kawm nruab nrab
- c = qhov ntev ntawm chav kawm (qhov dav ntawm lub sijhawm)
B. Cov Kauj Ruam Los Txheeb Xyuas Qhov Nruab Nrab ntawm Cov Ntaub Ntawv Pab Pawg
1. Tsim ib daim ntawv faib zaus thiab ntxiv ib kab zaus sib sau ua ke.
2. Xam n (tus naj npawb ntawm cov zaus) thiab txiav txim siab n/2.
3. Txheeb xyuas pawg nruab nrab, uas yog pawg uas suav nrog n/2 qhov chaw raws li qhov zaus sib sau ua ke.
4. Sau cov nqi rau hauv tus qauv nruab nrab rau cov ntaub ntawv pawg.
C. Tus Qauv Nruab Nrab rau Cov Ntaub Ntawv Pab Pawg
\[
Kuv = L + \left(\frac{n}{2} – F}{f}\right)\times c
\]
Cov mis no ua qhov kev sib txuas lus linear hauv chav kawm nruab nrab, xav tias cov ntaub ntawv tau faib sib npaug thoob plaws hauv chav kawm.
D. Piv txwv ntawm Median ntawm Cov Ntaub Ntawv Pab Pawg
Piv txwv li, cov ntaub ntawv qhab nia xeem hauv qab no:
| Tus nqi nruab nrab | Zaus (f) |
|—|—:|
| 40–49 | 5 |
| 50–59 | 8 |
| 60–69 | 12 |
| 70–79 | 10 |
| 80–89 | 5 |
1) Tag nrho cov zaus:
\[
n = 5 + 8 + 12 + 10 + 5 = 40
\]
2) Xam n/2:
\[
n/2 = 20
\]
3) Zaus sib sau ua ke:
– 40–49: 5
– 50–59: 5+8 = 13
– 60–69: 13+12 = 25
– 70–79: 35
– 80–89: 40
Txoj hauj lwm 20 yog nyob rau hauv chav kawm nrog thawj qhov qhab nia sib sau ua ke ≥ 20, uas yog 60–69. Yog li qhov no yog chav kawm nruab nrab.
4) Txheeb xyuas cov khoom sib xyaw:
– L = ntug qis ntawm chav kawm nruab nrab. Rau lub sijhawm 60–69, ntug qis yog 59,5 (yog tias cov ntaub ntawv yog tus nqi integer).
– F = zaus sib sau ua ke ua ntej chav kawm nruab nrab = 13
– f = zaus ntawm chav kawm nruab nrab = 12
– c = ntev ntawm chav kawm = 10
5) Sau rau hauv tus qauv:
\[
Kuv = 59,5 + \left(\frac{20 – 13}{12}\right)\times 10
\]
\[
Kuv = 59,5 + \left(\frac{7}{12}\right)\times 10
\]
\[
Kuv = 59,5 + 5,833… = 65,333…
\]
Yog li ntawd, qhov nruab nrab ntawm cov ntaub ntawv pawg yog kwv yees li 65,33.
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4. Tej Yam Yuam Kev Uas Feem Ntau Ua
Qee qhov yuam kev uas feem ntau tshwm sim thaum xam qhov nruab nrab:
1. Tsis cais cov ntaub ntawv rau ib qho xwb, yog li tus nqi nruab nrab tsis raug.
2. Txiav txim tsis raug qhov chaw ntawm nruab nrab thaum n yog tus lej txawm (yuav tsum siv qhov nruab nrab ntawm ob tus nqi nruab nrab).
3. Rau cov ntaub ntawv pawg, nws yog qhov tsis yog los xaiv chav kawm nruab nrab vim nws tsis tsim kom muaj zaus sib sau ua ke.
4. Siv qhov txwv qis ntawm chav kawm ntug qis (L) thaum cov ntaub ntawv yog cov lej txuas ntxiv / sib nrug.
5. Txiav txim siab tsis raug qhov ntev ntawm chav kawm (c), tshwj xeeb tshaj yog tias qhov sib nrug tsis sib xws.
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5. Xaus Lus
Tus lej nruab nrab yog ib qho kev ntsuas yooj yim tab sis muaj zog ntawm qhov sib txawv ntawm qhov nruab nrab, tshwj xeeb tshaj yog thaum cov ntaub ntawv muaj cov nqi siab heev. Rau cov ntaub ntawv ib leeg, tus lej nruab nrab yog txiav txim siab ncaj qha los ntawm qhov chaw nruab nrab tom qab cov ntaub ntawv raug txheeb xyuas, nrog rau kev kho sib txawv rau cov lej khib thiab tus lej txawm ntawm cov ntaub ntawv. Lub caij no, rau cov ntaub ntawv pawg, tus lej nruab nrab yog suav siv cov mis interpolation raws li chav kawm nruab nrab, zaus sib sau ua ke, thiab qhov ntev ntawm chav kawm.
Los ntawm kev nkag siab lub tswv yim thiab cov kauj ruam, koj tuaj yeem xam tus lej nruab nrab sai thiab raug, ob qho tib si ntawm cov ntaub ntawv yooj yim thiab ntawm cov ntaub ntawv sau ua ke hauv cov lus qhia. Hauv ntau qhov xwm txheej kev tshuaj xyuas, tus lej nruab nrab yog qhov kev xaiv sawv cev ntau dua li qhov nruab nrab, tshwj xeeb tshaj yog thaum cov ntaub ntawv faib tawm tsis sib xws lossis muaj cov outliers.
Yog tias koj xav tau, kuv kuj tuaj yeem ntxiv cov lus nug xyaum ua ke nrog kev sib tham kom txhawb koj txoj kev nkag siab txog qhov nruab nrab ntawm cov ntaub ntawv ib leeg thiab pawg.
