Piv txwv cov lus nug thiab kev sib tham: Kev hloov pauv thiab kev hloov pauv tus qauv ntawm cov ntaub ntawv ib leeg
Kev suav lej yog ib ceg ntawm kev suav lej uas cuam tshuam nrog kev sau, kev tshuaj xyuas, kev txhais lus, kev nthuav qhia, thiab kev teeb tsa cov ntaub ntawv. Ob lub tswv yim tseem ceeb hauv kev suav lej yog qhov sib txawv thiab qhov sib txawv ntawm tus qauv. Tsab xov xwm no yuav tham txog ntau yam yuav ua li cas xam qhov sib txawv thiab qhov sib txawv ntawm tus qauv ntawm ib pawg ntaub ntawv los ntawm ntau qhov teeb meem piv txwv.
Nkag Siab Txog Qhov Sib Txawv Thiab Qhov Sib Txawv Tus Qauv
Kev hloov pauv yog ib qho kev ntsuas uas ntsuas seb cov lej hauv cov ntaub ntawv tau nthuav dav npaum li cas ntawm qhov nruab nrab lossis qhov nruab nrab. Kev hloov pauv yog qhia ua cov chav plaub fab ntawm cov ntaub ntawv qub, ua rau nws feem ntau nyuaj rau txhais.
Tus qauv kev hloov pauv yog lub hauv paus plaub fab ntawm qhov sib txawv. Nws muab kev ntsuas yooj yim dua ntawm seb cov ntaub ntawv feem ntau hloov pauv deb npaum li cas ntawm qhov nruab nrab vim tias nws cov chav nyob zoo ib yam li cov chav nyob thawj ntawm cov ntaub ntawv.
Cov Qauv Dav Dav
Rau cov ntaub ntawv ib leeg, cov mis rau qhov sib txawv \(\sigma^2\) thiab qhov sib txawv ntawm tus qauv \(\sigma\) ntawm cov pej xeem yog raws li nram no:
1. Kev Hloov Pauv (σ²):
\( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2 \)
2. Kev Sib Txawv Txheem (σ):
\( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2} \)
Qhov twg:
- \( N \) yog tus naj npawb ntawm cov ntaub ntawv hauv cov pej xeem.
- \( x_i \) yog tus nqi ntawm cov ntaub ntawv ith.
– \( \mu \) yog qhov nruab nrab ntawm cov ntaub ntawv.
Yog tias peb xam qhov sib txawv thiab qhov sib txawv ntawm tus qauv, ces cov mis saum toj no raug hloov kho me ntsis:
1. Qauv Sib Txawv (s²):
\( s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i – x)^2 \)
2. Qauv Kev Sib Txawv Txheem (cov):
\( s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i – x)^2} \)
Qhov twg:
- \( n \) yog tus lej ntawm cov ntaub ntawv hauv cov qauv.
- \( \bar{x} \) yog qhov nruab nrab lossis qhov nruab nrab ntawm cov qauv.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Piv txwv lus nug 1:
Muab cov ntaub ntawv hauv qab no:
8, 10, 10, 10, 12, 14
Xam qhov sib txawv thiab qhov sib txawv ntawm cov ntaub ntawv!
Cov Kauj Ruam Kev Daws Teeb Meem:
1. Xam Qhov Nruab Nrab (Nruab Nrab) \( \mu \):
\[
\mu = \frac{8+10+10+10+12+14}{6} = \frac{64}{6} = 10.67
\]
2. Xam Qhov Sib Txawv Ntawm Cov Ntaub Ntawv thiab Qhov Nruab Nrab, Tom Qab Ntawd Ua Square:
\[
(8 – 10.67)^2 = 7.1289
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(12 – 10.67)^2 = 1.7689
\]
\[
(14 – 10.67)^2 = 11.1089
\]
3. Ntxiv Tag Nrho Cov Txiaj Ntsig Squared:
\[
\sum (x_i – \mu)^2 = 7.1289 + 0.4489 + 0.4489 + 0.4489 + 1.7689 + 11.1089 = 21.3534
\]
4. Xam Qhov Sib Txawv (σ²) ntawm Cov Ntaub Ntawv Pej Xeem:
\[
\sigma^2 = \frac{21.3534}{6} = 3.559
\]
Lus Cim: Vim tias cov ntaub ntawv no tau teev tseg ua cov ntaub ntawv pej xeem, peb faib los ntawm 6.
5. Xam Qhov Txawv Txawv (σ):
\[
\sigma = \sqrt{3.559} \approx 1.886
\]
Yog li, qhov sib txawv ntawm cov ntaub ntawv yog 3.559 thiab qhov sib txawv ntawm tus qauv yog 1.886.
Piv txwv lus nug 2:
Raws li cov ntaub ntawv piv txwv hauv qab no:
5, 6, 8, 9, 10, 11
Xam qhov variance thiab tus qauv deviation ntawm cov qauv!
Cov Kauj Ruam Kev Daws Teeb Meem:
1. Xam Qhov Nruab Nrab (Nruab Nrab) \( \bar{x} \):
\[
\bar{x} = \frac{5 + 6 + 8 + 9 + 10 + 11}{6} = \frac{49}{6} = 8.167
\]
2. Xam Qhov Sib Txawv Ntawm Cov Ntaub Ntawv thiab Qhov Nruab Nrab, Tom Qab Ntawd Ua Square:
\[
(5 – 8.167)^2 = 10.035
\]
\[
(6 – 8.167)^2 = 4.694
\]
\[
(8 – 8.167)^2 = 0.028
\]
\[
(9 – 8.167)^2 = 0.694
\]
\[
(10 – 8.167)^2 = 3.361
\]
\[
(11 – 8.167)^2 = 7.945
\]
3. Ntxiv Tag Nrho Cov Txiaj Ntsig Squared:
\[
\sum (x_i – x)^2 = 10.035 + 4.694 + 0.028 + 0.694 + 3.361 + 7.945 = 26.757
\]
4. Xam Qhov Sib Txawv ntawm Cov Qauv (s²):
\[
s^2 = \frac{26.757}{5} = 5.351
\]
Lus Cim: Vim qhov no yog cov ntaub ntawv piv txwv, peb faib los ntawm 5 (n-1).
5. Xam Qhov Txawv Txawv ntawm Tus Qauv (s):
\[
s = 5.351 x 2.313
\]
Yog li, qhov sib txawv ntawm cov ntaub ntawv qauv yog 5.351 thiab qhov sib txawv ntawm tus qauv yog 2.313.
Xaus
Kev xam qhov sib txawv thiab qhov sib txawv ntawm tus qauv yog qhov tseem ceeb rau kev nkag siab txog qhov kev nthuav dav ntawm cov ntaub ntawv hauv ib pawg. Txawm hais tias qhov sib txawv muab kev ntsuas theoretical ntawm kev sib kis hauv cov squares ntawm cov units qub, qhov sib txawv ntawm tus qauv txhais qhov kev ntsuas ntawm kev sib kis hauv cov units qub ntawm cov ntaub ntawv, ua rau nws yooj yim dua rau kev nkag siab. Hauv kev tshuaj xyuas cov ntaub ntawv, ob qho kev ntsuas no feem ntau siv los ntsuas qhov sib txawv ntawm cov ntaub ntawv thiab txiav txim siab txog kev suav lej.
Los ntawm kev nkag siab txog cov kauj ruam thiab cov qauv tsim nyog, peb tuaj yeem yooj yim xam qhov sib txawv thiab qhov sib txawv ntawm tus qauv rau ntau yam xwm txheej uas peb ntsib hauv kev sau thiab tshuaj xyuas cov ntaub ntawv txhua hnub. Vam tias, tsab xov xwm no muab kev nkag siab tob dua txog cov ntsiab lus ntawm qhov sib txawv thiab qhov sib txawv ntawm tus qauv hauv cov ntaub ntawv ib leeg.
