Mibvunzo yeMienzaniso neKukurukurirana: Kusiyana uye Kutsauka Kwakajairika kweData Rimwe Chete
Nhamba dzezviverengero ibazi remasvomhu rine chekuita nekuunganidza, kuongorora, kududzira, kuratidzwa, uye kurongeka kwedata. Pfungwa mbiri dzakakosha munhamba idzi imhando yevariance uye standard deviation. Chinyorwa chino chichakurukura zvakadzama kuti tingaverenga sei variance uye standard deviation yedata rimwe chete kuburikidza nematambudziko akati wandei emuenzaniso.
Kunzwisisa Kusiyana uye Kutsauka Kwakajairika
Kusiyana-siyana inzira yekuyera kuti nhamba dziri mudata dzakapararira kusvika papi kubva paavhareji kana kuti paavhareji. Kusiyana kunoratidzwa muzvikamu zvemasekori zvedata rekutanga, zvichiita kuti zviome kududzira.
Kutsauka kwakajairwa ndiko mudzi wepakati wevariance. Kunopa chiyero chakajeka chekuti data rinowanzo tsaukana sei kubva paavhareji nekuti mayuniti aro akafanana nemayuniti ekutanga edata.
Fomura Yakazara
Kune data rimwe chete, fomura yekusiyana \( \sigma^2 \) uye standard deviation \( \sigma \) yevanhu ndeiyi inotevera:
1. Kusiyana (σ²):
\( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2 \)
2. Kutsauka Kwakajairwa (σ):
\( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2} \)
Di mana:
– \( N \) ndiyo nhamba yedata muhuwandu hwevanhu.
– \( x_i \) ndiyo kukosha kwedata.
– \( \mu \) ndiyo avhareji kana kuti avhareji yedata.
Kana tikaverenga musiyano uye kutsauka kwakajairwa kwemuenzaniso, fomura iri pamusoro inochinjwa zvishoma:
1. Musiyano wemuenzaniso (s²):
\( s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2 \)
2. Muenzaniso weKutsauka Kwakajairwa:
\( s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2} \)
Di mana:
– \( n \) ndiyo nhamba yedata riri mumuenzaniso.
– \( \bar{x} \) ndiyo avhareji kana kuti avhareji yemuenzaniso.
Mibvunzo yemuenzaniso nekukurukurirana
Muenzaniso Mubvunzo 1:
Zvichienderana nedata rinotevera:
8, 10, 10, 10, 12, 14
Verenga musiyano uye kutsauka kwakajairwa kwedata!
Matanho Ekugadzirisa:
1. Kuverenga Avhareji (Avhareji) \( \mu \):
\[
\mu = \frac{8 + 10 + 10 + 10 + 12 + 14}{6} = \frac{64}{6} = 10.67
\]
2. Verenga Musiyano Uripo Pakati peData neAvhareji, Wobva Waverenga Sikweya:
\[
(8 – 10.67)^2 = 7.1289
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(12 – 10.67)^2 = 1.7689
\]
\[
(14 – 10.67)^2 = 11.1089
\]
3. Kuwedzera Mhedzisiro Yese Yakaganhurirwa:
\[
\sum (x_i – \mu)^2 = 7.1289 + 0.4489 + 0.4489 + 0.4489 + 1.7689 + 11.1089 = 21.3534
\]
4. Kuverenga Kusiyana (σ²) kweData reVanhu:
\[
\sigma^2 = \frac{21.3534}{6} = 3.559
\]
Cherechedza: Sezvo data iri richinzi data revanhu, tinogovanisa ne6.
5. Kuverenga Kutsauka Kwakajairwa (σ):
\[
\sigma = \sqrt{3.559} \inenge 1.886
\]
Saka, musiyano wedata i3.559 uye kutsauka kwakajairwa i1.886.
Muenzaniso Mubvunzo 2:
Zvichienderana neruzivo rwemuenzaniso runotevera:
5, 6, 8, 9, 10, 11
Verenga musiyano uye kutsauka kwakajairwa kwemuenzaniso!
Matanho Ekugadzirisa:
1. Kuverenga Avhareji (Mean) \( \bar{x} \) :
\[
\bar{x} = \frac{5 + 6 + 8 + 9 + 10 + 11}{6} = \frac{49}{6} = 8.167
\]
2. Verenga Musiyano Uripo Pakati peData neAvhareji, Wobva Waverenga Sikweya:
\[
(5 – 8.167)^2 = 10.035
\]
\[
(6 – 8.167)^2 = 4.694
\]
\[
(8 – 8.167)^2 = 0.028
\]
\[
(9 – 8.167)^2 = 0.694
\]
\[
(10 – 8.167)^2 = 3.361
\]
\[
(11 – 8.167)^2 = 7.945
\]
3. Kuwedzera Mhedzisiro Yese Yakaganhurirwa:
\[
\sum (x_i – \bar{x})^2 = 10.035 + 4.694 + 0.028 + 0.694 + 3.361 + 7.945 = 26.757
\]
4. Kuverenga Musiyano weSample (s²):
\[
s^2 = \frac{26.757}{5} = 5.351
\]
Cherechedza: Sezvo iyi iri data remuenzaniso, tinoigovanisa ne5 (n-1).
5. Kuverenga Kutsauka Kwakajairwa:
\[
s = \sqrt{5.351} \inenge 2.313
\]
Saka, musiyano wedata remuenzaniso i5.351 uye kutsauka kwakajairwa i2.313.
Mhedziso
Kuverenga musiyano uye kutsauka kwakajairwa kwakakosha kuti unzwisise kuti kupararira kwedata kuri sei mukati mechikamu chakapihwa. Kunyange zvazvo kusiyana kuchipa chiyero chedzidziso chekupararira muzvikwere zvemayuniti ekutanga, kutsauka kwakajairwa kunotsanangura chiyero chekupararira maererano nemayuniti ekutanga edata, zvichiita kuti zvive nyore kunzwisisa. Mukuongorora data, zviyero izvi zviviri zvinowanzo shandiswa kuongorora kusiyana kwedata uye kuita sarudzo dzehuwandu.
Nekunzwisisa matanho anodiwa uye mafomura, tinogona kuverenga zviri nyore kusiyana uye kutsauka kwakajairwa mumamiriro akasiyana-siyana atinosangana nawo mukuunganidza nekuongorora data zuva nezuva. Tinovimba kuti chinyorwa chino chichatipa kunzwisisa kwakadzama kwepfungwa dzekusiyana uye kutsauka kwakajairwa mudata rimwe chete.
