Claonadh àbhaisteach
Tha claonadh àbhaisteach cudromach airson a bhith a' tomhas sgaoileadh no sgapadh.
Bidh e ag innse dhuinn dè cho fada, gu cuibheasach, 's a tha na toraidhean bhon mheadhan.
Mar sin ma tha an claonadh àbhaisteach beag, tha sin ag innse dhuinn gu bheil na toraidhean faisg air a' mheadhan. Ma tha an claonadh àbhaisteach mòr, tha na toraidhean nas sgapte.
Mar eisimpleir, tha nàdar an dà sheata-dàta a leanas gu math diofraichte, ach tha an cuibheas, am meadhan agus an raon aca an aon rud.
Bhiodh e feumail tomhas àireamhail air choreigin a bhith againn a dhèanadh sgaradh eatarra.
| Seata-dàta 1 | 1 | 7 | 12 | 15 | 20 | 22 | 28 |
| Seata-dàta 2 | 1 | 15 | 15 | 15 | 15 | 16 | 28 |
| Seata-dàta 1 |
|---|
| 1 |
| 7 |
| 12 |
| 15 |
| 20 |
| 22 |
| 28 |
| Seata-dàta 2 |
|---|
| 1 |
| 15 |
| 15 |
| 15 |
| 15 |
| 16 |
| 28 |
Tha a' mhòr-chuid dhe na toraidhean ann an seata-dàta 2 faisg air a' chuibheas, ach an coimeas ris an sin, tha a' mhòr-chuid dhe na toraidhean ann an seata-dàta 1 nas fhaide bhon chuibheas.
Saoilidh sinn bhon seo gu bheil an claonadh àbhaisteach nas lugha ann an seata-dàta 2 na ann an seata-dàta 1.
Ann a bhith a' coimeas sgaoilidhean, tha e nas fheàrr tomhas sgaoilidh no sgapaidh a chleachdadh (leithid claonadh àbhaisteach no leth-raon eadar-chairtealach) a bharrachd air tomhas claonadh meadhain (leithid cuibheas, meadhan no mòd).
Tha dà fhoirmle ann airson claonadh àbhaisteach obrachadh a-mach, ach 's e am foirmle as trice a bhios air a chleachdadh airson claonadh àbhaisteach obrachadh a-mach:
\(\text{Claonadh àbhaisteach}\) \((\text{CÀ})\) \(SD = \sqrt {\frac{{\sum {{(X - \bar X)}^2}}}{{n - 1}}}\)
Far a bheil \(\sum\) a' ciallachadh 'sùim de'
'S e \({\bar X}\) an cuibheas
'S e \(\text{n}\) àireamh an dàta san t-sampall
Cleachd am fiosrachadh seo agus feuch na ceistean gu h-ìosal.
Question
Obraich a-mach an cuibheas (mean) is an claonadh àbhaisteach (SD) aig na h-àireamhan a leanas: \(4,\,7,\,9,\,11,\,13,\,15,\,18\)
\(SD = \sqrt {\frac{{\sum {{(X - \bar X)}^2}}}{{n - 1}}}\)
Airson obrachadh a-mach dè an luach a tha aig \(\sum {(X - \bar X)^2}\), bidh sinn a' cleachdadh clàr mar a chì thu gu h-ìosal:
| \(x\) | \(X - \bar X\) | \({(X - \bar X)^2}\) |
| 4 | \(4 - 11 = - 7\) | \({( - 7)^2} = 49\) |
| 7 | \(7 - 11 = - 4\) | \({( - 4)^2} = 16\) |
| 9 | \(9 - 11 = - 2\) | \({( - 2)^2} = 4\) |
| 11 | \(11 - 11 = 0\) | \({(0)^2} = 0\) |
| 13 | \(13 - 11 = 2\) | \({(2)^2} = 4\) |
| 15 | \(15 - 11 = 4\) | \({(4)^2} = 16\) |
| 18 | \(18 - 11 = 7\) | \({(7)^2} = 49\) |
| \(\sum {(X - \bar X)^2} = 138\) |
| \(x\) | 4 |
|---|---|
| \(X - \bar X\) | \(4 - 11 = - 7\) |
| \({(X - \bar X)^2}\) | \({( - 7)^2} = 49\) |
| \(x\) | 7 |
|---|---|
| \(X - \bar X\) | \(7 - 11 = - 4\) |
| \({(X - \bar X)^2}\) | \({( - 4)^2} = 16\) |
| \(x\) | 9 |
|---|---|
| \(X - \bar X\) | \(9 - 11 = - 2\) |
| \({(X - \bar X)^2}\) | \({( - 2)^2} = 4\) |
| \(x\) | 11 |
|---|---|
| \(X - \bar X\) | \(11 - 11 = 0\) |
| \({(X - \bar X)^2}\) | \({(0)^2} = 0\) |
| \(x\) | 13 |
|---|---|
| \(X - \bar X\) | \(13 - 11 = 2\) |
| \({(X - \bar X)^2}\) | \({(2)^2} = 4\) |
| \(x\) | 15 |
|---|---|
| \(X - \bar X\) | \(15 - 11 = 4\) |
| \({(X - \bar X)^2}\) | \({(4)^2} = 16\) |
| \(x\) | 18 |
|---|---|
| \(X - \bar X\) | \(18 - 11 = 7\) |
| \({(X - \bar X)^2}\) | \({(7)^2} = 49\) |
| \(x\) | |
|---|---|
| \(X - \bar X\) | |
| \({(X - \bar X)^2}\) | \(\sum {(X - \bar X)^2} = 138\) |
\(SD = \sqrt {\frac{{\sum {{(X - \bar X)}^2}}}{{n - 1}}}\)
\(\text{C}\text{À} = \sqrt {\frac{{138}}{{7 - 1}}}\)
\(\text{C}\text{À} = \sqrt {\frac{{138}}{6}}\)
\(\text{C}\text{À}=\sqrt{23}\)
\(\text{C}\text{À} = 4.796\,(gu\,3\,id)\)
Question
Tha cuideam seachdnar bhoireannach ann an cilegraman gu h-ìosal:
\(52,\,41,\,58,\,63,\,49,\,50,\,72\)
Obraich a-mach an cuibheas agus an claonadh àbhaisteach aig na cuideaman sin.
(b) B' e 60 agus 5.5, san òrdugh sin, an cuibheas agus an claonadh àbhaisteach aig buidheann fhireannach.
Dèan dà abairt a tha a' coimeas buidheann nam fireannach agus buidheann nam boireannach.
(a) \(\bar X = \frac{{52 + 41 + 58 + 63 + 49 + 50 + 72}}{7} = \frac{{385}}{7} = 55\)
| \(x\) | \(X - \bar X\) | \({(X - \bar X)^2}\) |
| 52 | \(52 - 55 = - 3\) | \({( - 3)^2} = 9\) |
| 41 | \(41 - 55 = - 14\) | \({( - 14)^2} = 196\) |
| 58 | \(58 - 55 = 3\) | \({( 3)^2} = 9\) |
| 63 | \(63 - 55 = 8\) | \({(8)^2} = 64\) |
| 49 | \(49 - 55 = - 6\) | \({(- 6)^2} = 36\) |
| 50 | \(50 - 55 = - 5\) | \({(- 5)^2} = 25\) |
| 72 | \(72 - 55 = 17\) | \({(17)^2} = 289\) |
| Iomlan | \(628\) |
| \(x\) | 52 |
|---|---|
| \(X - \bar X\) | \(52 - 55 = - 3\) |
| \({(X - \bar X)^2}\) | \({( - 3)^2} = 9\) |
| \(x\) | 41 |
|---|---|
| \(X - \bar X\) | \(41 - 55 = - 14\) |
| \({(X - \bar X)^2}\) | \({( - 14)^2} = 196\) |
| \(x\) | 58 |
|---|---|
| \(X - \bar X\) | \(58 - 55 = 3\) |
| \({(X - \bar X)^2}\) | \({( 3)^2} = 9\) |
| \(x\) | 63 |
|---|---|
| \(X - \bar X\) | \(63 - 55 = 8\) |
| \({(X - \bar X)^2}\) | \({(8)^2} = 64\) |
| \(x\) | 49 |
|---|---|
| \(X - \bar X\) | \(49 - 55 = - 6\) |
| \({(X - \bar X)^2}\) | \({(- 6)^2} = 36\) |
| \(x\) | 50 |
|---|---|
| \(X - \bar X\) | \(50 - 55 = - 5\) |
| \({(X - \bar X)^2}\) | \({(- 5)^2} = 25\) |
| \(x\) | 72 |
|---|---|
| \(X - \bar X\) | \(72 - 55 = 17\) |
| \({(X - \bar X)^2}\) | \({(17)^2} = 289\) |
| \(x\) | |
|---|---|
| \(X - \bar X\) | Iomlan |
| \({(X - \bar X)^2}\) | \(628\) |
\(SD = \sqrt {\frac{{\sum {{(X - \bar X)}^2}}}{{n - 1}}}\)
\(\text{C}\text{À} = \sqrt {\frac{{628}}{{7 - 1}}}\)
\(\text{C}\text{À} = \sqrt {\frac{{628}}{6}}\)
\(\text{C}\text{À} =\sqrt{104.67}\)
\(\text{C}\text{À} = 10.2\,(gu\,1\,id)\)
(b) Tha an cuibheasAn luach as fheàrr a riochdaicheas seata-dàta. Tha trì seòrsaichean cuibheis ann - an cuibheas, am meadhan agus am mòd. aig na boireannaich nas ìsle na tha e aig na fireannaich oir tha 55 < 60. Tha seo ag innse dhuinn gu bheil cuibheas cuideam nam boireannach nas lugha na na fireannaich.
Tha an claonadh àbhaisteach aig na boireannaich nas motha na tha e airson nam fireannach oir tha 10.2 > 5.5. Tha seo ag innse dhuinn gu bheil barrachd atharrachaidh ann an cuideam airson toraidhean nam boireannach na tha ann an toraidhean nam fireannach.
