ࡱ> X<YOh+'0 , H T `lt|%s2 = 1/(n – 1) ∑i (xi – Xbar)2 2 =john1/(ohnohn Normal.dotjohnl.d6hnMicrosoft Word 8.0@#@"zt@CG23vidc.M263vidc.M261msacm.msa՜.+,D՜.+,T hp  john(1 %s2 = 1/(n – 1) ∑i (xi – Xbar)2 Title 6> _PID_GUIDAN{F8CE89DF-E653-44EA-BE45-8F17042D05A8}sacm.msas2 = 1/(n  1) "i (xi  Xbar)2 Xbar = 1/n (x1 + x2 + ... + xn) s2 = 1/n "i (xi  x1/n  x2/n  ...  xn /n)2 s2 = 1/n "i 1/n (n xi  x1  x2  ... xn)2 s2 = 1/n "i 1/n [(xi  x1) + (xi  x2) + ... + (xi  xn)]2 s2 = 1/[n(n  1)] "i [(xi  x1) + (xi  x2) + ... (xi  xi) + ... + (xi  xn)]2 s2 = 1/[n(n  1)] "i "j (xi  xj)2  "&(*:<RTXZ\bdfxz| 248:<FHJLNTV6H* 6CJH*6CJEHH*CJH*6[LNRTVdfhjxz|~tv"(>@FLbdjVXv~2BFnprv6CJ>*6H*CJ5CJ 6H*6CJEHH*CJX>@68PR<F&(06&Z\z8HLtvx|0 2 4 8 L N P ` b h j r t !!!!!"&& & 6B*H*6CJ 56CJ 5CJ H*>*6H*6CJ>Why We Use n 1 to Compute Sample Variance Sample variance = (1) (2) (3) Substitue (3) for Xbar in (1): (4) Multiply times n/n: (5) Expand : (6) i ... + (7) BudhVX|~24NPxZ\$<F&(06&Z\z8HLtvx|0 2 4 8 L N P ` b h j r t !!!!!"&& &( 6B*H*6CJ 56CJ 5CJ H*>*6H*6CJ?Why We Use n 1 to Calculate Sample Variance Sample variance = (1) (2) (3) Substitute (3) for Xbar in (1): (4) Multiply times n/n: (5) Expand : (6) But one of the differences in this series is i: i ... + (7) And i = 0 of necessity. It can never contribute to an estimate of variability among observations. Also note from above we get this result: j (8)showing that sample variance can alternatively be calculated based on a sum of squared distances between all data points.  [(@(NormalCJmH <A@<Default Paragraph Font./ABqr=>RS&'xyCD"d! UnknownjohnADt i = 0 of necessity. It can never contribute to an estimate of variablity among observations. Also note from above we get this result: j (8)showing that sample variance can alternatively be calculted based on a sum of squared distances between all data points.8:TV~8 j 4!8!! [(@(NormalCJmH <A@<Default Paragraph Font{,-?@op:;OP78}&d Unknownjohn johnEC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of Document1.asdjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnOC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of sample variance n-1.asdjohnOC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of sample variance n-1.asdjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.doc@5]C$Eƀ56CJ b'C$Eƀb'C$Eƀ6b' #$+,-<>?@DEGefjkmnopyz{46789:JKLMNOPUVZ_imno  &(*-/2345678<=?BCDEFJV`ez{qqqqq(q6q8qDqFpTpVqXqvpzp|q~qqqqqqqqp<pqqqq@qqqqpq@qVqPq\qPqfqPq|qqqqpp0p2q4q>qqqqBqFqHqJpLpNqnqpqrqtppvqxqqqqqqqqq qqq0qqqqpqqqp4pqqqqq<qqHqXqq`qqjqqxqqqqqqqqqqpqqqqq<qqVqq`qqjqqrqqxqqqqqqqqqqqqqqqppqqrqqxqqqqqqq(q<qFqpqppPqq"qRqq`qq&qq(q0q2q4ppq6pGzTimes New Roman5Symbol3& zArial"1h='4=')! 4zs2 = 1/(n  1) "i (xi  Xbar)2johnjohn [(@(NormalCJmH <A@<Default Paragraph Font8Y@8 Document Map-D OJQJ ./ABrsABVW,-~IJ!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!./ABrsABVW,-~IJ////////////////////////////// &d! Unknownjohnmn<=GJBImns~<=WbGJjohnEC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of Document1.asdjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnOC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of sample variance n-1.asdjohnOC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of sample variance n-1.asdjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnEF:\shuttle\Lecture 09\Why We Use n-1 to Calculate Sample Variance.doc@BIr}S^ADjohnEC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of Document1.asdjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnOC:\DOCUME~1\ADMINI~1\LOCAL>@68PRtv*Nld$/ =!"#$%Why We Use n 1 to Compute Sample Variance Sample variance = (1) (2) (3) (4) (5) (6) i ... + (7) But i = 0 of necessity. It can never contribute to an estimate of variablity among observations. Also note from above we get this result: j (8)showing that sample variance can alternatively be calculted based on a sum of squared distances between all data points.S~1\Temp\AutoRecovery save of sample variance n-1.asdjohnOC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of sample variance n-1.asdjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.doc@5]C$Eƀ56CJ b'C$Eƀ**\\**@&C$Eƀ56CJ b'C$Eƀb'C$Eƀ6b'C$EƀB*b'C$EƀB*b' %&-./>@ABFGIghlmnpqrs|}~:<=>?@AQRSTUVW\]afptuv !#&'()*+,-1347:FGIJLNQSW[\^abcdefmoqtvyz{|}~8:=>?AEFGHIJ|111111&181:1<1H1J0X0Z1\1z0~011111111&110<01111@11&11101@1V1P1\1P1f1P1|11&111011 06081:1D1111H1&1L1N1P0R0T1t1v1x1z00|1~111111111 111011&111011104011111<11H1X11`11j11x11111111 &11101. 1r11x110 112 14 06 018 1@ 1D 1F 1<11V1J1L 1X11`11j11r 1 1x 1 1N 1 1P 11R 111111` 1b 1d 1f 00h 1j 1r11x11r 11t 1v 1~ 1 1 1 1 1!1.!02!14!06!0P1!1!1R11`11!11!1!1!1!001!1 "1"1"0GzTimes New Roman5Symbol3& zArial5& zaTahoma"0h='sBG2!4ds2 = 1/(n  1) "i (xi  Xbar)2johnjohnq Dbjbjt+t+ (AA]k;;;;;kYYYYu/!!!!!!!$0w2v!"v ;!"b'C$Eƀ6b'C$EƀB*b'C$EƀB*b' %&-./>@ABFGIghlmopqr{|} 79:;<=MNOPQRSXY]blpqr !"#$%&'+-.14@ACDFHKMQUVX[\]^_`giknpstuvwxy}~24789;?@ABCDvqqqqqq&q8q:q<qHqJpXpZq\qzp~pqqqqqqqqqp<pqqqq@qqqqpq@qVqPq\qPqfqPq|qqqqpqq p6p8q:qDqqqqHqLqNqPpRpTqtqvqxqzpp|q~qqqqqqqqq qqq0qqqqpqqqp4pqqqqq<qqHqXqq`qqjqqxqqqqqqqqqqpq. qrqqxqq0 qq2 q4 p6 p q8 q@ qD qF q<qqVqJqL qXqq`qqjqqr q qx q qN q qP qqR qqqqqq` qb qd qf pph qj qrqqxqqr qqt qv q~ q q q q q!q.!p2!q4!p6!pPq!q!qRqq`qq!qq!q!q!q!ppq!q "q"q"pGzTimes New Roman5Symbol3& zArial"1h='@='1!4ds2 = 1/(n  1) "i (xi  Xbar)2johnjohn4ds2 = 1/(n  1) "i (xi  Xbar)2johnjohn [(@(NormalCJmH <A@<Default Paragraph Font8Y@8 Document Map-D OJQJ./ABrsABVW,-~JK//////////////////////////////(d! Unknownjohnmn<=GJEFBImns~<=WbGJEFjohnEC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of Document1.asdjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variancDocumentSummaryInformation8CompObjjObjectPoolg=g=0Tableqe n-1.docjohnOC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of sample variance n-1.asdjohnOC:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\AutoRecovery save of sample variance n-1.asdjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnLC:\statistics\Statistics for Engineers\Lecture notes\sample variance n-1.docjohnEF:\shuttle\Lecture 09\Why We Use n-1 to Calculate Sample Variance.docjohnTC:\__website\dated_backup\2013_10_08\Why We Use n-1 to Calculate Sample Variance.doc@FF\\FE@&C$Eƀ56CJ b'C$Eƀ;; v;;!O];;;;!!;;! ^FY3!Root Entry FϪ=@FK@1Table2WordDocument6(SummaryInformation( ()  ' !"#$%&-+* ,7./012345H89:;=>?DLM  FMicrosoft Word Document MSWordDocWord.Document.89q՜.+,D՜.+,T hp  john(1 %s2 = 1/(n – 1) ∑i (xi – Xbar)2 Title 6> _PID_GUIDAN{F8CE89DF-E653-44EA-BE45-8F17042D05A8}Oh+'0 , H T `lt|%s2 = 1/(n – 1) ∑i (xi – Xbar)2 2 =john1/(ohnohn Normal.dotjohnl.d5hnMicrosoft Word 8.0@#@"zt@rFb'C$Eƀ6b'C$EƀB*b'C$EƀB*b' %&-./>@ABFGIghlmnpqrs|}~:<=>?@AQRSTUVW\]afptuv !#&'()*+,-1347:FGIJLNQSW[\^abcdefmoqtvyz{|}~8:=>?AEFGHIJK}111111&181:1<1H1J0X0Z1\1z0~011111111&110<01111@11&11101@1V1P1\1P1f1P1|11&111011 06081:1D1111H1&1L1N1P0R0T1t1v1x1z00|1~111111111 111011&111011104011111<11H1X11`11j11x11111111 &11101. 1r11x110 112 14 06 018 1@ 1D 1F 1<11V1J1L 1X11`11j11r 1 1x 1 1N 1 1P 11R 111111` 1b 1d 1f 00h 1j 1r11x11r 11t 1v 1~ 1 1 1 1 1!1.!02!14!06!0P1!1!1R11`11!11!1(1!1!1!001!1 "1"1"0GzTimes New Roman5Symbol3& zArial5& zaTahoma"1h='xBG2!4dEs2 = 1/(n  1) "i (xi  Xbar)2johnjohnq Tbjbjt+t+ *AA]qAAAAAU////KUE2$$$$$$$$13%5v$ A$AAs AA$UUAAAA$$AA$g (WGU/$Root Entry FϪ=(WG]@1Table2WordDocumentV*SummaryInformation( () '  !"#$%&-+* ,7./01234589:;=>@ABCEFGNOPQRSTUWZ\[DocumentSummaryInformation8CompObjjObjectPoolg=g=0Table5_PID_GUIDAN{F8CE89DF-E653-44EA-BE45-8F17042D05A8}Oh+'0 , H T `lt|%s2 = 1/(n – 1) ∑i (xi – Xbar)2 2 =john1/(ohnohn Normal.dotjohnl.d6hnMicrosoft Word 8.0@#@"zt@CG  FMicrosoft Word Document MSWordDocWord.Document.89q՜.+,D՜.+,T hp  john(1 %s2 = 1/(n – 1) ∑i (xi – Xbar)2 Title 6>