ࡱ> RTQ @ EbjbjVV (Hr<r<*,$8|, Zfhhhhhh$RW9"ff `hjBp <Rr0 ;.;,,0;|,, d ,,  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/Standard_deviation.wikipedia" Standard deviation is the measure of the spread of a series from its  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean value. It indicates the differences and variability which a set of numbers have.  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/Standard_deviation.wikipedia" Standard deviation is mathematically the square root of  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/variance.wikipedia" variance which is defined as the  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean of the square of the differences between the elements and their  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean: Suppose that you are in some course and have just received your grade on an exam. It is natural to ask how the rest of the class did on the exam so that you can put your grade in some context. Knowing the mean or median tells you the "center" or "middle" of the grades, but it would also be helpful to know some measure of the spread or variation in the grades. Lets look at a small example. Suppose three classes of 5 students each write the same exam and the grades are: Class 1Class 2Class 3828267788266708266584266424265Each of these classes has a mean (average) of 66 and yet there is great difference in the variation of the grades in each class. One measure of the variation is the range, which is the difference between the highest and lowest grades. In this example the range for the first two classes is 82 - 42 = 40 while the range for the third class is 67 - 65 = 2. The range is not a very good measure of variation here as classes 1 and 2 have the same range yet their variation seems to be quite different. One way to see this variation is to notice that in class 3 all the grades are very close to the mean, in class 1 some of the grades are close to the mean and some are far away and in class 2 all of the grades are a long way from the mean. It is this concept that leads to the definition of the standard deviation. The steps to calculate the standard deviation are: 1. Calculate the  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean of the series. 2. Calculate the differences between the elements and the  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean:  INCLUDEPICTURE "http://formula.algebra.com/cgi-bin/plot-formula.mpl?expression=x%5Bi%5D-mean&x=0003" \* MERGEFORMATINET for all elements. 3. Calculate the squares of the differences. 4. Calculate the  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean of the squares by adding all the squares and dividing by the number of elements. 5. Calculate the square root of the the  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean. This is the  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/Standard_deviation.wikipedia" standard deviation. Example: Calculate the standard deviation of the series {1,3,12,5,9}. Solution: The standard deviation can be calculated as follows: 1.  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" Mean of the series = (1+3+12+5+9)/5 = 30/5 = 6 2. The differences between the individual elements and  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean would be: {1-6, 3-6, 12-6, 5-6, 9-6} = {-5,-3,6,-1,3} 3. The squares of the differences would be {25,9,36,1,9}. 4. The  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean of the squares would be (25+9+36+1+9)/5 = 80/5 = 16 5. The square root of the  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson" mean of squares would be  INCLUDEPICTURE "http://formula.algebra.com/cgi-bin/plot-formula.mpl?expression=sqrt%2816%29&x=0003" \* MERGEFORMATINET = 4 Thus the  HYPERLINK "http://www.algebra.com/algebra/homework/Probability-and-statistics/Standard_deviation.wikipedia" standard deviation would be 4. For class 1 this gives  INCLUDEPICTURE "http://mathcentral.uregina.ca/rr/database/rr.09.95/weston25.gif" \* MERGEFORMATINET  Class 1 Test scoresDifference From Test Score to Mean(Difference From Test Score to Mean)2Sum of squared differencesMean of squared differencesStandard Deviation8216 (16)2 = 256256 + 144 + 16+ 64 + 576 = 1056Mean of squares = 1056 4 = 264Sd = "264 = 16.278121447041658-864A similar calculation gives a standard deviation of 21.9 for class 2 and 0.7 for class 3. So for class 3, where the grades are all close to the mean, the standard deviation is quite small, for class 1, where the grades are spread out betweeno    ! $ l m   x y ³³vfZZPZvvh0JOJQJjhOJQJUht10J>*B*OJQJphht1hOJQJjht1hOJQJUht1OJQJhOJQJ$hh0J>*B*OJQJphjhhOJQJUhhOJQJ*hh0J5>*B*OJQJ\phhh5OJQJ jhh5OJQJU  Hkd$$IfTFm 06    34ab pT $$Ifa$gdgdE ? @ %/?@˵˦˦˦˦˦˛ˏˏևssisht10JOJQJjht1OJQJUhHOJQJht1OJQJht1h5OJQJht1ht1OJQJht1hCJOJQJaJht1h5OJQJ\hFOJQJht1hOJQJhOJQJjht1hOJQJU$ht1h0J>*B*OJQJph) Tkd$$IfTFm 06    34ab pT $$Ifa$ ]TTT $$Ifa$kd$$IfTFm 06    34ab pT ]TTT $$Ifa$kd$$IfTFm 06    34ab pT ]TTT $$Ifa$kdt$$IfTFm 06    34ab pT ]]XXXXSSSgdt1gdkdQ$$IfTFm 06    34ab pTIJNOQR{|XY]^lm7@yz>?vw{|()-.CD;<NOYjm ht1OJQJUht16OJQJ]j.ht1OJQJUht10JOJQJht1OJQJjht1OJQJUFYZ^uvtv$(  %&DEEӸ⸇zhHht1OJQJUhHhHOJQJht1ht1>*OJQJht1ht1H*OJQJht1ht15H*OJQJht1ht15OJQJjht1ht1OJQJUjht1ht1OJQJUht1ht1OJQJht1OJQJht15OJQJ\]^,x  $$Ifa$gdt1gdt1 &dPgdt1 48,,, $$Ifa$gdt1kdH$$Iflֈ >N,"}q- t0644 la4t $Ifgdt1 $$Ifa$gdt1 7+++ $$Ifa$gdt1kd$$Ifl4ֈ >N,"}q```- t0644 la     +kd$$Ifl4ֈ >N,"}q   - t0644 la $$Ifa$gdt1  $$Ifa$gdt1"7+++ $$Ifa$gdt1kd$$Ifl4ֈ >N,"}q   - t0644 la"#$%&+kdl$$Ifl4ֈ >N,"}q   - t0644 la $$Ifa$gdt1n 42 and 82, the standard deviation is considerably larger and for class 2, where all the grades are far from the mean, the standard deviation is larger still. The standard deviation is the quantity most commonly used by statisticians to measure the variation in a data set. &Egd&1h:pH/ =!"#$%$$If!vh555#v#v#v:V 06,5/ 34pT$$If!vh555#v#v#v:V 06,5/ 34pT$$If!vh555#v#v#v:V 06,5/ 34pT$$If!vh555#v#v#v:V 06,5/ 34pT$$If!vh555#v#v#v:V 06,5/ 34pT$$If!vh555#v#v#v:V 06,5/ 34pT?Ddhx  S TAplot-formulax%5Bi%5D-meanRsv\}q,XKOr20FGv\}q,XKJFIF;CREATOR: gd-jpeg v1.0 (using IJG JPEG v62), quality = 80 C   %# , #&')*)-0-(0%()(C   (((((((((((((((((((((((((((((((((((((((((((((((((((L" }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz w!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?յX44I|˥6DPIC2\FI-K5WUR2 zauТ ;Y& hY\*F2Y$-7Sv iq.# U"+|; oZ\\iZiɨ>&$Q p. \4@tFQn*Uu#jۧE# \00B A .,..E2Xko"PB,"IBR#QUֹh$𮡣ioKYծ1THoI#<tU].;"MVg̖ݠ83`} 8OuM:SlK=g놛v7qnۏr+8 3ᵍSmk2EW;Kuc呾_*#u_ԮdW}NS塹Y .R#V@"?Ne}c_oQxݟں(XDdoLv  S RAplot-formulasqrt%2816%29Rmb?Q}1]gj 20Fbmb?Q}1]gJFIF;CREATOR: gd-jpeg v1.0 (using IJG JPEG v62), quality = 80 C   %# , #&')*)-0-(0%()(C   ((((((((((((((((((((((((((((((((((((((((((((((((((( " }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz w!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?'[]iEszeƪ׻U@%\)2nQ"lH"x2:,*ԼAֶ޴&Pn"gۓ#mx$:uHf׆hJv^(BcCF:z}Η^\E\ZkZa_)`P [T5_Mw+$(-pp DZrjڌ,w2KPc,cK1f<֡wORt_x}g=uݲ<]WH<+k>Glwvqg>F 2II-&HYf\Gs[Z4?y6ڤ^n۸g8E?jV^[XkŔr%` (¨A[TQ@كDdn}V  S 2Aweston25b:=TLy>' 20n:=TLy>'PNG  IHDR$:PLTEf3̙f3f3ffffff3f3333f333f3f3̙f3̙̙̙̙f̙3̙ffffff3f3333f333f3̙f3̙̙f3̙f3ff̙ffff3f33̙33f333̙f3ffffff3ffff̙fff3fffffff3ffffffffffff3fff3f3f3f3ff33f3ffffff3f3333f333333̙3f3333333f3333f3f3f3ff3f33f33333333f333333333f333f3̙f3f3ffffff3f3333f333f3wUD"wUD"wUD"ݻwwwUUUDDD"""tRNS@fbKGDH cmPPJCmp0712HsIDATXGX۲ tEQ"fiz²,` ޥz]?1cWuQ'.Rf꺿&$ F: K_ρ(2y3c@WCAZN^ mgX$ydY=Dv=.,&3M2֗d;\$u\E($ :sU`^'J7f(Z;Tv$ 9!E3 B`ِ>h72pzh4 lֺFU=^ ͟'tFuw>b^l}Q}|<.|Ғ%_הgG3-|Ixܚw!FO%\0{S(uGHm0^Oüy}*'Ϟ' Ŵg*phB^`B  Normal (Web)dd[$\$j@j 2 Table Grid7:V0* H  ]^<Ws     "#$%&,00000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00000000 0 0 0 0 0 0 0 0 0 0 000 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ^< "#$%&,n;0n;0n;0T.n{0(n{0(n{0f{0 (f{0 (n8>-D.\.(f{0 (f{0 T.f{0 (f{0 (f{0 (n{0(@R(n{0(n{0T.Rn{0T.Rn{0( n{0( n{0 (f{0(f{0(d{0(l;0, n;0T. YE ] 4  "&E !#Enlx?? I N Q  { X ] l y >v{(-C;Nu*XXXXXXXXCXXXXXXXCXC ,, - + 4 t u ,333333)),,T1(R(R? T1YX8?1\Gf wq * 1Z \ik,>f\,&J '(f-t1Uf;}=Q>P@Q@F$HP MjNS%V` X*Z$yZ[{\Uaya e0_eT"fh~OiMIrAvpzN{~w{Uc<$Y}H4u2pz/OY.3c?c}2A+Qt 9TJy:m79Z. 8HQ,hS<Ws     "#$%&,@)) )?),"4*@8@@UnknownG: Times New Roman5Symbol3& : ArialCFComic Sans MS"qhscGcG%( &( &!24  3HP)?9OStandard deviation is the measure of the spread of a series from its mean valuejarcurijarcuriOh+'0 $4 DP l x PStandard deviation is the measure of the spread of a series from its mean valuejarcuriNormaljarcuri2Microsoft Word 10.0@9+@ k @6p (՜.+,D՜.+,< hp  SD67&   PStandard deviation is the measure of the spread of a series from its mean value Title 8@ _PID_HLINKSA `= 3`http://www.algebra.com/algebra/homework/Probability-and-statistics/Standard_deviation.wikipedia%=-Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson%=*Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson%='Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson%=$Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson= !`http://www.algebra.com/algebra/homework/Probability-and-statistics/Standard_deviation.wikipedia%=Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson%=Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson%=Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson%=Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson%=Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson%= Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lessonUC Vhttp://www.algebra.com/algebra/homework/Probability-and-statistics/variance.wikipedia= `http://www.algebra.com/algebra/homework/Probability-and-statistics/Standard_deviation.wikipedia%=Ohttp://www.algebra.com/algebra/homework/Probability-and-statistics/mean.lesson= `http://www.algebra.com/algebra/homework/Probability-and-statistics/Standard_deviation.wikipedia  !"#$&'()*+,-./012456789:;<=>?@BCDEFGHJKLMNOPSRoot Entry F`hjBp UData %;1Table3;WordDocument(HSummaryInformation(ADocumentSummaryInformation8ICompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q