ࡱ> hjg'` _bjbj 4d2hhhh$,s"s"s"Q+S+S+S+S+S+S+$,h /lw+E#!|s"##w+C+$$$#"Q+$#Q+$$%'6q' `Bch#.=''\+0,E',/$|/q'/q's"0""$""s"s"s"w+w+$s"s"s",#### Cumulative Frequency Example Here are the marks for 80 students for a recent mathematics exam.  MarksFrequency0 d" m < 10010 d" m < 20320 d" m < 30730 d" m < 403540 d" m < 502150 d" m < 601060 d" m < 704 To look clearly at the totals we create a cumulative frequency table. Instead of the frequency we look at the running totals. MarksFrequency< 100< 203< 3010< 4045< 5066< 6076< 7080 We can plot these cumulative frequencies on a graph.  Here are the heights of 48 students. Height (cm)Frequency140 d" h < 1452145 d" h < 1504150 d" h < 1558155 d" h < 1609160 d" h < 16512165 d" h < 1707170 d" h < 1754175 d" h < 1802 Complete the cumulative frequency table below. Height (cm)Cumulative Frequency< 145 Draw the cumulative frequency curve for these results on the graph paper provided. Using a Cumulative Frequency Curve From our curve we can find useful values MEDIAN - Halfway mark, at this point 50% of the results are above and 50% below LOWER QUARTILE quarter mark, 75% of the results are above and 25% below UPPER QUARTILE three quarter mark, 25% of the results are above and 75% below INTERQUARTILE RANGE the spread of the middle 50%, we work this out by upper quartile lower quartile (*  6 8 T X t x # d e   l ǿ{woghVmH sH hmH sH hh$,jhDh;iUh$,mH sH h[DmH sH jhgQUmHnHujh$,UmHnHuh>*mH sH hmH sH hj mH sH hUmH sH jhj UmHnHuh]mH sH hN4>*mH sH h]>*mH sH %*:_V $Ifgd]zkd$$Ifl0   t0644 la $$Ifa$gdU $$Ifa$gd]gd]$a$gd] F]_   {o $$Ifa$gdU $Ifgd]zkd_$$Ifl0   t0644 la  6 : {o $$Ifa$gdU $Ifgd]zkd$$Ifl0   t0644 la: < T Z {o $$Ifa$gdU $Ifgd]zkd$$Ifl0   t0644 laZ \ t z {o $$Ifa$gdU $Ifgd]zkd|$$Ifl0   t0644 laz | {o $$Ifa$gdU $Ifgd]zkd$$Ifl0   t0644 la {o $$Ifa$gdU $Ifgd]zkd:$$Ifl0   t0644 la ] ^ d o sg $$Ifa$gd* $$Ifa$gdgd]zkd$$Ifl0   t0644 lao p u w xl $$Ifa$gd* $$Ifa$gdzkd$$Ifl0   t0644 law x }  xl $$Ifa$gd* $$Ifa$gdzkdW$$Ifl0   t0644 la xl $$Ifa$gd* $$Ifa$gdzkd$$Ifl0   t0644 la xl $$Ifa$gd* $$Ifa$gdzkd$$Ifl0   t0644 la xl $$Ifa$gd* $$Ifa$gdzkdt$$Ifl0   t0644 la xl $$Ifa$gd* $$Ifa$gdzkd$$Ifl0   t0644 la xl $$Ifa$gd* $$Ifa$gdzkd2$$Ifl0   t0644 la    % zzzzzznb $$Ifa$gd* $$Ifa$gdVgd]gdzkd$$Ifl0   t0644 la % &   zn $$Ifa$gd* $$Ifa$gdVykdT$$Ifl0p t044 la  8 < zn $$Ifa$gd* $$Ifa$gdVykd$$Ifl0p t044 la< > Z ^ zn $$Ifa$gd* $$Ifa$gdVykd $$Ifl0p t044 la^ ` | zn $$Ifa$gd* $$Ifa$gdVykdh$$Ifl0p t044 la zn $$Ifa$gd* $$Ifa$gdVykd$$Ifl0p t044 la zn $$Ifa$gd* $$Ifa$gdVykd $$Ifl0p t044 la zn $$Ifa$gd* $$Ifa$gdVykd|$$Ifl0p t044 la  zn $$Ifa$gd* $$Ifa$gdVykd$$Ifl0p t044 la  l n uu $$Ifa$gd*gd]ykd4$$Ifl0p t044 la zz $$Ifa$gd*ykd$$Ifl0pH t044 la zz $$Ifa$gd*ykd$$Ifl0pH t044 la zz $$Ifa$gd*ykdH$$Ifl0pH t044 la zz $$Ifa$gd*ykd$$Ifl0pH t044 la zz $$Ifa$gd*ykd$$Ifl0pH t044 la zz $$Ifa$gd*ykd\$$Ifl0pH t044 la zz $$Ifa$gd*ykd$$Ifl0pH t044 la zz $$Ifa$gd*ykd$$Ifl0pH t044 la JKno45gd]ykdp$$Ifl0pH t044 lal ,IJKn+-5CY\\\D]F]X^^^^^^ιŰymiahgQmH sH hj hj hj 5mH sH h5mH sH hj 5mH sH hj mH sH hhmH sH hw4mH sH hmH sH Uh5mH sH hJhJ5mH sH hJ5mH sH hJmH sH hW^5mH sH hW^>*mH sH hW^mH sH hzmH sH h+KmH sH "h\j\\\\\B]D]F]]]^^V^X^^^_____$a$gdj gd & FgdgdJWe can also use our curve to work out other totals. The number of students who are taller than 160cm The number of students who are between 145cm and 170cm. How many students got less than 30? 0 + 3 + 7 = 10 How many got more than 50? 10 + 4 = 14 How many got less than 60? 80  4 = 76 Sometimes we are interested in the totals The final cumulative number is the total frequency Notice that the Marks column now only shows the upper class boundary. ^_________ǼhhmH sH h*h Sh$,h$,h$,mH sH h$,mH sH hgQhgQhgQ5mH sH hgQ5mH sH  ___gd21h:p0 / =!"#$% ]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5]$$If!vh5 5#v #v:Vl t65 5d Dd _0  # A" ̛Q W4 4 @= ̛Q W4p3iG5 x[}lo}:9P9&B(h#Yb[%UpaVjPI4R#%JB R&SuFnDRUq{sޭԕ{yo潝[#8bsVZ6 L @5? ^QD_TMjJ%+2:ƷIJPzRd%e*q 8ԚP !S LIyrr*\An),J d'¢~l)Ϧ?H9EM6q"uK+|wϸx0>v::B\Ytxr2oݏ~۞>{\6e{Ug俇|^#4κ?{^}*mkȸ  iV|jn.~Uع_si]>W0~ji*c1g,zcttt>#r/O*W݋{i 5PTO=ԃ kڜǜުueOQٓuyVt%]taFcf   C\3~07 >αl$O݆=zb9߸aql:m>b#nYٸeUoM_=X\J\.Op Ǭxs/_TJ6aܿXG }:h |Wxq ZˆkMF\k츖0Zdǵa&zco:>d>n:uG |ٯu1(܄=ږJn2bu|ҋY15U챦S{qQ%*?K0I( QYɲL;N=N[.=qBYv#^g֎Wa;~m`3B`1 ;mp9&:\AXm9O8zwXfBOx ߃ay?yo%| 6d~5(WZAkVZ/?-@6wc,9X}'ُ;kXn{vךo_4"ƃN^:2HAo ~Dǩ岓5t\D2 P$:֐/!ϭMRntIr(MLF ʯ>ԋQt&)?AH&>MaGvŨ%1:S|_j|%P}WtŨ:ӱ*; :[ONt,ZN =z>)o%հa tB+7Bmr$= A3ҏ CRʹjFZq{#"*q+;Ul\CLԜedH9mAs|70q ),v8G5λ`c{wEjsݣR_L,;\>ɩVďSc^W Iy~ Y1lp*ùX\x .K}~Ã]{eB7)];&c>x=!ƒs=}{Ѕwޕ»ׅ75bwRl=^OyMYs; ~©M?\x/uM 1wNU؍Ħ6ׅ'[lzKr}'[lL鼟uMktߧ\x6T~oi/cz#T9B1c1| ξB2$yYk9/~R}9VС ]_I:ep W會ߍ{0ן:"ƿ:cIGx&4WNn㣇5uh=bnuls;יodtUG|;n:zڢn޹?Ig,\Ŀw: ]ioCiCNVÿzxNgg΢PG ]'ǹwXaYce?W>TeZuNboc.=+Է&!^ΙOfn60u3ML8RkΔ^HK?d$:OZ$$If!vh55#v#v:Vl t55Z$$If!vh55#v#v:Vl t55Z$$If!vh55#v#v:Vl t55Z$$If!vh55#v#v:Vl t55Z$$If!vh55#v#v:Vl t55Z$$If!vh55#v#v:Vl t55Z$$If!vh55#v#v:Vl t55Z$$If!vh55#v#v:Vl t55Z$$If!vh55#v#v:Vl t55Z$$If!vh55 #v#v :Vl t55 Z$$If!vh55 #v#v :Vl t55 Z$$If!vh55 #v#v :Vl t55 Z$$If!vh55 #v#v :Vl t55 Z$$If!vh55 #v#v :Vl t55 Z$$If!vh55 #v#v :Vl t55 Z$$If!vh55 #v#v :Vl t55 Z$$If!vh55 #v#v :Vl t55 Z$$If!vh55 #v#v :Vl t55 @@@ NormalCJ_HaJmH sH tH DAD Default Paragraph FontRi@R  Table Normal4 l4a (k(No Listj@j ] Table Grid7:V001 014d_agqr}]^dopuwx}%&467EGHVXYgijx{|      st]^ TU!"KLwx00000 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@0@00x 0x 0| @0 @0 @0 @0 @0 @0 @0 0 @0 @0 0 @0 @0 0 @0 @0x @0x @0| @0x @0x @0| @00000000 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 0 0000 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 @0 @0000000000000000000 0000 00000000000H0000%&67GHXYij{|    K00cI0I0I0K0/@ @ K0/@ @K0/@ K0/@  K0 /@ K0  /@K0d,K0  @10DK0 I0 @ K0/I0 K0/I0 K0/I0 K0/I0 K0 /I0  K0  /I0 K0,K0  I0 K0 K0 @ K0 K0 @] K00l ^_,0 : Z z o w  %  < ^ __  !"#$%&'()*+-1_ 8@(  \  3 )"` \  3 "` NB  @ S DB     B S  ?_d% t $ Ctpt /tyA, zA<<{A,&9*urn:schemas-microsoft-com:office:smarttagsplace=*urn:schemas-microsoft-com:office:smarttags PlaceName=*urn:schemas-microsoft-com:office:smarttags PlaceType 33a^t :h\2^`o() ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.:h         +K0 j N4$,Z ;[D S]W^;izL $hIkD5U*'"JgQw4Vagqr}^dopuwx}%&467EGHVXYgijx{|     """@\70@)) PP@P P@P\UnknownGz Times New Roman5Symbol3& z Arial"1h+f+f  Y42QHX ?]2Cumulative Frequencyghentypc user Oh+'0|  8 D P\dltCumulative FrequencyghentyNormalpc user2Microsoft Office Word@F#@K@c@K@c՜.+,0 hp   hone ' Cumulative Frequency Title  !"#$%&'()*+,-./012456789:;<=>@ABCDEFGHIJKLMNOPQRSTUVXYZ[\]^`abcdefiRoot Entry F@BckData 31Table?/WordDocument4dSummaryInformation(WDocumentSummaryInformation8_CompObjq  FMicrosoft Office Word Document MSWordDocWord.Document.89q