ࡱ> @ RrbjbjFF ,,.c,,,,,8 ,4Ħ(llle3g3g3g3g3g3g3$5R 839Vh993,,BYx49,~ (e39e32 n"ܥ @p);a".#404"R&9&9\",,,,,,&9"l0"_lll33,,М,,М Unit 20 : Measures of Central Tendency and Dispersion Learning Objectives The students should be able to: determine the mean, median and mode from ungrouped data determine the mean, median and mode from grouped data determine the range, inter quartile range and standard deviation. Activities Teacher demonstration and student hand-on exercise. Use MS Excel spreadsheet, internal functions and data analysis to measure central tendency and dispersion. Reference Suen, S.N. (1998) Mathematics for Hong Kong 5A; Rev. Ed.; Canotta Measures of Central Tendency and Dispersion 1. Measure of central tendency: mean, median and mode from grouped and ungrouped data For a set of data, we determine a quantity used to summarise the whole set of data. This quantity is termed a measure of central tendency. The most commonly used measures are mean, medium and mode. 1.1 mean For ungrouped data,  EMBED Equation.3  Example 1 Find the mean for the set data: 3, 7, 2, 1, 7 Solution  EMBED Equation.3  = For grouped data,  EMBED Equation.3  Example 2 a) Find the mean of the set of data: 25, 36, 42, 38, 36 Find the mean from the set of grouped data Class mark10.530.550.570.590.5110.5Frequency1963212 Solution a) mean = b) xfx f10.519199.530.5650.5370.5290.51110.52sum33 mean = Example 3 The HK Consumer Price Index B from 1997 to 2003 was as following: 1996 99.7 1997 105.5 1998 108.5 1999 103.4 99.4 97.7 94.7 92.1 Calculate the average consumer price index B: a) For the first 4 years, (1997 2000). b) For the next 3 years, (2001 2003) c) For all 7 years d) Suppose the original data was lost, and only the 4- and 3-year averages in a) and b) were available. Would it still be possible to calculate the overall 7-year average? How? Solution a) From 1997 - 2000, n = 4. The average price index = (105.5 + 108.5 + 103.4 + 99.4) ( 4 = b) From 2001- 2003, n = 3. The average price index = (97.7 + 94.7 + 92.1) ( 3 = c) From 1997 - 2003, n = 7. The average price index = (105.5 + 108.5 + 103.4 + 99.4 + 97.7 + 94.7 + 92.1) ( 7 = d) The average price index over 7 years = ( ( 4 + ( 3 ) ( (4 + 3) = 1.2 Median For ungrouped data, Median = the middle datum, when n is odd. Median = the mean of the two middle data, when n is even. e.g.1 For the set of data  2, 4, 7, 9, 21 middle datum median = 7e.g.2 For the set of data 3, 5, 7, 7  middle of two data median = (5 + 7) ( 2 = 6 For grouped data, Step 1: Draw the cumulative frequency polygon. Step 2: The median is the datum corresponding to the middle value of the cumulative frequency. Example 4 Find the median of 2, 3, 10, 12, 999. Find the median of 2, 3, 10, 12, 22, 123. The cumulative frequency polygon for maths marks of a class is given below, find the median mark.  Solution Median = Median = Total frequency = 40 The rank of median = 40/2 = From the cumulative polygon, median = Example 5 The provisional figures on the population by age group in Hong Kong as at 9/2001 are tabulated below. Draw a cumulative frequency polygon and determine the median age for the population. Age group0 - 910 - 1920 - 29 30 - 3940 - 4950 - 5960 - 69> 70Population ( 000)676885100012671208677503499  Solution Age group xPopulation ('000)cumulative population ( 000 000)x < 106760.67610 f" x < 208851.56120 f" x < 30100030 f" x < 40126740 f" x < 50120850 f" x < 6067760 f" x < 7050370 < x 499 The rank of median = 6.715/2 = The median age = 1.3 mode For ungrouped data, mode is the datum that has the highest frequency. For grouped data, modal class is the class that has the highest frequency. Example 6 Find the mode of the data: 1, 2, 2, 2, 3, 3, 9 Find the modal class Class10 - 1415 - 1920 - 2425 - 29Frequency2873 Solution The mode is The modal class is Example 7 The temperature in degree Celsius each day cover a three week period were follow: 17, 18, 20, 21, 19, 16, 15, 18, 20, 21, 21,,22, 21, 19, 20,19, 17,16,16,17. Compute the mean, median, and mode of these raw dates by using two-degree intervals starting with 15-16. Draw a cumulative frequency polygon. Solution Temperature (!)TallyFrequency fClass mark xf x15 - 1617 - 1819 - 2021 - 2223 - 24Sum Temperature (!)cumulative frquency < 14.5< 16.5< 18.5< 20.5< 22.5< 24.5 The mean temperature = The modal class of temperature is The median temperature is Remark Mean seems to be the most commonly used (and often misused) quantity for measuring central tendency. If the distribution of the data set shows a strong degree of skewness, then mean is not a reliable measure as it is strongly affected by the extreme values. In this case, medium may be a better choice. Mode is used when there is reason to choose the most commonly occurring data value as the representative for the whole data set. Measure of dispersion: Range, Inter-quartile range and Standard deviation Apart from using a measure of central tendency to summarise a set of data, we need a quantity to measure the degree of dispersion of the set of data (so that we can determine the reliability of the set of data). Range is a measure that is very simple to use but it provides relatively little information on dispersion. Quartile deviation is used in association with the median whereas standard deviation goes with the mean. 2.1 Range For ungrouped data, the range is the difference between the largest datum and the smallest datum. For grouped data, the range is the difference between the highest class boundary and the lowest boundary. Example 8 a) Find the range of the data: 1, 2, 2, 2, 3, 3, 9 b) Find the range of the grouped data Class10 - 1415 - 1920 - 2425 - 29Frequency2873 Solution a) The range = b) The range = 2.2 Inter quartile range Inter quartile range = Q3 Q1 where Q1, Q2, Q3 are called quartiles which divide the data (which have been ranked, i.e. arranged in order) into four equal parts. Moreover, Q2 is the median of the whole set of data, Q1 is the median of the lower half, Q3 is the median of the upper half. Quartile deviation, Q.D. = (Q3 - Q1) Example 9 Find the inter quartile range of 1, 2, 3, 5, 11, 12, 13. 1, 2, 3, 4, 11, 12, 13, 14. Solution inter-quartile range = inter-quartile range = Example 10 The following frequency distribution gives the life hours of a sample of 50 light bulbs: Life hours (000)FrequencyLife hours Up to (000)Cumulative frequency0.60.6 to under 0.720.70.7 to under 0.840.80.8 to under 0.960.90.9 to under 1.0141.01.0 to under 1.1131.11.1 to under 1.271.21.2 to under 1.341.3 Find the median and the inter-quartile range of the data.  The rank of median = 50 = The median of life hours is hrs. The rank of upper quartile = 50 = = 38 , to the nearest integer The upper quartile Q3 is hrs The rank of lower quartile = 50 = = 13 , to the nearest integer The lower quartile Q1 is hrs. The inter-quartile range = Q3 - Q1 = Quartile deviation = (Q3 - Q1) = Example 11 Find the range, inter-quartile range and quartile deviation for the data in example 4 and example 7 respectively. 2.3 Standard deviation For ungrouped data x1, x2,,xn, with a mean  EMBED Equation.3 , the standard deviation (() is  For grouped data with class marks x1, x2,,xn; corresponding frequencies f1,f2,,fn, and a mean  EMBED Equation.3 , the standard deviation (() is  Example 12 Find the standard deviation for the ungrouped data 8, 9, 10, 10, 11 the grouped data x17222732374247f2478742 Solution a) mean  EMBED Equation.3  = standard deviation  EMBED Equation.3  = Calculator key-in method: Model 3600Model 3900Model 506RSet Statistic modeMODE 3MODE 22ndF MODE 3 0 Clear memoryKACKAC2ndF CAKey-in data8 DATA 9 DATA 10 DATA 10 DATA 11 DATA8 DATA 9 DATA 10 DATA 10 DATA 11 DATA8 DATA 9 DATA 10 DATA 10 DATA 11 DATAmeanSHIFT 1SHIFT 42ndF 4s.d.SHIFT 2SHIFT 52ndF 6b) mean  EMBED Equation.3  = standard deviation  EMBED Equation.3  = Calculator key-in method: Model 3600Model 3900Model 506RSet Statistic modeMODE 3MODE 22ndF MODE 3 0 Clear memoryKACKAC2ndF CAKey-in data17 ( 2 DATA 22 ( 4 DATA 27 ( 7 DATA 32 ( 8 DATA 37 ( 7 DATA 32 ( 4 DATA 47 ( 2 DATA17 ( 2 DATA 22 ( 4 DATA 27 ( 7 DATA 32 ( 8 DATA 37 ( 7 DATA 32 ( 4 DATA 47 ( 2 DATA17 2 DATA 22 4 DATA 27 7 DATA 32 8 DATA 37 7 DATA 42 4 DATA 47 2 DATAmeanSHIFT 1SHIFT 42ndF 4s.d.SHIFT 2SHIFT 52ndF 6 Example 13 The life hours of 50 light bulbs has the following frequency distribution. Complete the table with class marks. Calculate the mean and standard deviation. Life hours (000)Class markFrequency0.6 to under 0.70.6520.7 to under 0.80.7540.8 to under 0.90.8560.9 to under 1.00.95141.0 to under 1.11.05131.1 to under 1.21.1571.2 to under 1.31.254 Solution mean  EMBED Equation.3  = standard deviation  EMBED Equation.3  = Example 14 The height of Basil team members at the 2002 FIFA World Cup is listed as following: Marcos 1.93 Cafu 1.76 Lucio 1.88 Roque Junior 1.86 Edmilson 1.85 Carlos 1.68 Richardino 1.76 Silva 1.85 Ronaldo 1.83 Rivaldo 1.86 Ronaldinho 1.80 Calculate the average height, and the standard deviation: Solution Example 15 Find the mean and standard deviation for the data given below: Age group xPopulation ('000)567615885251000351267451208556776550375499 Solution Practice 1. The Hong Kong unemployment rate in the year of 4/2003 4/2004 was as following: 5/2003 8.3 6/2003 8.68$ P Q x y     İ럕{saQ{HhmHnHujhEHUmHnHu#jdK= hUVmHnHo(uhB*phjhUmHnHuh>*B*CJphhB*CJph h>*jh5EHU&jK= h5UVmHnHo(u h5jh5U hCJ h5CJh5CJ mH sH hh} mH o(sH hmH sH 789MNOPpqr$ % & 1 2 f g # %%%%%%%%%%%%%%%%%%%%%%%%%% & F$a$89r# P Q o x   % = > %%%%%%%%%%%%%%%%%%% x1$7$8$H$^ WDX`gd`*` 1$7$8$H$`` ^` >^`>$a$   $ % & 9 : ; < = ? J    $ % - . 6 7 ? @ H I R S [ \ ˻ݫ{f{R{R{R{R{R{R{R&hB*CJOJQJhnH phtH )hB*CJH*OJQJhnH phtH hB*CJhnH phtH !h6B*CJhnH phtH  h>*hB*CJphh>*B*CJmH phsH jhEHUmHnHu#jԟK= hUVmHnHo(u h>*jhUmHnHu hCJh h`*o( > ? I J %%%:%%%XXXXX $$Ifa$$If 1$7$8$H$^ & F1$7$8$H$ 1$7$8$H$` x1$7$8$H$ ,&X $$Ifa$$Ifkd$$IfTl   ֞L|   0 0 0 0 0  t0  4 laT XXXX $$Ifa$ *(%&%&%kd$$IfTl   ֞L|   0 0 0 0 0  t0  4 laT         $ %%^^Zkdi $$IfF,     4 a $$Ifa$ $$Ifa$ $ % * , - . 3 ^7^Zkdw $$IfF,     4 a $$Ifa$ $$Ifa$Zkd $$IfF,     4 a3 5 6 7 < > ? ^ $$Ifa$Zkd $$IfF,     4 a $$Ifa$? @ E G H I O ^7^Zkd $$IfF,     4 a $$Ifa$ $$Ifa$Zkd $$IfF,     4 aO Q R S W Z [ ^ $$Ifa$Zkd $$IfF,     4 a $$Ifa$[ \ ] e f g q r %%%%%%z%l%l%l%l% 77$8$H$^7 77$8$H$`77$8$H$ 77$8$H$`77$8$H$ ^`Zkd $$IfF,     4 a \ e f q r ?@CGghno89ABCNO $8<>HJbdfϿ htqCJo( hCJo( jhCJhtqhCJOJQJhmH sH  h1"CJh1"hCJo(h1"CJmH o(sH hCJmH sH  h1"CJo(h h>* hCJ h>*: !Jq89BC JL%%%%%%%%%%%%%%A%%A%%A%%A7$8$H$7$8$H$(07$8$H$^(`07$8$H$^` & F 7$8$H$ & F7$8$H$fhpr JZ[`jklqr"6789Owkwhm*B*CJo(phhm*B*CJphjhm*6UmHnHu hm*5CJhm*5CJo( htqhtq htqo(h5CJo( h5CJh htqCJ jhCJ h`*CJo( hCJo( jhCJ htqCJo( hCJhtqhCJOJQJ%{6%%%%%A%%%%%%%%%%%$G$Ifgdm*gdtqG$^` G$^^X7$8$H$^`X(07$8$H$^(`06PQ^_pA$1$7$8$H$IfWD`gdm*$1$7$8$H$If^`gdm*$If^`gdm* $Ifgdm*$1$7$8$H$Ifgdm* OP]^_eoptuvw ǻ֩֕sf^XQ h5CJ hCJhtqhtqo( jhm*B*CJphhm*6B*CJphhm*B*CJo(phhm*B*CJph'jhm*B*CJUmHnHphuhm*hm*B*mH o(phsH hm*B*mH phsH  htqo(hm*B*o(phhm*B*phhm*6B*phhm*6mH sH hm*B*CJmH phsH   ;%y%n%_%_%]%]%]%X% & F.G$^.` G$^^gdtqzkd $$If0"` t0644 la 7ETY[\fgprs|}~Xpr~"$.RTVXjlh>*mH sH "jh`*CJUmHnHsH uhOJQJhmH sH  h`*o(hH+ ho(jh`*UmHnHsH uh`*%jh`*>*CJUmHnHsH u htqo( hH+o(h h>*7Z[]fgqr}~Xl%%%%%%%%%%%%%%%%%%  7n$If 7n VD^gdH+gdH+ & F & Flx $.8@HPRTXjAAAAAAAAA%%% 7nFf  7n$IfFf 7n$If^`$ 7n$Ifa$jl%::2:2:`:2:2:kd$$IfF$ Bnn t0    4 a$ 7n$7$8$H$Ifa$ 7n l&0<DFHJT`bjlnpz  +->?Birż ho( h>*hmH o(sH h`*mH sH h`*mH o(sH  hOJo(hmH sH h6mH sH hmHnHu h6hD(0<zgg2:g2:$ 7n$7$8$H$Ifa$kd$$IfF$ Bnn t0    4 a<>V`bzgg2:g2:$ 7n$7$8$H$Ifa$kd?$$IfF$ Bnn t0    4 abd|zgg2:g2:$ 7n$7$8$H$Ifa$kd$$IfF$ Bnn t0    4 azgg2:g2:$ 7n$7$8$H$Ifa$kd$$IfF$ Bnn t0    4 azgg2:g2:$ 7n$7$8$H$Ifa$kdF$$IfF$ Bnn t0    4 azgg2:g2:$ 7n$7$8$H$Ifa$kd$$IfF$ Bnn t0    4 a zgg2:g2:$ 7n$7$8$H$Ifa$kd$$IfF$ Bnn t0    4 a   +,-?@ABKLzs%q%q%q%q%q%q%q%o%q%q% 7nkdM$$IfF$ Bnn t0    4 a -.4<DLT%%%%%%%%t\\\\ $$Ifa$h^h & F TU_acegVMtM\M\M\M\ $$Ifa$kd$$Ifl   r6j L 4 4 4 4 t0  4 laVghirsVT%T%T%O%J%O%T%gd`* & F kd$$Ifl   r6j L 4 4 4 4 t0  4 laV$&,.0268>@FHJLPRXZ`bdhjlrtzhOJQJmH sH h6mHnHuhmHnHu h6hOJmH o(sH hOJmH sH h>*mH sH %jh2>*CJUmHnHsH uhmH sH  hCJ h>*hh`*8%%%%%%%::::::$ 7n$7$8$H$Ifa$7$8$H$.0246TA_A:A:A:A:$ 7n$7$8$H$Ifa$kd$$Ifr$3 GBS t04 a68HJLNPTA_A:A:A:A:$ 7n$7$8$H$Ifa$kdg$$Ifr$3 GBS t04 aPRbdfhjTA_A:A:A:A:$ 7n$7$8$H$Ifa$kdN$$Ifr$3 GBS t04 ajl|~TA_A:A:A:A:$ 7n$7$8$H$Ifa$kd5$$Ifr$3 GBS t04 aTA_A:A:A:A:$ 7n$7$8$H$Ifa$kd$$Ifr$3 GBS t04 aTA:A:A:A:A:$ 7n$7$8$H$Ifa$kd $$Ifr$3 GBS t04 aTA:A:A:A:A:$ 7n$7$8$H$Ifa$kd $$Ifr$3 GBS t04 a    !!!##$$?%I%K%L%X%Y%Z%%%%%%%%%%%j&p&&& ''T''''''''hOJQJmH sH hH*mH sH  hH* h2o( ho( h5hH+ h>* hH+>*o(hOJmH sH h2mH o(sH hOJmH o(sH hmHnHuhmH sH h: TR%?2:?2?2:$ 7n$7$8$H$Ifa$kd!$$Ifr$3 GBS t04 a  z2:z2:$ 7n$7$8$H$Ifa$qkd"$$If0Pnn t04 a,.z2:z2:$ 7n$7$8$H$Ifa$qkdO#$$If0Pnn t04 a.0>@z2:z2:$ 7n$7$8$H$Ifa$qkd#$$If0Pnn t04 a@BPRz2:z2:$ 7n$7$8$H$Ifa$qkd}$$$If0Pnn t04 aRTbdz2:z2:$ 7n$7$8$H$Ifa$qkd%$$If0Pnn t04 adftvz2:z2:$ 7n$7$8$H$Ifa$qkd%$$If0Pnn t04 avxz     !!%%%%%%%%%|%%$^a$^qkdB&$$If0Pnn t04 a !""####3$$$$$$$%% %%%"%*%%%%%%%%%%%%%%%t\\\\ $$Ifa$^^$a$ & F *%+%5%7%9%;%=%VMtM\M\M\M\ $$Ifa$kd&$$Ifl   r4 hL 4 4 4 4 t0  4 la =%>%?%I%Y%Z%j%k%%VP%P%P%P%P%N%L%^kd'$$Ifl   r4 hL 4 4 4 4 t0  4 la %%%%R&T&h&& 'T''''''(3(4(=(>(V(W(o(p({(|((%%%%%%%%%%%%%%%%%%%%%% 7ngd2 & F  & F ^'''''''( (#(1(4(=(U(V(W(p({((((()) ) )!)#)()))<)@)A)B)R)S)U)Y)Z)[)k)l)n)r)s)t)))))))))))))))))))))))))*꾵ƾƾƾƾƾƾƾƾƾƾƾƾƾƾh2mH o(sH h2mH sH h2 ho( h6 h5 h>* h3G>* h3G>*o(hhmH sH hH*mH sH F((((( ) )!)")#)%aQkd_($$If\ }.04 a$ 7n$Ifa$ 7n #)')())):)<)@)A)aXakd+)$$If\ }.04 a$ 7n$Ifa$A)B)S)U)Y)Z)gXXXXa$ 7n$Ifa$kd)$$If\ }.04 aZ)[)l)n)r)s)gXXXXa$ 7n$Ifa$kd*$$If\ }.04 as)t)))))gXXXXa$ 7n$Ifa$kd+$$If\ }.04 a))))))gXXXXa$ 7n$Ifa$kd[,$$If\ }.04 a))))))gXXXXa$ 7n$Ifa$kd'-$$If\ }.04 a))))))gXXXXa$ 7n$Ifa$kd-$$If\ }.04 a)))**+*:*g*h****g`%`%`%`%`%`%`%`%`%`% 7nkd.$$If\ }.04 a **0*1*2*3*4*V*b**********+++++ +@+A+E+R+u+v+,,,,v,x,~,,,,,,,,------....Ȯ趮蝙 hH* h6h h>*hOJQJmH sH h2mH sH hH*mH sH h2mH o(sH h2OJmH o(sH hOJmH sH hOJmH o(sH hmH sH jh2UmHnHsH u5**** +,+X+Y+ ,.,J,L,,,,,----J.L.M.N.....%%%%%%%%%%%&%%%%%%%%%$%%$a$ 7n..'.(.).*.D.E.J.K.p.q.r.t.u.v.y.z.{.......................K/M/c/e/u/}////h, CJmH sH h, CJmH o(sH  h>*jc1hEHU$j]; hCJUVmHnHu hH* h6jhUmHnHu jshj/hEHU$j]; hCJUVmHnHuhjhU1.../9/J/K/M/P/S/V/Y/\/_/b/%%%%%%, $$Ifa$ & Fb/c/e/, $$Ifa$kd;3$$Ifl   ִDH@ 80 | | | | | | | t0      4 lae/g/i/k/m/o/q/s/ $$Ifa$s/t/u/%kd4$$Ifl   ִDH@ 80 | | | | | | | t0      4 lau/~///////00%%%x %%   $Ifgd,  //////////////////////////0#0絡琀p_ZUPUPUKF h=o( hL+o( hH_o( hO-o( ho(!h, B*CJOJQJho(phh, B*CJhnH phtH j#7h, CJEHUmH sH  jXc%> h, UVmH o(sH &h, B*CJOJQJhnH phtH  h, CJhf-CJmH o(sH j4h, CJEHUmH sH  jc%> h, UVmH o(sH h, CJmH sH jh, CJUmH sH 00#0+030N0R0^XX ;X ;K ;K : <$IfYDgd=$Ifkd9$$If\3+#%   t0644 la#0'0)0*0+0/01020307090=0?0@0A0M0N0P0Q0R0`0c0d0g0h0l0n0p0~00000000000000000000㳩ىxx hH_CJo(h=h=aJo(h=aJeh o(r h=CJeh o(r hH_hI`+CJo( h=CJo( h=aJo($h=h=CJeh o(r h=o(hH_h=CJo(h4h=CJo($h1bh=CJeh o(r /R0S0`0d0h0q0^XX ;X ;O ; $Ifgd=$Ifkd+:$$If\3+#%   t0644 laq0r0~000000^XI ;@ ;@ ;I ;X ; $IfgdL+d$IfWD2`dgdH_$Ifkd:$$If\3+#%   t0644 la0000000000000000000000000000000000000000001111 1 1 11111 1!1"1#1)1*1+1,1-11121314151<1B1C1D1E1صصӵh=CJeh o(r h=CJo( hH_CJo( h=o($h1bh=CJeh o(r h4h=CJo(hH_h=CJo(G000 1111#1 ; ; ; ;@ ;kd ;$$If\3+#%   t0644 la $Ifgd=d$IfWD2`dgdH_$If#1-16171<1F1P1Y1 ; ;O ; ; ;kdx;$$If\3+#%   t0644 la $Ifgd=$IfE1F1L1M1N1O1P1T1U1V1W1X1Z1]1b1c1v1w1x1y1z1{1}1111111{{weT jXc%> h, UVmH o(sH "h, B*OJQJhnH phtH h, hf-mH o(sH jV<h, EHUmH sH  jc%> h, UVmH o(sH jh, UmH sH h, mH sH  h, o(h=CJeh o(r hH_h=CJo( h=CJo(h4h=CJo($h1bh=CJeh o(r h=o(Y1Z11111111^Y%x W%W%NN N N  $IfgdI`+gd, kd;$$If\3+#%   t0644 la11111111112222 2 2 22222222(2)2,2Ŷ||ib[|Kb|hI`+CJeh o(r hI`+CJo( hI`+aJo($h=hI`+CJeh o(r hH_hI`+CJo(h4hI`+CJo($h1bhI`+CJeh o(r hI`+o( ho( h, o(hh, B*OJQJho(phh, B*hnH phtH "h, B*OJQJhnH phtH jh, UmH sH j>h, EHUmH sH 11122)2-2^UU ;U ;H ;H : <$IfYDgdI`+ $IfgdI`+kd!A$$If\3+#%   t0644 la,2-2;2>2?2B2C2G2I2K2Y2\2]2^2_2`2a2c2g2h2i2k2m2n2o2p2r2s2w2x2z2|2}2~22222222222222222ó𩟌ݟ~~ݳ~ݟ~~ݳ hI`+CJo( h, CJo($ jhI`+CJeh o(r h4hI`+CJo(hI`+hI`+CJo(hI`+CJeh o(r hI`+aJo($h=hI`+CJeh o(r $h1bhI`+CJeh o(r hI`+o(h=hI`+aJo(1-2.2;2?2C2L2^UU ;U ;U ; $IfgdI`+kdA$$If\3+#%   t0644 laL2M2Y2i2x22222^UU U U U U L  $IfgdH_ $IfgdI`+kdA$$If\3+#%   t0644 la222222222222222222222222222222222ܸծxn^nYnRn h, CJo( hI`+o(hI`+CJeh o(r hI`+hI`+CJo(hH_CJeh o(r $h1bhH_CJeh o(r $ jhH_CJeh o(r h4hH_CJo(hI`+hH_CJo($h1bhI`+CJeh o(r hH_CJo( hI`+CJo(h4hI`+CJo($ jhI`+CJeh o(r 2222233"313404L4f4444         $ $ $ $ $ $ $IfgdH_ $IfgdI`+222222222222222222222222222233333 3 3 3 333333333$ jhH_CJeh o(r h4hH_CJo(hI`+hH_CJo( hH_CJo(hI`+CJeh o(r $h1bhI`+CJeh o(r $ jhI`+CJeh o(r hI`+hI`+CJo( hI`+CJo(h4hI`+CJo(,3!3"3$3&3'3(3)3*3,303134444 4444444 4$4&4.404446484:4<4@4H4J4L4P4R4T4ɶɓvoveeoveeveveoveeoveh4hH_CJo( hH_CJo(hI`+hH_CJo($hI`+hI`+CJeh o(r $h1bhI`+CJeh o(r hI`+CJeh o(r $ jhI`+CJeh o(r h4hI`+CJo(hI`+hI`+CJo(hH_CJeh o(r $h1bhH_CJeh o(r 'T4V4X4\4d4f4j4l4n4r4v4~444444444444444444444444444444444555ѵwhI`+CJeh o(r hH_hI`+CJo( hI`+CJo(h4hI`+CJo($h1bhI`+CJeh o(r hI`+o( h, CJo(hH_CJeh o(r $h1bhH_CJeh o(r h4hH_CJo(hI`+hH_CJo( hH_CJo(.444444^UU ;U ;U ; $IfgdI`+kdnB$$If\3+#%   t0644 la445505B5^UU ;U ;U ; $IfgdI`+kdB$$If\3+#%   t0644 la5555(5*5,5.50585:5<5>5@5D5F5H5^5`55 6.676J6L6^6i6s6t6666666666666666666666666鷰h!h!mH o(sH h!mH sH  h!CJo( hCJhmH sH  h, >*o( h>*hI`+CJeh o(r h4hI`+CJo($h1bhI`+CJeh o(r hI`+o(hH_hI`+CJo( hI`+CJo(4B5D5F5H5^5`5K6L6^\%\%\%Q%:QQ%:  9r 7$8$H$kdLC$$If\3+#%   t0644 laL6^6i6s6t6666ahakdC$$If   F<  t0      4 aX$ 7n$Ifa$66666whhha$ 7n$Ifa$kdwD$$If   F<  t0      4 aX66666whhha$ 7n$Ifa$kd3E$$If   F<  t0      4 aX66666whhha$ 7n$Ifa$kdE$$If   F<  t0      4 aX66666whhha$ 7n$Ifa$kdF$$If   F<  t0      4 aX667 7 7whhha$ 7n$Ifa$kdgG$$If   F<  t0      4 aX677 7 7 777"7$7%7&7.7/7075767I7J7K7L7M7N7P7R7d7e7f7y7ĺuo[&h, B*CJOJQJhnH phtH  h, CJhf-CJmH o(sH h, CJmH o(sH jIh, CJEHUmH sH  jc%> h, UVmH o(sH jh, CJUmH sH h, CJmH sH h!>*mH o(sH h>*mH o(sH h>*mH sH hmH sH h!mH o(sH h!h!mH sH  7 77"7$7whhha$ 7n$Ifa$kd#H$$If   F<  t0      4 aX$7%7&7/707777777wp%n%n%in%n%n%ga%gd,  7nkdH$$If   F<  t0      4 aX y7z7{7|7}7777777778888888889(9)9=9ѽ|r|l_RrrlKl h6CJhf-B*CJho(phhB*CJho(ph hCJhCJmH sH h5>*CJh5CJmH sH hhmH sH !h, B*CJOJQJho(phh, B*CJhnH phtH &h, B*CJOJQJhnH phtH jh, CJUmH sH jKh, CJEHUmH sH  jXc%> h, UVmH o(sH 77777 88,8:8K8X8f8t8888888888888889%%%%%%%%%%%%%%%%:%%%%%%%%:%%:% 77$8$H$^77$8$H$99*9<9=9?9C9%rtkdfN$$If07j ;3 t04 aU$ 7n$7$8$H$Ifa$7$8$H$=9>9?9f9g99999999999999999999999:00000 0 0 0000000000!0$0%0&0'0+0,0.0102030408090;0>0?0@0A0E0F0H0K0L0M0Q0R0U0X0Y0Z0;U hCJo( hDCJ hDCJo( h5CJhmH sH h hCJ hCJ]hCJ]mH sH KC9D9G9K9xx$ 7n$7$8$H$Ifa$tkd O$$If07j ;3 t04 aUK9L9O9T9xx$ 7n$7$8$H$Ifa$tkdO$$If07j ;3 t04 aUT9U9X9]9xx$ 7n$7$8$H$Ifa$tkdUP$$If07j ;3 t04 aU]9^9a9f9xx$ 7n$7$8$H$Ifa$tkdP$$If07j ;3 t04 aUf9g9j9n9xx$ 7n$7$8$H$Ifa$tkdQ$$If07j ;3 t04 aUn9o9r9v9xx$ 7n$7$8$H$Ifa$tkdDR$$If07j ;3 t04 aUv9w9z9~9xx$ 7n$7$8$H$Ifa$tkdR$$If07j ;3 t04 aU~9999999990 00%%%%q%e%ee%e% 7$8$H$VDG^X7$8$H$WD^X`7$8$H$tkdS$$If07j ;3 t04 aU 7/2003 8.7 8/2003 8.6 9/2003 8.3 10/2003 8.0 11/2003 7.5 12/2003 7.3 1/2004 7.3 2/2004 7.2 3/2004 7.2 4/2004 7.1 5/2004 7.0 Calculate the average, median, mode and the standard deviation of unemployment rate: a) For 5/2003 12/2003 b) For1/2004 5/2004 c) For all 13 months. 2. Find the mean, median, mode of the following: 10, 13, 14, 14, 14, 15, 15,16, 17, 22 3. Which student has the highest average mark? StudentABCEnglish786355Chinese808572Mathematics597195 4. The frequency distribution of the lengths of 100 leaves from a certain species of plant is given below: length (mm)Frequency20 24625 291030 341835 392540 442245 491550 544 5. The following table shows the distribution of heights of 50 students: Height (cm)Frequency160 1648165 16912170 17414175 1797180 1846185 1893 Find the range and standard deviation of heights. 6. The mean of one set of six numbers is 9 and the mean of a second set of eight numbers is 12.5. Calculate the mean of the combined set of fourteen numbers. 7. The mean of the numbers a, b, c, d is 8 and the mean of the numbers a, b, c, d, e, f, g is 11. What is the mean of the numbers e, f, g? 8. Find the mean and standard deviation of the 5 numbers in term of x: x-5, x-3, x-2, x+1, x+4. 9. The mean of the five numbers 6, 9, 2, x, y is 5 and the standard deviation is EMBED Equation.3 . Find the values of x and y. Answer: 1.a) 8.16: 8.3; undefined; 0.52 b) 7.16: 7.2; 7.2; 0.11 c) 7.78: 7.5; undefined; 0.65 2) 15: 4.5; 4 3) 72; 73; 74 4) 37.4 5) 30; 7.14 6) 11 7) 15 8) x-1; 3.16 9) (3, 5); (5, 3) CMV6120 Mathematics Unit 20 Central tendency & dispersion Page  PAGE 13 of  NUMPAGES 13 Temperature (!) Age group  EMBED Equation.3 0%020?0L0Y0f0s000001-1.1_111111111%%%%%%%%%%%%%%:%%%%%0 $$Ifa$^x7$8$H$ 7$8$H$VDG^Z0^0_0d0e0f0g0k0l0q0r0s0t0x0y0~000000000000 1 1111#1$1-111111233u4v4x4y4{4|4~4444444444444444,5666 66ͻͻͻͻͻͻͻͻͻͻͻͻͻhOJQJmH sH h6mH sH h.nmH o(sH hmH sH h h{CJ h{CJo( hCJ hCJo( hDCJo( hDCJE111111lccc0c $$Ifa$kd3T$$Ifl\@ Jpb04 la<111111lccc0c $$Ifa$kdT$$Ifl\@ Jpb04 la<111111lccc0c $$Ifa$kdU$$Ifl\@ Jpb04 la<112k2l2x22l`%`%^%UU  $$Ifa$ 0x^`0kd|V$$Ifl\@ Jpb04 la<2222  $$Ifa$lkd?W$$Ifl0<4 04 la2222  $$Ifa$lkdW$$Ifl0<4 04 la2222  $$Ifa$lkdmX$$Ifl0<4 04 la2222  $$Ifa$lkdY$$Ifl0<4 04 la2222  $$Ifa$lkdY$$Ifl0<4 04 la2222  $$Ifa$lkd2Z$$Ifl0<4 04 la2222  $$Ifa$lkdZ$$Ifl0<4 04 la222 3!3-373%%  $$Ifa$lkd`[$$Ifl0<4 04 la7383B3D3  $$Ifa$lkd[$$Ifl0T< `04 laD3E3O3R3  $$Ifa$lkd\$$Ifl0T< `04 laR3S3]3`3  $$Ifa$lkd%]$$Ifl0T< `04 la`3a3k3m3  $$Ifa$lkd]$$Ifl0T< `04 lam3n3x3z3  $$Ifa$lkdS^$$Ifl0T< `04 laz3{333  $$Ifa$lkd^$$Ifl0T< `04 la33333Z4406:7<7>7%%%%t%tt%R%o%$a$ 0x^`0 0^`0`lkd_$$Ifl0T< `04 la 66666&6(66666666666(7*74767:7<7>7@7L7N7P7T7V7X7\7^7d7f7h7n7r7777777õⱨxoxxxxh{mH o(sH hf-mH o(sH hyUmH o(sH hf-mH sH  ho( hyUo( hf-o( h5CJhf-5CJo(hj`hEHUmH sH $jfZ; hCJUVmHnHujhUmH sH hmH sH hOJQJmH sH h6mH sH +>7@7P788888999999999%mh%%%A%( r %$d%d&d'dNOPQ+ 9r %$d%d&d'dNOPQgdf-$a$$&dPa$gdf-7777777777777777778888 8888 8"888888888899999\9^9j9̻̫xq hyU0J6jhyU0J6UhyUhyU6mH o(sH hyU6mH sH h|r6mH nH o(sH tH hhf-OJQJmH sH hf-6mH sH hmH sH hf-mH o(sH hf-mH sH hmH o(sH hyUmH o(sH h{mH o(sH h{mH sH ,j9l9p9r9z9|999999999999999999:ppppppppҺұqgV j_; hyUUVmHnHujbhyUEHU jn; hyUUVmHnHuUjhyUUhyUOJmH o(sH hyUOJmH sH hyUmH sH h|rhyUhyU6mH sH h|r0J6CJmHnHuhyU0J6CJjhyU0J6CJU hyU0J6h|r0J6mHnHujhyU0J6U99ppppp0p1p6p7p*CJOJQJhyU>*CJOJQJo(hyUCJmH sH  hHLhyU hyU>*CJhyU>*CJo( hyUCJhyUB*mH phsH hyUmH sH hyUjhyUUj_ehyUEHU7ppppppppppppppppppppppppppppgdHL1$7$8$H$pqqq qqqqqqq q!q%q&q0q1q4q5q8q9q*CJo( hyU>*CJhyU5B*mH phsH h`*hyUB*mH phsH hH+hyUB*mH phsH  hyUCJhyUhyUB*mH phsH DIqLqMqQqRqVqWqZq[qqqqqqqqqqqqqqqqqqqqq1$7$8$H$qqqqqqqqqqqqqqqqqqqqqqqrrr r rr1$7$8$H$qqqqqqqrrr r rrrrrrrrr)rDrnrorzr}rrrrrrrrrrrrrrrrrrhhyU>*CJo( hyU>*CJ hyUCJhyUB*mH phsH hyU*rrrrrrrrFrnrorzr{r|r}rrrrrrrrrrrrrr`1$7$8$H$rrrrrrrrrrrrrrrrrrrrrrrrrrrrr$a$1$7$8$H$(&P . A!n"n#$n%77`!Bs&E=cX,hL ``/`\xTkQ7._a׻ n˱8 !G ;^@.J:Cl vV[lDU"yoYcٙyO@&Z$KBhF`$iP3(Rڸ2NӐP zVTiJOnl]tN8EAל~0k KZYIr˖۩XYݫ.ƫp1~5#WQH}V>ƻ#L.nCSZ O"\(E>NC8 v=8~`!/LFg2"<$ 8p`xMkQ=3iI|tNji4֕*E [tg!R!I 8*. HF\(f%qҟ"pՂ~rfc' s\ c07e h$s,NK;y:Y$r[nW2 &wݝ^Ct2n{xc4o=i-ڗ֞RWkʔjEi׌5Pm@ȍ#1AmTtKh*bGL|jtSn8V&ubvDl;` o+V={ 9aO8n 審qvpQn_<S34ur)N??ee7zGM@#jpw"jiʙeojqIb.PDyۦ$'h(?zR'%ͯjam,\ AhZfyGG wbԿϔ7辞ǕݬWȖnAy3d"I}s~S4̢:N/MP*uZv u[ gq0Do6ZpqCjDd lB  S A ? 2= c9 6"U~D`!= c9 6"U~^PtxKP]RۤVAJ kG:ֹL+]QC'nɹ{g(}%ݻw ,,d &VLY'4`rK|;(ʍn; crm&/J&I.}R8:4W?Q&[=N;F nTd -}!ۖb-_hn:a~\OrMɖ=qbkd`V5#uȌՍTKۧksttK|K0\Ƿ.D 3_"RT>cmh7!ڵigզT=d)Qf&.j~7X[ͦ-W:B\ M5CUzC'Z!~HuEDd XlB  S A? 2r8ypƜ*J`!r8ypƜ*Jv  7Qxcdd``f ĜL  312Ec21BUs30=`>-d3H1ibY;a3PT obIFHeA*P - :@@ l.#t_ `0*c`J7S? `01d++&1TBMfdF\=3@M*)7>qȯ-J.X W8mæZl*`tg)L*~5NnsIw== p q&| W|ޮXe(a3Ȁ3]2ӂt|fmuPȕmhssqjR{9BY+lqSᄛ/8u ogͭ,^$keue^ &'vgAȯl`eC|#L~e|d+u|1d' i~'<,OzPt WY9AWNswc) 77$]6C)\57$$If!vh550505050505#v#v0#v:V l t0  5505/ 4T$$If!vh550505050505#v#v0#v:V l t0  5505/ 4T$$If!vh555#v#v#v:V 55544 a$$If!vh555#v#v#v:V 55544 a$$If!vh555#v#v#v:V 55544 a$$If!vh555#v#v#v:V 55544 a$$If!vh555#v#v#v:V 55544 a$$If!vh555#v#v#v:V 55544 a$$If!vh555#v#v#v:V 55544 a$$If!vh555#v#v#v:V 555/ 44 ab$$If!vh55`#v#v`:V t655`a$$If!v h555555555 #v#v :V  t0  55 /  / /  44 akd$$If   V  v> #  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entry7 F=)Data ]hWordDocument6ObjectPool9@on)=)_1028366223"F@on)@on)Ole CompObjfObjInfo #&),-0589:;<=>ADEFGIJKLMNOQRSTUW FMicrosoft Equation 3.0 DS Equation Equation.39qLޤ$IwI Mean(x)=x 1 +x 2 +x 3 +...+x n nEquation Native _1028366180' F@on)@on)Ole CompObj f FMicrosoft Equation 3.0 DS Equation Equation.39qLLrIoI x=3+7+2+1+75 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native  h_1028366292F@on)@on)Ole  CompObjfObjInfoEquation Native d_998399397F@p)@p)LH$IwI Mean(x)=x 1 f 1 +x 2 f 2 +x 3 f 3 +...+x n f n f 1 +f 2 +f 3 +...+f nOle CompObjfObjInfoEquation Native 0 FMicrosoft Equation 3.0 DS Equation Equation.39qmIxI x FMicrosoft Equation 3.0 DS Equation Equation.39q_998399443, F@p)@p)Ole CompObjfObjInfoEquation Native  0_1042637708F@p)@p)Ole !CompObj "fI(oI x FMicrosoft Equation 3.0 DS Equation Equation.39qJLI`I x=x i n i "ObjInfo!$Equation Native %h_1042637656$F@p)@p)Ole 'CompObj#%(fObjInfo&*Equation Native +_998464102)F@p)@p) FMicrosoft Equation 3.0 DS Equation Equation.39qJtdIdI = (x i "x) 2i " n  FMicrosoft Equation 3.0 DS EqOle .CompObj(*/fObjInfo+1Equation Native 24uation Equation.39qmIxI  6  FMicrosoft Equation 3.0 DS Equation Equation.39q4I(oI  (x 1_9983737421.F@p)@p)Ole 3CompObj-/4fObjInfo06Equation Native 7_9983737273F@p)@p)Ole ?CompObj24@f "x) 2 f 1 +(x 2 "x) 2 f 2 +(x 3 "x) 2 f 3 +...+(x n "x) 2 f n f 1 +f 2 +f 3 +...+f n FMicrosoft Equation 3.0 DS Equation Equation.39qxI`I  (x 1 "x) 2 +(x 2 "x) 2 +(x 3 "x) 2 +...+(x n "x) 2 n ObjInfo5BEquation Native C41Table9SummaryInformation(8H      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ t0  $$$$4 a$$If!v h555555555 #v#v :V  t0  55 / / 44 akd$$If   V  v> # t0  $$$$4 a$$If!vh5B5n5n#vB#vn:V  t05B5n44 $$If!vh5B5n5n#vB#vn:V  t05B5n44 $$If!vh5B5n5n#vB#vn:V  t05B5n44 $$If!vh5B5n5n#vB#vn:V  t05B5n44 $$If!vh5B5n5n#vB#vn:V  t05B5n44 $$If!vh5B5n5n#vB#vn:V  t05B5n44 $$If!vh5B5n5n#vB#vn:V  t05B5n44 $$If!vh5B5n5n#vB#vn:V  t05B5n44 $$If!vh5B5n5n#vB#vn:V  t05B5n44 $$IfV!vh5L54545454#vL#v4:V l t0  5L54/ 4aV$$IfV!vh5L54545454#vL#v4:V l t0  5L54/ 4aV$$If!vh5B5555S#vB#v#v#v#vS:V  t05B5555S44 $$If!vh5B5555S#vB#v#v#v#vS:V  t05B5555S44 $$If!vh5B5555S#vB#v#v#v#vS:V  t05B5555S44 $$If!vh5B5555S#vB#v#v#v#vS:V  t05B5555S44 $$If!vh5B5555S#vB#v#v#v#vS:V  t05B5555S44 $$If!vh5B5555S#vB#v#v#v#vS:V  t05B5555S44 $$If!vh5B5555S#vB#v#v#v#vS:V  t05B5555S44 $$If!vh5B5555S#vB#v#v#v#vS:V  t05B5555S44 $$If!vh5n5n#vn:V  t05n44 $$If!vh5n5n#vn:V  t05n44 $$If!vh5n5n#vn:V  t05n44 $$If!vh5n5n#vn:V  t05n44 $$If!vh5n5n#vn:V  t05n44 $$If!vh5n5n#vn:V  t05n44 $$If!vh5n5n#vn:V  t05n44 $$If !vh5L54545454#vL#v4:V l t0  5L54/ 4a $$If !vh5L54545454#vL#v4:V l t0  5L54/ 4a $$If!vh55.55#v#v.#v#v:V 055.5544 a$$If!vh55.55#v#v.#v#v:V 055.5544 a$$If!vh55.55#v#v.#v#v:V 055.5544 a$$If!vh55.55#v#v.#v#v:V 055.5544 a$$If!vh55.55#v#v.#v#v:V 055.5544 a$$If!vh55.55#v#v.#v#v:V 055.5544 a$$If!vh55.55#v#v.#v#v:V 055.5544 a$$If!vh55.55#v#v.#v#v:V 055.5544 a$$If!vh55.55#v#v.#v#v:V 055.5544 aDd TB  S A? 2BB诚-0/`!B诚-0H@ XJx5O1n@]C V0(^"D&@T2ҧ =ͮf>@tUYQMlC\mgSn]'(額>J! W'l5K9Ypa؋CrO11tWwo<5F΋|':~(s.@A"d>Fӿy]{Ŗ$nR/Dd TB  S A? 2BEl 7I81`!El 7I8H@ XJx5O1n@]C V0(^"D&@!@|ЀgVڻȁ.>+I3m˲3l3bMZ==G"2$tf3^4 .Z{qH=)&:Ζ_'Scȗq㷏"[΅А{( R'4Ȳ1޷U^`ؒ3 ]/3$$If!vh55|5|5|5|5|5|5|#v#v|:V l t0  55|/ 4a$$If!vh55|5|5|5|5|5|5|#v#v|:V l t0  55|/ 4a2Dd B  S A? 2//\5%:Q#x55`!p//\5%:Q#V` P>xڅQJAۨ<4#(h/*b x^#! Z ~Hϙdݛvf #PY%e&De,S9"tZtc Va2AwolSq9U8oZC0ZBVCo<@^A|$#GB3 ̼" QKb oDEӱaL"`|*VgUgsRX٪lJ=O2 0ccQ% EΰgF2t R9&ewŦcm# F-x/js[qnTͬ3k 2΄'Oݩ!έX8/ 6\A>Jmyf7L1}ș(I?oɩV'Gz9NSTwYC,#ٿvf#|2YigEg "i:tcǐFw=zZ+qۣy-m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 2Dd B   S A? 2//\5%:Q#x<`!p//\5%:Q#V` P>xڅQJAۨ<4#(h/*b x^#! Z ~Hϙdݛvf #PY%e&De,S9"tZtc Va2AwolSq9U8oZC0ZBVCo<@^`!WSL@3 X HL xڕS=KA}{\C Xm?ZZD!`Kaa`i!!?@"Uj *6*v K̽qw @>A|$#GB3 ̼" QKb oDEӱaL"`|*VgUgsRX٪lJ=O2 0ccQ% EΰgF2t R9&ewŦcm# F-x/js[qnTͬ3k 2΄'Oݩ!έX8/ 6\A>Jmyf7L1}ș(I?oɩV'Gz9NSTwYC,#ٿvf#|2YigEg "i:tcǐFw=zZ+qۣy-m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 m$$If!vh55 5 5 #v#v :V t655 $$IfX!vh555#v#v#v:V  t0  55544 aX$$IfX!vh555#v#v#v:V  t0  55544 aX$$IfX!vh555#v#v#v:V  t0  55544 aX$$IfX!vh555#v#v#v:V  t0  55544 aX$$IfX!vh555#v#v#v:V  t0  55544 aX$$IfX!vh555#v#v#v:V  t0  55544 aX$$IfX!vh555#v#v#v:V  t0  55544 aX$$IfX!vh555#v#v#v:V  t0  55544 aX2Dd B   S A?  2//\5%:Q#xI`!p//\5%:Q#V` P>xڅQJAۨ<4#(h/*b x^#! Z ~Hϙdݛvf #PY%e&De,S9"tZtc Va2AwolSq9U8oZC0ZBVCo<@^A|$#GB3 ̼" QKb oDEӱaL"`|*VgUgsRX٪lJ=O2 0ccQ% EΰgF2t R9&ewŦcm# F-x/js[qnTͬ3k 2΄'Oݩ!έX8/ 6\A>Jmyf7L1}ș(I?oɩV'Gz9NSTwYC,#ٿvf#|2YigEg "i:tcǐFw=zZ+qۣy-$$IfU!vh535#v3#v:V  t053544 aU$$IfU!vh535#v3#v:V  t053544 aU$$IfU!vh535#v3#v:V  t053544 aU$$IfU!vh535#v3#v:V  t053544 aU$$IfU!vh535#v3#v:V  t053544 aU$$IfU!vh535#v3#v:V  t053544 aU$$IfU!vh535#v3#v:V  t053544 aU$$IfU!vh535#v3#v:V  t053544 aU$$IfU!vh535#v3#v:V  t053544 aU$$If<!vh5p5b55#vp#vb#v#v:V l05p5b554a<$$If<!vh5p5b55#vp#vb#v#v:V l05p5b554a<$$If<!vh5p5b55#vp#vb#v#v:V l05p5b554a<$$If<!vh5p5b55#vp#vb#v#v:V l05p5b554a<$$If!vh5 5#v #v:V l05 54a$$If!vh5 5#v #v:V l05 54a$$If!vh5 5#v #v:V l05 54a$$If!vh5 5#v #v:V l05 54a$$If!vh5 5#v #v:V l05 54a$$If!vh5 5#v #v:V l05 54a$$If!vh5 5#v #v:V l05 54a$$If!vh5 5#v #v:V l05 54a$$If!vh5 5`#v #v`:V l05 5`4a$$If!vh5 5`#v #v`:V l05 5`4a$$If!vh5 5`#v #v`:V l05 5`4a$$If!vh5 5`#v #v`:V l05 5`4a$$If!vh5 5`#v #v`:V l05 5`4a$$If!vh5 5`#v #v`:V l05 5`4a$$If!vh5 5`#v #v`:V l05 5`4aDd |hB  S A ?  2cEJ+!H/ф?\``!7EJ+!H/ф`@0|x]1OA߼C"11V :M41Xx).Jq@ Pg,bc#j&}4@ 'p@APaUU\ȩ)@7Wh֯ +B[ϥf_Z L q\,2`z7}[={\{rm1o=Z*魔{N[yw)9t{MhyLnur7'3;_ť Ᏹ g;vwMn#d/-,{EzF6NDd4<  C A? 2/LFg2" n`!/LFg2"<$ 8p`xMkQ=3iI|tNji4֕*E [tg!R!I 8*. HF\(f%qҟ"pՂ~rfc' s\ c07e h$s,NK;y:Y$r[nW2 &wݝ^Ct2n{xc4o=i-ڗ֞RWkʔjEi׌5Pm@ȍ#1AmTtKh*bGL|jtSn8V&ubvDl;` o+V={ 9aO8n 審qvpQn_<S34ur)N??ee7zGM@#jpw"jiʙeojqIb.PDyۦ$'h(?zR'%ͯjam,\ AhZfyGG wbԿϔ7辞ǕݬWȖnAy3d"I}s~S4̢:N/MP*uZv u[ gq0Do6ZpqCjDd<  C A? 2ns&E=cX,hL JNn`!Bs&E=cX,hL ``/`\xTkQ7._a׻ n˱8 !G ;^@.J:Cl vV[lDU"yoYcٙyO@&Z$KBhF`$iP3(Rڸ2NӐP zVTiJOnl]tN8EAל~0k KZYIr˖۩XYݫ.ƫp1~5#WQH}V>ƻ#L.nCSZ O"\(E>NC8 v=8~<@< Normal_HmH nHsH tHH@"H Heading 1$<@& 5CJ KHD@D Heading 2$<@&5CJDD Heading 3$<@&5CJb2b Heading 4 $x1$7$8$@&H$`>*B*CJnHtHu^@2^ Heading 5$1$7$8$@&H$`>*B*CJnHtHu^@^ Heading 6)$(07$8$@&H$VD,WD^(`0>*CJF@F Heading 7$7$8$@&H$>*CJDA@D Default Paragraph FontVi@V  Table Normal :V 44 la (k(No List $O$ FDCJ4@4 Header  9r 4 @4 Footer  9r .)@!. Page Number>@2> Normal Indent ^VP@BV Body Text 21$7$8$H$B*CJ(mH phsH uRB@RR Body Text1$7$8$H$B*CJ(mH phsH ubC@bb Body Text Indent#(07$8$H$VD,WD^(`0CJF"@F Caption 7$8$H$>*CJmH sH uj@j tq Table Grid7:V06OPbhntz"(.4:@FLRWbfjnrvz~ !&+016;@EJO1G*d'&%$#"! cboyxvutsrqponmlkjihgfedc19:a`_^]\[ZYXWVUTSRQ]Y[\2^_`abcd e f g h ijklmnZ6OPbhntz"(.4:@FLRWbfjnrvz~ !&+016;@EJO      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmno1789MNOPpqr$%&12fg#PQox %=>?IJ $%*,-.3567<>?@EGHIOQRSWZ[\]efgqr!Jq89BC_`0kno + e f g i  5 G H I [ \  H  * 4 5 ? @    % - 5 = B C U Y ] b g l p t x y z | !&'(489:FJKLTXYZ[{|}-.89Th}~E !+7;AKMXZ^_ghijkltuvwxy    %&IJefgho !lm !-STZbjrz{yz!"-.Og&'9CN[pqrswxy  "$()*+eg{./K[|! #&)*,.02468:;<EF   ' + / 8 9 E Z o x ! !! !!!t!u!!!!!!!!!!!!!"" """ "0"?"O"^"n"}"""""""""##!#.#<#I#V#W#\#f#p#y#z##########F$G$Y$d$n$o$$$$$$$$$$$$$$$$$$$$$$%%%%%% %!%*%+%{%|%}%~%%%%%%%&&'&5&F&S&a&o&&&&&&&&&&&&&&&''%'7'8':'>'?'B'F'G'J'O'P'S'X'Y'\'a'b'e'i'j'm'q'r'u'y'z'{''''''''(( (-(:(G(T(a(n({(((()()))Z))))))))))))))))))))))))))f*g*s*}*~***********************++(+2+3+=+?+@+J+M+N+X+[+\+f+h+i+s+u+v+++++++U,,@------*.q........../ ///,/-/E/F/G/X/Y/^/_/d/e/j/k/p/q/v/w/|/}///////////////////////////000000$0%0*0+000106070<0=0B0C0H0I0M0N0X0Y0\0]0`0a0d0e0h0i0l0m0p0q0t0u0y0z0~0000000000000000000000000000001111 1 1111111!1"1&1'1(1,1-111216171;1<1@1A1E1F1n1111111111111111111111111111111111111111111111100000090909090909 0909 0909 090900%0&0&0&0&00000000Q0Q000x0x0x0x0xH0xH0x000000H00?0? 0?0?0?0?0?0?0?0?0?0?0?0?0?0?0?0?0?0?0?H0H0H0H0H000000000000 000000000000000000000H0H00]0]0]0]0]0]0]0]0] 0] 0] 0] 0]0]0]0]0]]0]X]090909090909090909090909090909090909090000000e 0e 0e 0e 0e 0e 0e 0e 0e 0e 0e 0e 0e e 0e 0e 0e 00000 0 0  0  0  0 0 0 0 0  0 0  0 0  0 0 0 0 0 0 x0   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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000 00 00000000000000000 00 00000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000x00x000000 00!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!0!0!0!0!0!000000000000000 0 0000 00 0000000 0 000 0 0 0 00 0 0 0 000 0 0 000 0 0 000 0 0 000 0 0 000 0 0 000 0 0 000 00000x000000000000000000000x0000x0000000000 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 00x0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 00 00 00 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 0000000 0000000 00000000 @0 @0 @0 @0 @0 @0 @0 @0 @0 @0 00 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 0 0 0 0 0 0 00x0x0 00000x0%x0% 0% 0% 0% 0%0%0%0%0% 0%0%0%(0%(0%(0%h0%(0&0&0&0&0& 0&(0&h0%(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& 0& 0& 0& 0& 0&0&(0& 0&0&x&&&&&&&&&&&0&(0(0&(0&(0&(0&0&(00(0&(0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 00x0&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&0&x0& 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&0&(0&(0&0&0(0(0x0&(0&0(@0@0@0@00Ԣ600x000000000x00x00x00x00x00x00x00x0x00x00x00x00x00x00x00x00x0000000000000000000000000000000000000000000000000H0000000000000000000000000000000000000000H00x00H000x00x00x00x00x00x00x0x00x00x00x00x00x00x00x00x00x00x000000000000000000000000 0)*  8 9 !!!!!!!""V#W#y#z###n$o$..1<008|00 |004 |00|00|00|00 fw |00@03t |00G7 |00@0% |00 |00|00|00|00 fw |00@03 |00G7 |00@0% <000}  0@0hY  0ؗbbbe \ fOl'*./#00E11,2223T456y7=9Z067j9pqqr!,.137CLYcekmpsuwz|}~# > $ 3 ? O [ 6lj<b Tg6Pj .@Rdv!*%=%%(#)A)Z)s)))))*.b/e/s/u/0R0q00#1Y11-2L2244B5L666666 7$779C9K9T9]9f9n9v9~9011112222222273D3R3`3m3z33>79ppIqqrrr "#$%&'()*+-/0245689:;<=>?@ABDEFGHIJKMNOPQRSTUVWXZ[\]^_`abdfghijlnoqrtvxy{r%9;uNbd~)!=!?!Y!m!o!0%D%F%`%t%v%---1::::::::::::BILQ\_e!136JL::P  o2$M7LG6@^~4h$2$V >Kqϳ0$2$s&E=cX,hL J*2$/LFg2"t@W p\    V( n V  C A ?"V  C A ?"8n  ' L3 m~#" f  b M3 s q$`B N c $DjJ bb`B O c $DjJ  b`B P c $D) ) b`B Q c $D2 2 b`B R c $D; ; b`B S c $DEEb`B T c $DNNb`B U c $DWWb`B V c $D``b`B W c $Diib`B X c $Drrb`B Y c $D{{b`B Z c $D G G `B [ c $D > > `B \ c $D 4 4 `B ] c $D ++`B ^ c $D ""`B _ c $D `B ` c $D Y Y `B a c $D P P f` g$! b#    c </cf"`k$! / d <.df"`pM . e <-ef"`: - f <,ff"` , g <+gf"` + h <*hf"` * i <)if"` ) j <(jf"`M ( k <'kf"`g ' l 6&lf"`{  & m NZ%m?f ?"` % n NZ$n?f ?"`T $ o NZ#o?f ?"`> # p NZ"p?f ?"`= " q NZ!q?f ?"`$ ! r NZ r?f ?"` {   s NZs?f ?"`M  t NZt?f ?"`9!  u NZu?f ?"` #  v NZv?f ?"`"2%  w S BLCLDE$F,fo LD$,@`<DL@"`< #h x 3 x"` '  h y 3 y"`8+    Z ?m~"` h q*V#; 43 m~"$L ,M"D9 3,M"D9j` (!=9 0#" ,!D9fB  s *DjJ (=9+fB  s *D (=9+fB  s *D (=9+fB  s *D (=9+fB  s *D (=9+fB  s *D (=9+fB   s *D !(!=9+VL -M"8 2-M"8fB  s *DjJ 8M"8#fB  s *D q3M"q3$fB  s *D y1M"y1%fB  s *D /M"/%fB  s *D -M"-&fB  s *D 7M"7$fB  s *D 95M"95&  BGf "`%8 9 G` (,v7 1#" ,/8  BFf "`5vw7 F  Bf "`3d5   Bf "`13   Bf "`01   Bf "`'.>/   Bf "`(,.    < f "`/7 n ! C !"`Mq*"A-  " TZ "?f ?"`u8:   # TZ #?f ?"`A8:   $ TZ $?f ?"`+8:   % TZ %?f ?"`*8:   & TZ &?f ?"`8w!u:   ' TZ'?f ?"` 8V#u: n * C *"`96!; T V(* o# 2L &( n&(fB = s *DjJ(fB > s *Dww(fB ? s *D^^(fB @ s *DEE(fB A s *D//(fB B s *D!!(fB C s *D""(fB D s *D$$(fB E s *D&&(F` ))8 i# '(fB < s *DjJ8)8fB H s *D3)3fB I s *D*1)*1fB J s *D@/)@/fB K s *DY-)Y-fB L s *Dr+)r+fB M s *D)))fB N s *D6)6fB O s *D4)4` Tq%}6 P# )v) Q <YQf56 Y R <XRf2}W4 X S <WSf0D2 W T <VTf.0 V U <UUf,. U V <TVf+E, T W <SWf) * S X <RXfW' ) R Y <QYfq%& Q~ Z 6PZfT(F@2 P` [ C O[{'< O \ TZN\?f ?') N ] TZM]?f ?q') M ^ TZL^?f ?[') L _ TZK_?f ?Z' ) K ` TZJ`?f ?A '") J a TZIa?f ?'"'$) I b TZb?f ?#'j&)  c TZc?f ?%'V() ` d C dH)<$* T T2#;&K8 # f  b 3  @$ &d5`B  c $DjJ bb`B  c $DjJ  b`B  c $D) ) b`B  c $D2 2 b`B  c $D; ; b`B  c $DEEb`B  c $DNNb`B  c $DWWb`B  c $D``b`B  c $Diib`B  c $Drrb`B  c $D{{b`B  c $D G G `B  c $D > > `B  c $D 4 4 `B  c $D ++`B  c $D ""`B  c $D `B  c $D Y Y `B  c $D P P L Tq%}6 Tq%}6  BEf56 E  BDf2}W4 D  BCf0D2 C  BBf.0 B  BAf,. A  B@f+E, @  B?f) * ?  B>fW' ) >  B=fq%& =  <<fT(F@2 <`  C 1W2#;&(% 1  NZ?f ?!'57   NZ0?f ?5^ 7 0  NZ4?f ?5H7 4  NZ5?f ?%5H7 5  NZ6?f ?Oh+'0( 4@ \ h t 6Unit 23 : Measures of Central tendency and dispersionenit W.K. ChanMeEthanhathaFD1vtc6cMicrosoft Word 10.0@@vU@\EW@'DocumentSummaryInformation8PdCompObjVj՜.+,04 px  FD MathVTCT}. 6Unit 23 : Measures of Central tendency and dispersion Title  FMicrosoft Word Document MSWordDocWord.Document.89q/5/#7 6  NZ7?f ?;5/7 7  NZ8?f ?u,5 7 8  NZ9?f ?a 5"7 9Z  3 {6!K8 TB } c $DjJ  ~  BlCDEFjJgll @ C"B S  ?H0(   z !e1}t~p-;4L6 S%lT Tz*a%Ct4 %To[,%Pt;744e4<9$4l|49$z '1  '19*urn:schemas-microsoft-com:office:smarttagsplace "  +,=>89PQRS@Bu !z#}#%%%%& && &5&?&S&Z&a&h&o&y&--.........///+/-/D/11 sw  ]ai n  - 2 * 0 +,=>ps_ceuIM$!(!+%/% &&t)w),,,,%-&-C------5.9.u.x......../+/-/D///N0W0n1q1113333333333333ssssss333333333333333333333  %km 7 \ * @ U z !&48FJ}!+y!z".'9[qswy  "$(}M]euF = E o s!!0"^"""#W####G$!%+%z%S&o&8':'?'J'P'S'Y'\'b'e'j'm'r'{''(((C------*............./ ///+/-/D/Y/]/_/c/e/i/k/o/q/u/w/{/}//////////////////////00000#0%0)0+0/0105070;0=0A0C0G0I0L0N0W0Y0[0]0_0a0c0e0g0i0k0m0o0q0s0u0x0z0}0000000000000000000000001111 1 1111111 1"1%1(1+1-101215171:1<1?1A1D11111111111111111111...1 7J >,R(@p6HsB+)(dog`+ l3 i4rMsl3_6fRLlB+)AeN60HD)_g`+Bk7Ji45f<*SR(@Ms<6Ο        $#D{:|rG1b`*m*H+I`+O-f-3GHL)RyUrYqU`Tbk-otqb{H_^} , U1"2=L+8!4{.n $%*,-.3567<>?@EGHIOQRSWZ[\0i G H ? @    % - 5 = B C U Y ] b g l p t x y !&'(489:FJKLTXYZ}h}~!+;AMZ^_ghijkltuvwxy   Jg-STZbjrz{z&'9C[pqrswxy  "$()* #&)*,.02468:;F  ' + / 8 9 E x ! !! !!!$!!!!!!!!!!!!"" """ """V#W#\#f#p#y#z########F$G$Y$d$n$o$$$$$$$$$$$$$$$$$$$$$$%%%%%% %+%&''%'7'8':'>'?'B'F'G'J'O'P'S'X'Y'\'a'b'e'i'j'm'q'r'u'y'z'))))))))))))))))))))))f*g*s*}*~**********************++(+2+3+=+?+@+J+M+N+X+[+\+f+h+i+s+u+v++++-..1=0""a0u/i@jPPP P PPPPPP"#'(-(.(/1@@@@$@@@@4@@@@@@@&@P@@,@\@@4@l@@`@@6@8@@UnknownGz Times New Roman5Symbol3& z ArialC.e0}fԚPMingLiU"1Xh &j,&M9+'T'T!xx4d}.}.3QH(?} 3C:\Program Files\Microsoft Office\Templates\FD1.dot5Unit 23 : Measures of Central tendency and dispersionEthan W.K. Chanvtc\