Measures of Central Tendency

Transcription

1 Chapter 6 Measures of Cetral Tedecy Defto of a Summary Measure (page 185) A summary measure s a sgle value that we compute from a collecto of measuremets order to descrbe oe of the collecto s partcular characterstcs. 1

2 Defto of Measures of Cetral Tedecy (page 185) Notes: A measure of cetral tedecy s a sgle value that ca be used to represet all the other values the collecto. Some people refer to ths measure as the average. Ths measure tells us where the ceter of the dstrbuto les. The use of ths measure wll also facltate the comparso of two or more collectos of measuremets. Defto of the Arthmetc Mea (page 19) Defto 6.: The arthmetc mea s the sum of all the values the collecto dvded by the total umber of elemets the collecto. The populato mea for a fte populato wth N elemets, deoted by the lowercase Gree letter, s N 1 where s the measure tae from the th elemet of the populato. The sample mea for a sample wth elemets, deoted by (read as -bar ), s 1 where s the measure tae from the th elemet of the sample. N

3 Example 6.3a (page 193) Fve judges gve ther scores o the performace of a gymast as follows: 8, 9, 9, 9, ad 10. Fd the mea score of the gymast. Soluto: We compute for the populato mea. Let be the score gve by the th judge the populato. Add the scores gve by the 5 judges. 5 = = 45 1 The dvde the sum of the scores by the umber of judges. We have N = 5 judges ad get μ The mea score of the gymast s 9. Example 6.3b (page ) A cereal compay selects a sample of 8 cartos of cereals for a qualty cotrol chec o the specfed weght. The weghts grams are as follows: 56.1, 55., 55.0, 55.3, 55.9, 56., 55.8, ad Fd the mea weght for the sample of cereal cartos. Questo: Ca we compute for? Soluto: Let be the weght of the th cereal carto the sample. Addg the weghts of the 8 cereal cartos, we have, 8 = = We dvde the sum of weghts by the umber of cereal cartos the sample. We have = 8 cereal cartos ad get The mea weght of the cereals the sample s grams. Note; The computed mea s ot oe of the measuremets the collecto. 3

4 The Mea as a Ceter of Mass Fgure 6.1 (page 194) The The loads that that we we place ow o place the o the seesaw are are the the observatos,, :, : 8,9,9,9,10 7,9,9,9, To balace the seesaw, the fulcrum must fulcrum be at what must value be at of µ= µ? 9. be at what value of µ? µ=8.8 µ=06.8 What would happe f ths the tme frst f measuremet the last measuremet had bee had 7 stead bee 1000 of 8? stead of 10? Effect of a Outler o the Mea (pages ) Defto: Outlers are data values that are maredly dfferet from the rest of the data tems. Sce the mea s the ceter of mass the ts value s gravely affected by outlers. A outler wll pull the value of the arthmetc mea ts drecto ad away from the locato of majorty of the observatos. Wth the presece of outlers, the mea mght ot be a sutable measure of cetral tedecy because t may ot be a good represetatve of the observatos the collecto. 4

5 Example 6.6 (page 197) a) Let us cosder the mothly salares of 5 employees: P9,500.00, P10,00.00, P9,000.00, P10,500.00, ad P11, Fd the mea. 9,500 10,00 9,000 10,500 11,000 Soluto: P10,040/mo. 5 Questo: Is ths a good represetatve of the values the collecto? b) Suppose the mothly salary of the ffth employee s P60, stead of P11, What wll be the mea salary? Soluto: 9,500 10,00 9,000 10,500 60,000 P19,840/mo. 5 Questo: Is ths stll a good represetatve of the values the collecto? Propertes of the Mea (pages , 15) The mea s the ceter of mass. It uses all the observed values the calculato. It may or may ot be a actual observed value the data set. We may treat ts formula algebracally. Its value s gravely affected by outlers. The mea of a fte collecto always exsts ad s uque. Data values should be measured usg at least a terval scale. 5

6 Mathematcal Propertes (page 15) Theorem 1: The frst cetral momet about the N 1 1 mea s 0, that s, 0 ad 0 N Proof: ( ) Mathematcal Propertes (page 16) Theorem. 1 ( c) Ths s mmum whe s mmum whe c. Proof: ( ) ( ) c c c c c c c ( ) 1 1 ( c c ) 1 ( ) 0,,. ( c) c that s c 6

7 Mathematcal Propertes (page 16) Theorem 3: If we add a costat c to all orgal observatos, the the mea of the ew observatos wll crease by the same amout costat c. That s, c. ew orgal Proof: Orgal Data = { 1,,, } Trasformed Data = {Y 1, Y,, Y } where Y = + c Y ( c) c Y c 1 1 c c. Mathematcal Propertes (page 17) Theorem 4: If we subtract a costat c to all orgal observatos, the the mea of the ew observatos wll decrease by the same amout costat c. That s, c. ew orgal 7

8 Mathematcal Propertes (page 17) Theorem 5: If we multply a costat c to all orgal observatos, the the mea of the ew observatos s the orgal mea multpled by the costat c. That s, xc. ew orgal Proof: Orgal Data = { 1,,, } Trasformed Data = {Y 1, Y,, Y } where Y = c Y ( c ) c Y c Mathematcal Propertes (page 17) Theorem 5: If we dvde all the orgal observatos by a costat c, the the mea of the ew observatos s the orgal mea dvded by the costat c. That s, c. ew orgal 8

9 Example I a sample of 4 days, the temperature ( cetgrade) 8.5, 7.0, 31., C. 4 If observatos were coverted to Fahrehet (F= (C x 9/5) + 3)): 83.30, 80.60, 88.16, Y F. 4 OR 9 9 Y x x F. 5 5 Approxmatg the Mea for Grouped Data (FDT) page 195 Populato Mea: Sample Mea: μ 1 f N f 1 where f = the frequecy of the th class = the class mar of the th class = total umber of classes f 1 N or = total umber of observatos = 9

10 Example 6.4 (page 195) The table below gves us the weght pouds of a sample of 75 peces of luggage. Approxmate the sample mea weght of the luggage. Weght No. of Luggage Class Mar ( pouds) f f f 75 f f pouds 7 75 f 1 Modfcatos of the mea: Weghted Mea (page 00) Used whe observatos are ot of equal mportace If we assg a weght to each observato, where = 1,,,, ad s the umber of observatos the sample, the the weghted sample mea s gve by w 1 1 W W W1 1 W... W W W... W 1 10

11 Example 6.9 (pages 00 01) Suppose a govermet agecy gves scholarshp grats to employees tag graduate studes. Courses graduate studes ear credts of 1,, 3, 4, or 5 uts. They ca get a partal scholarshp for the ext semester f they get a weghted average of 1.5 to 1.75 ad a full scholarshp f the average s better tha 1.5, whch meas a average of 1.0 to What d of scholarshp wll the employees get gve ther grades for the prevous semester? Employee A Employee B Subjects Uts Grade Subjects Uts Grade A A 1.0 B 1.5 B 1.75 C C D D E 5.0 E Soluto: We let the uts be the weghts W ad the grades be the. (1)(1) ()(1.5) (3)(1.5) (4)(1.75) (5)() 5 Weghted average of employee A: W Weghted average of employee B: W (1)() ()(1.75) (3)(1.5) (4)(1.5) (5)(1) Modfcatos of the Mea: Combed Mea or the Mea of Meas (page 01 0) Suppose that fte populatos havg N1,N,...,N measuremets, respectvely, have meas μ 1,μ,...,μ. The combed populato mea, c, f we combe the measuremets of all the populatos s μ c 1 N μ N1μ1 N μ... N μ N1 N... N N 1 If samples of sze 1,,...,, selected from these populatos, have the sample meas,, respectvely, the combed sample mea, c, f we combe the measuremets all the samples s c

12 Example 6.10 (page 0) Three sectos of a statstcs class cotag 8, 3, ad 35 studets averaged 83, 80, ad 76, respectvely, o the same fal examato. What s the combed populato mea for all 3 sectos? Soluto: We let N 1 = 8, N = 3, N 3 = 35, 1 = 83, = 80, 3 = 76 C (8)(83) (3)(80) (35)(76) Thus, the mea grade of the 3 sectos s Modfcatos of the Mea: Trmmed Mea (page 0) Objectve: remove the fluece of possble outlers Choose (Gree letter alpha: the proporto of observatos that wll be deleted), 0 < < 1. To fd the (/)(100)%-trmmed mea for a gve data set, we frst order the data accordg to magtude, say, lowest to hghest. The, we remove (/)(100)% of the observatos both the lower ad upper eds of the array. We the calculate the arthmetc mea for the remag observatos. 1

13 Example 6.11 (pages 0-03) Compute for the arthmetc mea ad 5% trmmed mea for the gve data set: Soluto: The arthmetc mea s We wll otce that the values 500 ad 54 are outlers. These outlers pulled the value of the mea ther drecto. As a result, the computed mea of s ot a good represetatve of the observatos the data set because t s so much hgher tha the values of majorty of the observatos. Example 6.11 cot d (pages 0-03) To compute for the 5% trmmed mea, we delete the bottom 5% ad the top 5% of the observatos the array. Sce there are 40 observatos ad 5% of 40 s the we would have to delete the frst two observatos the array ad the last two observatos. Ths wll leave us wth 36 observatos. These are: The 5% trmmed mea s the mea of these remag observatos. Thus, we have Ths value s a better summary measure to represet the observatos the orgal data set. 13

14 Assgmet (pages 04-06) Exercse 3. (No eed to terpret.) Exercse 5. Exercse 6. Exercse 8. (No eed to compare.) If you were ased to compute for the weghted mea, what do you suggest that we use as weghts? Defto of the Meda (page 06) Defto 6.3. The meda dvdes the array to two equal parts. 14

15 Fdg the Meda (page 06-07) Step 1: Arrage the observatos a array. We let () be the th observato the array, where = 1,,...,. Thus, (1) s the smallest observato whle () s the largest observato. Step : Determe the meda, Md. Case 1: If the umber of observatos s odd, Md 1 Case : If the umber of observatos s eve, Md 1 Example 6.13a (page 07) The followg are the total recepts of 7 mg compaes ( mllo pesos): Soluto: Array: Notato: (1) () ( 3) (4) ( 5) (6) (7) Sce =7 s odd, Md = 1 = 7 1 How may observatos are to the left of the meda? rght of the meda? = ( 8/) = ( 4 ) = 7.3 mllo pesos How may observatos are to the A meda of 7.3 mllo pesos dcates that compaes wth total recept of less tha 7.3 mllo pesos belog the lower half of the array; whereas, compaes wth total recept greater tha 7.3 mllo pesos belog the upper half of the array. 15

16 Example 6.13b (pages 07-08) The followg are the umber of years of operato of 8 mg compaes: Soluto: Arrage the observatos from lowest to hghest. Array: Notato: (1) () (3) ( 4) (5) ( 6) (7) ( 8) Two mddle observatos Sce =8 s eve, Md = = 4 (5) = 16.5 How may observatos are to the left of the meda? How may observatos are to the rght of the meda? A meda of 16.5 dcates that compaes operatg for less tha 16.5 years belog the lower half of the array whle compaes operatg for more tha 16.5 years belog the upper half of the array. Iterpretato of the Meda (page 08) The exact terpretato of the meda s as follows: At least half of the observatos are less tha or equal to the meda ad at the same tme at least half of the observatos are greater tha or equal to the meda. Ths geeral terpretato ca hadle all types of data sets, cludg those wth ted values the mddle of the array. Example: =1 Array: 3, 4, 4, 4, 4, 5, 5, 5, 5, 7, 8, 9 Meda = 5 How may are less tha 5? How may are greater tha 5? A meda of 5 meas that at least half of the observatos are less tha or equal to 5 ad at the same tme at least half are greater tha or equal to 5. The terpretato wll smplfy to half of the observatos are less tha the meda ad half are greater tha the meda f the meda s ot oe of the observed values, that s, s eve ad there are o tes (see prevous example). 16

17 Effect of Outlers o Meda Example 6.16 (page 11) Let us cosder the data set Example 6.6 o the mothly salares of 5 employees ( pesos): 9,500 10,00 9,000 10,500 60,000 The mea mothly salary s P19,840. Let us ow determe the meda. Array: Notato: (1) () (3) (4) (5) Md 10, The meda mothly salary s P10,00. Whch s a better measure of cetral tedecy? (3) Characterstcs of the Meda (pages 10, 15) The meda s the ceter of the array. The meda s also a measure of posto/locato. A observato whose value s smaller tha the meda belogs the lower half of the array whle a observato whose value s hgher tha the meda belogs the upper half of the array. Ule the mea, t uses oly the mddle value/s the array for ts computato. Ule the mea, the meda s ot affected by outlers (observatos whose values are extremely dfferet from the others the data set). Ule the mea, the meda s ot ameable to algebrac mapulato. Ule the mea, the meda s stll terpretable whe the level of measuremet s as low as ordal. The meda wll always exst ad s uque. 17

18 Approxmatg the Meda for Grouped Data (FDT) (page 09) Step 1. Calculate /, where s the umber of observatos= f 1 Step. Costruct the < cumulatve frequecy dstrbuto (< CFD). Step 3. Startg from the top, locate the value the <CFD colum that s greater tha or equal to / for the frst tme. The class terval correspodg to that value s the meda class. Step 4. Approxmate the meda usg the formula gve below. / CFD Md = LCB Md + C Md1 f Md where LCB Md s the lower class boudary of the meda class C s the class sze of the meda class s the umber of observatos <CF Md-1 s the less tha cumulatve frequecy of the class precedg the meda class f Md s the frequecy of the meda class Example (page 10) The table below gves us the weght pouds of a sample of 75 peces of luggage. Approxmate the meda weght of luggage. Weght Class Boudares No. of Luggage < CF ( pouds) LCB - UCB f The meda class s sce /= 75/ =37.5 ad the <CF of the terval, , s 53 whch s the value that s greater tha or equal to 37.5 for the frst tme from the top. / CFD Md LCB Md + C Md 1 = f Md 3 18

19 Ratoale / Less tha Ogve A C B Md Cosder the pots A, B ad C. Pot A: (UCB Md-1, <CFD Md-1 ) Pot B: (Md, /) Pot C: (UCB Md, <CFD Md ) These three pots all belog o the same le ad the slope of ths le ca be determed by ay two of these three pots. Thus, equatg the formula for the slope usg Pots A ad B ad the formula for the slope usg Pots A ad C, we have: / CFDMd CFDMd CFDMd Md UCBMd 1 UCBMd UCBMd 1 / CFDMd 1 fmd Md LCB C 1 1 Md Md LCB / CFDMd C 1 Md fmd Defto of the Mode (page 11) The mode s the observed value that occurs wth the greatest frequecy a data set. 19

20 Example 6.17 (pages 11-1) Determe the mode. b) Gve the umber of dogs owed by 3 studets: 0, 0, 0, 1, 1, 1, 1,,,,,,,,,,,,,,, 3, 3 d) Cosder the scores of 15 studets a quz: 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 0, 0, 0 c) Cosder the shoe sze of 4 faculty members: 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11 Multmodalty Detecto of multmodalty ca provde sghts to the uderlyg ature of the data set. For example: A metal alloy maufacturer was cocered wth customer complats about the lac of uformty the meltg pots of oe of the frm's alloy flamets. Sxty flamets were selected from the producto process ad ther meltg pots determed. The resultg frequecy polygo showed that the dstrbuto s bmodal. Closer examato of the data revealed that 6 of the flamets were produced by the frst shft ad the other 34 by the secod shft. The frequecy polygos for each shft were costructed. Ths revealed that the bmodal ature of the orgal frequecy dstrbuto s the result of a dfferece the postos of the dstrbutos for the two shfts. They the dscovered that the secod shft was usg the wrog alloy mxture specfcato. Correcto of the mxture resulted producto of flamets wth more uform meltg pots. Number of Flamets Number of Flamets Etre Data Set Frst shft Meltg Pot ( Celcus) Separato by Shft Meltg Pot ( Celcus) Secod shft 0

21 Example 6.0 (page 14) Gve the frequecy dstrbuto of the cvl status data of 4 employees, what s the mode? Cvl Status Number of Faculty Sgle.. 15 Marred 8 Wdowed.. 0 Separated.. 1 Total. 4 Characterstcs of the Mode (pages 19,1) 1. The mode s the ceter the sese that t s the most typcal value a set of observatos.. Outlers do ot affect the mode. 3. The mode wll ot always exst; ad f t does, t may ot be uque. A data set s sad to be umodal f there s oly oe mode, bmodal f there are two modes, trmodal f there are three modes, ad so o. 4. The value of the mode s always oe of the observed values the data set. 5. We ca get the mode for both quattatve ad qualtatve types of data; that s, the mode s terpretable eve f the level of measuremet s as low as omal. 6. The mode s geerally ot as useful a measure of cetral tedecy as the mea ad the meda whe the data cosst of oly a few umbers. For example, for the umbers 7, 1, 18,, 31, 31, the mode s 31 sce t appears twce ad all other umbers oly oce. But 31 caot be cosdered a good measure of cetral tedecy for these data sce t s fact at the extreme hgh ed of the values ad ts frequecy exceeds the frequecy of the other values by oly 1. 1

22 Approxmatg the Mode for Grouped Data (page 1) Step 1. Locate the modal class. For frequecy dstrbuto wth equal class szes, the modal class s the class terval wth the hghest frequecy. Otherwse, compute for adjusted frequeces frst. Step. Approxmate the mode usg the followg formula: f Mo f1 Mo = LCB Mo + C f Mo f1 f where LCB Mo C f Mo f 1 f s the lower class boudary of modal class s the class sze s the frequecy of the modal class s the frequecy of the class precedg the modal class s the frequecy of the class followg the modal class Example 6.18 (page 13) The table below gves us the weght pouds of a sample of 75 peces of luggage. Approxmate the modal weght of the luggage. Weght Class Boudares No. of Luggage ( pouds) LCB - UCB f Modal class f Mo f1 Mo = LCB Mo + C f Mo f1 f 3 4 = (3) 4 14

23 Ratoale A F C A E B A G A AEC ad BED are smlar tragles. Thus, AC EF BD EG f Mo f1 Mo LCBMo fmo f LCB Mo ( f f )( LCB Mo) ( Mo LCB )( f f ) Mo 1 Mo Mo ( LCB )( f f1) ( LCBMo )( f f) Mo(( f f) ( f f1)) Mo Mo Mo Mo ( LCBMo C)( fmo f1) ( LCBMo )( f Mo f ) Mo( fmo f1 f ) C( f f ) ( LCB )( f f f ) Mo( f f f ) Mo 1 Mo Mo 1 Mo 1 D C( f Mo f1) LCB ( f f f ) Mo 1 Mo Mo Mo Assgmet 1. Usg the data o heght ( meters) of a sample of 50 trees page 183, o. 5, determe the mea, meda ad mode.. Usg the dstrbuto of scores page 05, o. 4, approxmate the meda ad the mode for the sample of grls ad the sample of boys. 3