17/03/015 School of Psychology Dpt. Expermetal Psychology Desg ad Data Aalyss Psychology I Eglsh group (A) Salvador Chacó Moscoso Susaa Saduvete Chaves Mlagrosa Sáchez Martí Lesso 3 Group ad dvdual dexes 1
17/03/015 Group ad dvdual dexes Group dexes Idvdual dexes Cetral tedecy Varablty (Dsperso) Bas or Skewess (Asymmetry) Posto Decles (D ) Percetles (P ) Quartles (Q ) Raw scores ( ) Kurtoss Dfferetal scores (x ) Stadard scores (Z ) 3 To descrbe a data dstrbuto we eed at least two statstcs: 1. Oe that reflects the cetral tedecy: value whch represets the group.. Aother that reflects the dsperso aroud ths ceter. It determes how far or together the data are from each other. 4
17/03/015 Whch value does represet the group? Cetral tedecy dexes Mode Meda Mea 5 Cetral tedecy dexes 6 3
17/03/015 Defto They are a bref descrpto of a mass of data, usually obtaed from a sample. They serve to descrbe, drectly, the populato from whch the sample was extracted. 7 1. Whch s the most repeated value? Mode (Mo) 8 4
17/03/015 1. Mode (Mo) Defto: The most repeated value. The value most frequetly observed a sample or populato. The value of a varable wth the hghest absolute frequecy. It s symbolzed by Mo (Fecher ad Pearso) 9 1. Mode (Mo) 1.1. Type I dstrbutos: small data set a) Umodal dstrbuto: there s oly oe mode. Example: 8 8 11 11 15 15 15 15 15 17 17 17 19-19 10 5
17/03/015 1. Mode (Mo) 1.1. Type I dstrbutos: small data set a) Umodal dstrbuto: Example: 8 8 11 11 15 15 15 15 15 17 17 17 19-19 Mo 15 5 4,5 4 3,5 3,5 1,5 1 0,5 0 5 3 8 11 15 17 19 11 1. Mode (Mo) 1.1. Type I dstrbutos: small data set b) Amodal dstrbuto: more tha 80% of the values of preset the hghest absolute frequecy. Example: 8 8 8 11 11 11 15 15 15 17 17 17 19 19 19 1 6
17/03/015 1. Mode (Mo) 1.1. Type I dstrbutos: small data set b) Amodal dstrbuto: Example: 8 8 8 11 11 11 15 15 15 17 17 17 19 19 19 3 3 3 3 3 3,5 Wthout mode 1,5 1 0,5 0 8 11 15 17 19 13 1. Mode (Mo) 1.1. Type I dstrbutos: small data set c) Bmodal dstrbuto: There are two modes. Example: 8 9 9 10 10 10 10 11 11 13 13 13 13 15 14 7
17/03/015 1. Mode (Mo) 1.1. Type I dstrbutos: small data set c) Bmodal dstrbuto: Example: 8 9 9 10 10 10 10 11 11 13 13 13 13 15 Mo 1 10 Mo 13 4 3,5 3,5 1,5 1 0,5 0 4 4 1 1 8 9 10 11 13 15 15 1. Mode (Mo) 1.1. Type I dstrbutos: small data set d) Multmodal dstrbuto: there are more tha two modes. Example: 8 8 9 9 9 10 11 11 11 1 1 13 13 13 14 15-15 16 8
17/03/015 1. Mode (Mo) 1.1. Type I dstrbutos: small data set d) Multmodal dstrbuto: Mo 11 Example: 8 8 9 9 9 10 11 11 11 1 1 13 13 13 14 15-15 Mo 1 9 3 3 3 3 Mo 3 13,5 1,5 1 0,5 1 1 0 8 9 10 11 1 13 14 15 17 1. Mode (Mo) 1.. Type II dstrbutos: bg data set Frequecy table a) Umodal dstrbuto: example: f 1 3 14 5 16 10 18 5 18 9
17/03/015 1. Mode (Mo) 1.. Type II dstrbutos: bg data set Frequecy table a) Umodal dstrbuto: example: Mo 14 f 1 3 14 5 MOST REPEATED VALUE 16 10 18 5 19 1. Mode (Mo) 1.. Type II dstrbutos: bg data set b) Bmodal dstrbuto: example f 10 4 5 6 10 8 7 0 10
17/03/015 1. Mode (Mo) 1.. Type II dstrbutos: bg data set b) Bmodal dstrbuto: example: Mo 1 ad Mo 6 f 10 MOST REPEATED VALUES 4 5 6 10 8 7 1 1. Mode (Mo) Example: Complete the table kowg that the modes are: -, -1 ad 5; ad f 3 f 4 f rf % - 5-1 0 O.1 5 0 6 11
17/03/015 1. Mode (Mo) Example: soluto f rf % - 5 0. 0-1 5 0. 0 0 3 O.1 1 3 0.1 1 5 5 0. 0 6 4 0.16 16 Σ 5 1 100 3. Whch s the value exceeded by the half of the partcpats? Meda (Md) 4 1
17/03/015. Meda (Md) Deftos: It s the pot of the dstrbuto that dvdes t to equal parts. It s the value wth the property that the umber of observatos smaller tha tself s equal to the umber of observatos hgher tha tself. It s the value that holds the cetral pot of a ordered seres of data. The 50% of the values s above ad the other 50% s below t. 5. Meda (Md).1. Graphc represetato The Md s ot defed as a data or partcular measure, but a pot (a value). A pot whose value does ot ecessarly have to match ay observed value. 50% 50% Md 6 13
17/03/015. Meda (Md).. No-grouped data Type I dstrbutos: small data set ODD data set: Example: 7 11 6 5 7 1 9 8 10 6 9 1st) Data s sorted from the lowest to the hghest. d) Cetral value s obtaed: +1 7. Meda (Md).. No-grouped data Type I dstrbutos: small data set ODD data set: 1st) Data s sorted from the lowest to the hghest: 5 6 6 7 7 8 9 9 10 11 1 d) Cetral value s obtaed: + 1 11+ 1 6 8 14
17/03/015. Meda (Md).. No-grouped data Type I dstrbutos: small data set 1º 5 º 6 3º 6 4º 7 5º 7 6º 8 7º 9 Md 8 8º 9 9º 10 10º 11 11º 1 9. Meda (Md).. No-grouped data Type I dstrbutos: small data set EVEN data set: Example 3 35 43 9 34 41 33 38 38 3 1st) Data s sorted from the lowest to the hghest: d) Md CetralValue + CetralVal 1 ue 30 15
17/03/015. Meda (Md).. No-grouped data Type I dstrbutos: small data set EVEN data set: 1st) Data s sorted from the lowest to the hghest: 3 9 3 33 34 35 38 38 41 43 d) Md 34 + 35 34.5 31. Meda (Md).. No-grouped data Type I dstrbutos: small data set Md 34.5 1º º 3º 4º 5º 6º 7º 8º 9º 10º 3 9 3 33 34 35 38 38 41 43 3 16
17/03/015. Meda (Md).3. Grouped data Type II dstrbutos: bg data set Procedure: 1. Calculate F : cumulatve frequeces. Calculate / 3. Determe L : the lower exact lmt from the terval that cludes / (the 50% of the observed data) Exact lmts value ± 0.5 x measuremet ut 4. Determe the f that terval 5. Determe the F before that terval 6. Calculate the terval ampltude: I max m (exact lmts of the terval) 7. Calculate the formula: I Md L + F f f 16-1 4-7 9 8-33 1 34-39 40-45 16 46-51 11 5-57 7 58-63 8 64-69 33. Meda (Md).3. Grouped data Type II dstrbutos: bg data set 1. F. / 50 3. L : 34 0.5 x 1 33.5 4. f 4 5. F 3 34 6. I 39.5 33.5 6 7. 6 100 Md 33.5 + 34 37.86 f 1 F 16-1 4 4-7 9 13 8-33 1 34 34-39 56 40-45 16 7 46-51 11 83 5-57 7 90 58-63 8 98 64-69 100 34 17
17/03/015 3. Whch s the average score? Arthmetc mea ( ) 35 3. Arthmetc mea Defto: It s the cetral tedecy dex most commoly used. It s the sum of all the observed values dvded by the total umber of them. 36 18
17/03/015 3. Arthmetc mea 3.1. No-grouped data Type I dstrbutos: small data set k 1 + + +... + 1 3 k Example: The 10 umbers below are the tems remembered by 10 chldre a mmedate memory task 6 5 4 7 5 7 8 6 7-8 Calculate the arthmetc mea 37 3. Arthmetc mea 3.1. No-grouped data Type I dstrbutos: small data set 10 1 6 + 5 + 4 + 7 + 5 + 7 + 8 + 6 + 7 + 8 63 10 10 6. 3 38 19
17/03/015 3. Arthmetc mea 3.1. No-grouped data Type I dstrbutos: small data set Example: 3 10 8 4 7 6 9 1 4 Calculate the mea 39 3. Arthmetc mea 3.1. No-grouped data Type I dstrbutos: small data set 10 1 3 + 10 + 8 + 4 + 7 + 6 + 9 + 1 + + 4 10 65 10 65. 40 0
17/03/015 3. Arthmetc mea 3.. Grouped data Type II dstrbutos: bg data set Possblty 1: MEAN f f + f + f +... + f 1 1 3 3 k k FREQUENCY TABLE 41 3. Arthmetc mea 3.. Grouped data Type II dstrbutos: bg data set f 0 3 1 6 7 3 3 4 1 4 1
17/03/015 3. Arthmetc mea 3.. Grouped data Type II dstrbutos: bg data set f f 0 3 3 x 0 0 1 6 6 x 1 6 7 7 x 14 3 3 3 x 3 9 4 1 1 x 4 4 f (3x0) + (6x1) + (7x) + (3x3) + (1x 4) 33 1.65 0 0 43 3. Arthmetc mea 3.. Grouped data Type II dstrbutos: bg data set Possblty : f rf Example: f 0 3 1 6 7 3 3 4 1 44
17/03/015 3. Arthmetc mea 3.. Grouped data Type II dstrbutos: bg data set rf 0 + 0.30 + 0.70 + 0.45 + 0.0 1. 65 f rf rf 0 3 0.15 0.00 1 6 0.30 0.30 7 0.35 0.70 3 3 0.15 0.45 4 1 0.05 0.0 0 1 1.65 45 3. Arthmetc mea 3.. Grouped data Type II dstrbutos: bg data set f rf Calculate the arthmetc mea usg the two formulas Itervals f 64-69 58-63 8 5-57 7 46-51 11 40-45 16 34-39 8-33 1-7 9 16-1 4 Example: 46 3
17/03/015 3. Arthmetc mea 3.. Grouped data Type II dstrbutos: bg data set Itervals f f rf rf 64-69 66.5 133 0.0 1.33 58-63 60.5 8 484 0.08 4.84 5-57 54.5 7 381.5 0.07 3.815 46-51 48.5 11 533.5 0.11 5.335 40-45 4.5 16 680 0.16 6.8 34-39 36.5 803 0. 8.03 8-33 30.5 1 640.5 0.1 6.405-7 4.5 9 0.5 0.09.05 16-1 18.5 4 74 0.04 0.74 100 3950 1 39.5 47 3. Arthmetc mea 3.. Grouped data Type II dstrbutos: bg data set Possblty 1: f 3950 39.5 100 Possblty : rf 39.5 48 4
17/03/015 3. Arthmetc mea 3.3. Propertes: 1. Sum of the devato of all the values from ther arthmetc mea s 0: ( ) 0 Example: 9, 3, 6, 7, 5 49 3. Arthmetc mea 3.3. Propertes: Example: 30 5 6 ( ) 9 3 3-3 6 0 7 1 5-1 ( ) 3 + ( 3) + 0 + 1 + ( 1) 0 50 5
17/03/015 3. Arthmetc mea 3.3. Propertes:. Sum of the devatos square wth regard to the arthmetc mea s less tha wth regard to ay other value or average: Example: c 3 ( ) < ( ( c ) 9, 3, 6, 7, 5 c) 51 3. Arthmetc mea 3.3. Propertes: 3 ( + ( 3) ) + 0 (9 6) + 1 + ( 1) + (3 6) + (6 6) + (7 6) 9 + 9 + 0 + 1+ 1 0 + (5 6) 6 ( + (0) c) + 3 (9 3) + 4 + () + (3 3) + (6 3) + (7 3) 36 + 0 + 9 + 16 + 4 65 + (5 3) ( ) < ( c) 0 < 65 5 6
17/03/015 3. Arthmetc mea 3.3. Propertes: 3. If every value of the varable s creased by a costat, the arthmetc mea wll be creased by the same costat: ( + a) a + Example: a 9, 3, 6, 7, 5 53 3. Arthmetc mea 3.3. Propertes: +a 9 11 3 5 6 8 7 9 5 7 Σ30 Σ40 ( + a) 40 8 5 + a 6 + 8 ( + a) + a 8 8 54 7
17/03/015 3. Arthmetc mea 3.3. Propertes: 4. If every value of the varable s multpled by a costat, the arthmetc mea wll be multpled by the same costat: Example: a4 ( * a) a * 9, 3, 6, 7, 5 55 3. Arthmetc mea 3.3. Propertes: *a 9 36 3 1 6 4 7 8 5 0 Σ30 Σ10 ( * a) 10 4 5 * a 6*4 4 ( * a) * a 4 4 56 8
17/03/015 3. Arthmetc mea 3.3. Propertes: 5. If 1 ad are the meas of the two groups computed from the values 1 ad, the the mea s gve by the formula: 1 1 + + 1 57 3. Arthmetc mea 3.3. Propertes: Example: Three groups wth 3, 4 ad 5 partcpats respectvely are coformed order to check the effect of a drug a perceptve task. The frst group receve a placebo; the secod group, 1 mg of the drug; ad the thrd oe, mg. The results are preseted the table below: Group 1 Group Group 3 4 3 1 6 5 5 4 4 0 Calculate the arthmetc mea usg the two formulas below: 1 1 + + 3 + + 1 3 3 1 58 9
17/03/015 3. Arthmetc mea 3.3. Propertes: Group 1 Group Group 3 4 3 1 6 5 5 4 4 0 1 Σ15 Σ1 Σ10 15 3 1 1 4 10 5 3 5 3 + + + + 1 3 3x5 + 4x3 + 5x 15 + 1 + 10 37 3 + 4 + 5 1 1 1 1 3 3 3.083 4 + 6 + 5 + 3 + 5 + 4 + 0 + 1+ + 4 + + 1 37 3.083 1 1 59 3. Arthmetc mea 3.3. Propertes: 6. The arthmetc mea of a varable that s a leal combato of other varables s equal to the leal combato of the arthmetc meas of those varables. Whether: The: a1 1 + a +... + a k k a 1 + a +... + a k k 1 60 30
17/03/015 3. Arthmetc mea 3.3. Propertes: Example: The data preseted the table below are the scores obtaed by 4 partcpats three tellgece tests ( 1, ad 3 ): 1 3 4 5 3 5 6 4 6 4 3 5 1 From these varables, a ew oe s coformed: 1 - +3 3. Calculate: a) The value of for each partcpat. b) The arthmetc mea of the varable usg the two formulas below: a + a +... + a k 1 1 k 61 3. Arthmetc mea 3.3. Propertes: 1 3 1-3 3 a) 4 5 4-4 15 15 3 5 6 6-5 18 19 4 6 4 8-6 1 14 3 5 1 6-5 3 4 Σ1 Σ0 Σ16 Σ5 1 4 0 4 16 4 1 3 3 5 4 a 1 + a +... + a k 1 + 3 3 x3 5 + 3x4 1 6 5 + 1 13 k 5 13 4 6 31
17/03/015 Comparso betwee measures of cetral tedecy We usually prefer the mea: Other statstcs are based o the mea. It's the best estmator of ts parameter. We oly prefer the meda: Whe the varable s ordal. Whe there are very extreme data. Whe there are ope tervals. We oly prefer the mode: Whe the varable s qualtatve or omal. Whe the ope terval cludes the meda. 63 Example 1 The degree of agreemet cosderg "shoutg" as a sg of aggresso a sample s preseted the table below: f 1 3 6 3 5 4 6 5 1 Calculate the most approprate cetral tedecy dex. 64 3
17/03/015 Example 1 f F 1 3 3 6 9 3 5 14 4 6 0 5 1 3 Σ 3 + 1 3 + 1 33 16.5 Cetral values correspodg to postos 16th ad 17th cetralvalue + cetralvalue Md 4 + 4 1 4 65 Example The table below represets the umber of rtuals that studets do before a exam: f 1 3 8 4 10 5 1 6 19 7 9 8 9 1 1. Calculate the mode, the meda ad the arthmetc mea.. Whch s the most approprate cetral tedecy dex? 66 33
17/03/015 Example f F f 1 1 3 8 9 4 4 10 19 40 5 1 40 105 6 19 59 114 7 9 68 63 8 70 16 9 1 71 9 Σ 71 373 1. Mo 5 Md: + 1 71+ 1 7 36 Posto 36th Md 5 373 f 5.53 71. The most approprate cetral tedecy dex s the arthmetc mea (for quattatve varables). 67 Group ad dvdual dexes Group dexes Idvdual dexes Cetral tedecy Varablty (Dsperso) Bas or Skewess (Asymmetry) Posto Decles (D ) Percetles (P ) Quartles (Q ) Raw scores ( ) Kurtoss Dfferetal scores (x ) Stadard scores (Z ) 68 34
17/03/015 Quatles Md Quartles Percetles Decles 69 Posto dexes 70 35
17/03/015 Posto measures are used to provde formato about the relatve posto of a case wth respect to ts data set. They are dvdual dexes. 71 Quatles Defto: They dvde the dstrbuto k parts wth the same amout of data. Md: dvdes the dstrbuto parts: k Quartles (Q ): dvde the dstrbuto 4 parts: Q 1, Q, Q 3 : k 4 Decles (D ): dvde the dstrbuto 10 parts : D 1, D,..., D 9 : k 10 Percetles or Cetles(P or C ): dvde the dstrbuto 100 parts: P 1, P,..., P 99 : k 100 7 36
17/03/015 Quatles Graphc represetato 5% 5% 5% 5% D 1 P 10 D D3 D4 D5 D6 D7 D8 D9 P0 P30 P40 P50 P60 P70 P80 P90 Q 1 Q Q 3 M d C 50 P 50 Q D 5 Md 73 Percetle. Defto Percetle rak dcates the partcpat's stadg relatve to other partcpats the orm group. For example, a studet's percetle rak o a orm-refereced test tells us what proporto of studets the orm group scored the same or lower tha a target studet. 74 37
17/03/015 Procedure 1. Posto: ( + 1) k I + D the quartle, decle or percetle study. k umber of dvsos. I teger. D decmals.. Value: I + D( I+1 - I ) I value of the posto of the teger part. I+1 value of the followg posto to the teger part. 75 Example 1: Q 3 f 3 3 11 4 15 5 30 6 3 7 1 8 4 9 1 76 38
17/03/015 Example 1: Q 3 1. Posto: ( + 1) 3(99 + 1) 75 k 4. Value: Q 3 6 (the result s a teger) f F Postos 3 3 1 3 3 11 14 4 14 4 15 9 15 9 5 30 59 30 59 6 3 8 60 8 7 1 94 83 94 8 4 98 95 98 9 1 99 99 77 Examples ad 3: D 6 ad P 13 f 1 3 8 4 10 5 1 6 19 7 9 8 9 1 78 39
17/03/015 Example : D 6 1. Posto: ( + 1) 6(71 + 1) 43. k 10. Value: D 6 6 f F Postos 1 1 1 3 8 9 9 4 10 19 10 19 5 1 40 0 40 6 19 59 41 59 7 9 68 60 68 8 70 69 70 9 1 71 71 79 Example 3: P 13 1. Posto: ( + 1) 13(71+ 1) 9.36 k 100. Value: 9 I; 0.36 D I + D( I+1 - I ) P 13 3 + 0.36(4-3)3.36 f F Postos 1 1 1 3 8 9 9 4 10 19 10 19 5 1 40 0 40 6 19 59 41 59 7 9 68 60 68 8 70 69 70 9 1 71 71 80 40
17/03/015 Example 4: C 50 P 50 Q D 5 Md f 1 30 15 3 5 4 6 5 4 81 Example 4: C 50 P 50 Q D 5 Md f F 1 30 30 15 45 3 5 70 4 6 76 5 4 80. Value: C 50 P 50 Q D 5 Md 1. Posto: C 50 P 50 ( + 1) 50(80 + 1) 40.5 k 100 Q ( D 5 + k 1) (80 + 1) 40.5 4 ( + 1) 5(80 + 1) 40.5 k 10 Md ( + k 1) 1(80 + 1) 40.5 8 41
17/03/015 Example 5: C 5, C 75, C 38, C 90, C 95 f 1 30 15 3 5 4 6 5 4 83 Example 5: C 5, C 75, C 38, C 90, C 95 C 5 1 f F ( + k 1) 5(80 + 1) 100 05 100 0.5 1 30 30 15 45 3 5 70 4 6 76 5 4 80 C 75 3 ( + 1) 75(80 + 1) k 100 6075 100 60.75 C 38 I + D( I+1 - I )1+0.78(-1)1.78 ( + k 1) 38(80 + 1) 100 3078 100 30.78 84 4
17/03/015 Example 5: C 5, C 75, C 38, C 90, C 95 f F 1 30 30 15 45 3 5 70 4 6 76 C 90 4 ( + 1) 90(80 + 1) k 100 790 100 7.9 C 95 I + D( I+1 - I ) 4+0.95(5-4)4.95 ( + k 1) 95(80 + 1) 100 7695 100 76.95 5 4 80 85 Example 6: The tables below preset the results obtaed from a sample of wome ad me the provce of Sevlle a psychologcal test: Wome Me f f 19 5 0 4 0 15 1 8 1 6 4 10 5 6 1 5 1 7 8 9 8 9 8 Calculate Q 3 wome sample ad P 84 me sample 86 43
17/03/015 Example 6: Wome f F 19 5 5 ( + 1) k 3(60 + 1) 4 183 4 45.75 0 15 40 1 6 46 5 51 Q 3 1 5 1 5 9 8 60 87 Example 6: Me f F 0 4 4 ( + 1) k 84(50 + 1) 100 484 100 4.84 1 8 1 4 10 6 1 34 7 8 4 P 84 I + D( I+1 - I )7+0.84(9-7) 7+0.84x7+1.688.68 9 8 50 88 44
17/03/015 Procedure whe there are tervals It s aalogous to the Md calculato. Steps: 1. Calculate F : cumulatve frequeces. Calculate */k 3. Determe L : the lower exact lmt from the terval that cludes */k Exact lmts value ± 0.5 x measuremet ut 4. Determe the f that terval 5. Determe the F before that terval 6. Calculate the terval ampltude: I max m (exact lmts of the terval) 7. Calculate the formula: I * P, C, D, Q L + F f k 89 Example: Itervals f 16-1 4-7 9 8-33 1 34-39 40-45 16 46-51 11 5-57 7 58-63 8 64-69 Calculate C 90 90 45
17/03/015 Example: Itervals f 1 F 16-1 4 4-7 9 13 8-33 1 34 34-39 56 40-45 16 7 46-51 11 83 5-57 7 90 58-63 8 98 64-69 100 1. F. */k 90 x 100/100 90 3. L : 5 0.5 x 1 51.5 4. f 7 7 5. F 6 83 6. I 57.5 51.5 6 7. C 90 I * L + F f k 6 90 *100 51.5 + 83 7 100 51.5 + 0.857 (90 83) 51.5 + 0.857 * 7 51.5 + 5.99 57.499 91 Group ad dvdual dexes Group dexes Idvdual dexes Cetral tedecy Varablty (Dsperso) Bas or Skewess (Asymmetry) Posto Decles (D ) Percetles (P ) Quartles (Q ) Raw scores ( ) Kurtoss Dfferetal scores (x ) Stadard scores (Z ) 9 46
17/03/015 How s the data arraged wth respect to the dstrbuto ceter? How far or together are the data from each other? Varablty or dsperso dexes Total ampltude (Ta) Mea devato (MD) Varace (S ) Stadard devato (S) Semterquartle ampltude (Q) Pearso varato coeffcet (VC) 93 Varablty or dsperso dexes 94 47
17/03/015 Itroducto A 4 10 1 14 0 B 10 11 1 13 14 C 104 110 11 114 10 A 1 B 1 C 11 V V V A B C A C B 95 Itroducto A 4 10 1 14 0 B 10 11 1 13 14 C 104 110 11 114 10 96 48
17/03/015 Itroducto 5 5 5 5 5 5 5 5 5 5 V 0 5 97 Itroducto 1 1 1 5 5 8 8 9 10 V 0 1 5 8 9 10 98 49
17/03/015 Itroducto INDEES T A MD S S Q 1. TOTAL AMPLITUDE. MEAN DEVIATION 3. VARIANCE 4. STANDARD DEVIATION 5. SEMIINTERQUARTILE AMPLITUDE vc 6. PEARSON VARIATION COEFFICIENT 99 1. Total ampltude (or rage) Defto: It s the dstace betwee the maxmum ad mmum value of a data set. T A Max M Advatage: easy to calculate. Dsadvatages: It s ustable. It oly uses two data from the sample, so t s very sestve to extreme values ad sestve to average values. It s ot depedet of the sample szes (T A obtaed samples of dfferet szes are ot drectly comparable). 100 50
17/03/015 1. Total ampltude (or rage) Example: A) 3 7 8 9 10 11 1 13 B) 7 7 8 9 10 11 1 13 C) 7 10 10 10 10 10 10 13 101 1. Total ampltude (or rage) Example: A) 3 7 8 9 10 11 1 13 T A 13 3 10 B C B) 7 7 8 9 10 11 1 13 T A 13 7 6 A > B ad C C) 7 10 10 10 10 10 10 13 T A 13 7 6 10 51
17/03/015 1. Total ampltude (or rage) A) 3 7 8 9 10 11 1 13 B) C) 7 8 9 10 11 1 13 7 10 13 103. Mea devato Defto: arthmetc mea of all the devatos take wth postve sg. f MD Iterpretato: the hghest MD, the most devato. Advatage: It s depedet of the sample szes. Dsadvatage: statstc dffculty due to the absolute values. 104 5
17/03/015. Mea devato Example: f 4-6 7-9 8 10-1 10 13-15 8 16-18 Calculate the MD. 105. Mea devato f f f 4-6 5 10-6 6 1 7-9 8 8 64-3 3 4 10-1 11 10 110 0 0 0 13-15 14 8 11 3 3 4 16-18 17 34 6 6 1 30 Σ330 7 f 330 11 30 f MD 7.4 30 106 53
17/03/015 3. Varace ad 4. Stadard devato Varace: S ( ) f f S S rf Stadard devato: S S Quas-varace: ˆ S S 1 Quas stadard devato: S ˆ ˆ S They estmate more accurately the varace ad the stadard devato of the populato 107 3. Varace ad 4. Stadard devato Example 1: 3 6 9 1 15 Calculate the varace usg the formulas: S ( ) f f S 108 54
17/03/015 3. Varace ad 4. Stadard devato ( ) 3-6 36 6-3 9 9 0 0 1 3 9 15 6 36 Σ 0 90 45 9 5 S f ( ) 90 18 5 109 3. Varace ad 4. Stadard devato 3 9 6 36 9 81 1 144 15 5 45 495 45 9 5 S f 495 9 5 99 81 18 110 55
17/03/015 3. Varace ad 4. Stadard devato Example : f 3 1 4 7 5 6 6 3 7 3 S S Calculate the varace usg the formulas: ( ) f rf S f Calculate the stadard devato, the quas-varace ad the quas stadard devato 111 3. Varace ad 4. Stadard devato f f ( ) ( ) f( ) 3 1 3-4 4 4 7 8-1 1 7 5 6 30 0 0 0 6 3 18 1 1 3 7 3 1 4 1 Σ 0 100 0 10 6 f f ( ) 100 5 0 S 6 1.3 0 11 56
17/03/015 3. Varace ad 4. Stadard devato f f f 3 1 3 9 9 4 7 8 16 11 5 6 30 5 150 6 3 18 36 108 7 3 1 49 147 Σ 0 100 56 f 100 5 0 S f 56 5 6.3 5 1.3 0 113 3. Varace ad 4. Stadard devato f f rf rf 3 1 3 0.05 9 0.45 4 7 8 0.35 16 5.6 5 6 30 0.3 5 7.5 6 3 18 0.15 36 5.4 7 3 1 0.15 49 7.35 Σ 0 100 1 6.3 S rf f S S 6.3 5 100 5 0 1.3 1.14 1.3 0 S 1.3 1.3 x1.053 1 19 ˆ S 1.369 ˆ ˆ S S 1.369 1.17 114 57
17/03/015 3. Varace ad 4. Stadard devato Example wth tervals: f 60-64 0 55-59 30 50-54 100 Calculate the varace usg the formula: S ( ) f 45-49 30 40-44 0 115 3. Varace ad 4. Stadard devato f f ( ) ( ) f ( ) 60-64 0 6 140 10 100 000 55-59 30 57 1710 5 5 750 50-54 100 5 500 0 0 0 45-49 30 47 1410-5 5 750 40-44 0 4 840-10 100 000 Σ 00 10400 0 5500 f f ( ) 10400 5 00 S 5500 7.5 00 116 58
17/03/015 5. Semterquartle ampltude Defto: t s the semdstace betwee quartle 3 ad quartle 1. Q Q Q It s usually calculated: 3 1 Whe we oly wat to cosder the cetral scores of the dstrbuto. Whe the cetral tedecy dex recommeded s the meda. 117 5. Semterquartle ampltude A B A f A B f B 1 35 1 5 40 0 3 75 3 40 4 30 4 80 5 0 5 55 Example: The tables preset the frequecy dstrbuto to two dfferet tems (A ad B). 1. Calculate Q both tems A ad B.. Whch s the tem that presets more homogeety ther aswers? 118 59
17/03/015 5. Semterquartle ampltude A Q Q Q 3.75 1.75 3 1 0.875 A f A F A 1 35 35 40 75 Q 3 1. Posto. Value ( + k 1) 3(00 + 1) 150.75 4 I + D( I+1 - I ) Q 3 3 + 0.75 (4-3) 3.75 3 75 150 4 30 180 Q 1 1. Posto. Value Q 1 ( + k 1) 1(00 + 1) 50.5 4 5 0 00 119 5. Semterquartle ampltude Q Q B Q 5 3 3 1 1 B f B F B 1 5 5 0 5 3 40 65 Q 3 1. Posto. Value Q 3 5 Q 1 1. Posto. Value Q 1 3 ( ( + k + k 1) 3(00 + 1) 150.75 4 1) 1(00 + 1) 50.5 4 4 80 145 5 55 00 Q A 0.875 < Q B 1 The tem that presets more homogeety ther aswers s tem A 10 60
17/03/015 5. Semterquartle ampltude Example wth tervals: Itervals f 50-59 5 60-69 6 70-79 18 80-89 31 90-99 9 100-109 7 110-119 3 10-19 1 Steps: 1. F. */k 3. L : value - 0.5 x measuremet ut 4. f 5. F 6. I max m (exact lmts of the terval) 7. I * Q L + F f k Calculate the semterquartle ampltude 11 5. Semterquartle ampltude Q Q Itervals f 1 F 50-59 5 5 60-69 6 11 70-79 18 9 80-89 31 60 90-99 9 69 100-109 7 76 110-119 3 79 10-19 1 80 Q 89.513 74.495 Q 3 15.018 3 1 7.509 1. F. */k 3 x 80/4 60 3. L : 80 0.5 x 1 79.5 4. f 4 31 5. F 3 9 6. I 89.5 79.5 10 7. I * 10 3*80 Q3 L + F 79.5 + 9 f k 31 4 79.5 + 0.33(60 9) 79.5 + 10.013 89.513 1 61
17/03/015 5. Semterquartle ampltude Q 1 Itervals f 1 F 50-59 5 5 60-69 6 11 70-79 18 9 80-89 31 60 90-99 9 69 100-109 7 76 110-119 3 79 10-19 1 80 1. F. */k 1 x 80/4 0 3. L : 70 0.5 x 1 69.5 4. f 3 18 5. F 11 6. I 79.5 69.5 10 7. I * 10 1*80 Q1 L + F 69.5 + 11 f k 18 4 69.5 + 0.555(0 11) 69.5 + 0.555x9 69.5 + 4.995 74.495 13 6. Pearso varato coeffcet S VC 100 It s useful to compare two stadard devatos of dfferet samples or dfferet varables. It also measures the represetatveess of the arthmetc mea. The bgger the VC s, the less represetatve the arthmetc mea s. 14 6
17/03/015 6. Pearso varato coeffcet Example: We studed the reacto tme to two stmul, A ad B. The results were as follows: A B 50 600 S 5 6 1. Whch stmulus does preset more varato?. Whch arthmetc mea s more represetatve? 15 6. Pearso varato coeffcet VC VC A B S S A A B B 100 100 5 100 50 6 100 600 10 1 VC A 10 > VC B 1 1. A presets more varato.. The arthmetc mea of B s more represetatve. 16 63
17/03/015 Group ad dvdual dexes Group dexes Idvdual dexes Cetral tedecy Varablty (Dsperso) Bas or Skewess (Asymmetry) Posto Decles (D ) Percetles (P ) Quartles (Q ) Raw scores ( ) Kurtoss Dfferetal scores (x ) Stadard scores (Z ) 17 How are the data arraged wth respect to the rest? Are data pled at oe sde? Bas, Skewess or Asymmetry dexes (A s ) Pearso asymmetry Fsher asymmetry Iterquartle asymmetry 18 64
17/03/015 Bas, skewess or asymmetry dexes 19 Itroducto A s A s < 0 Asymmetrc egatve < Md < Mo A s 0 Symmetrc Md Mo A s > 0 Asymmetrc postve > Md > Mo 130 65
17/03/015 Itroducto 1. PEARSON ASYMMETRY INDEES (A s ). FISHER ASYMMETRY 3. INTERQUARTILE ASYMMETRY 131 1. Pearso asymmetry A s Mo S It oly ca be calculated whe the dstrbuto s umodal. 13 66
17/03/015 1. Pearso asymmetry Wth the data below: Itervals f 1-3 4-6 7 7-9 13 10-1 18 13-15 10 1. Calculate Pearso asymmetry dex.. Is the dstrbuto asymmetrc egatve, symmetrc or asymmetrc postve? 133 1. Pearso asymmetry Itervals f f f 1-3 4 4 8 4-6 5 7 35 5 175 7-9 8 13 104 64 83 10-1 11 18 198 11 178 13-15 14 10 140 196 1960 50 481 5153 A s Mo S 9.6 11 3.4 1.38 0.43 3.4 134 67
17/03/015 1. Pearso asymmetry f 481 9.6 50 S S S f 10.5 3.4 5153 9.6 50 103.06 9.54 10.5 135 1. Pearso asymmetry A.43 A < 0 s 0 s Asymmetrc egatve 136 68
17/03/015. Fsher asymmetry It s the best of the asymmetry dexes. A s f ( S 3 3 ) 137. Fsher asymmetry Example: wth the same example used before: Itervals f 1-3 4-6 7 7-9 13 10-1 18 13-15 10 Calculate Fsher asymmetry dex. 138 69
17/03/015. Fsher asymmetry A s 3 3 Itervals f ( ) f( ) 1-3 -7.6-44.45-884.9 4-6 5 7-4.6-98.61-690.7 7-9 8 13-1.6-4.5-55.5 10-1 11 18 1.38.63 47.34 13-15 14 10 4.38 84.03 840.3 50-74.78 f ( S ) 3 3 74.78 / 50 3 3.4 14.86 34.01 0.44 9.6 S 3.4 139 3. Iterquartle asymmetry Advatage: values betwee -1 ad +1. A s ( Q Q Q 3 ) ( 1) 3 Q Q 1 Q 140 70
17/03/015 3. Iterquartle asymmetry Example: wth the same example used before: Itervals f 1-3 4-6 7 7-9 13 10-1 18 13-15 10 Steps: 1. F. */k 3. L : value - 0.5 x measuremet ut 4. f 5. F 6. I max m (exact lmts of the terval) 7. I * Q L + F f k Calculate the terquartle asymmetry dex. 141 3. Iterquartle asymmetry A s ( Q Q ) ( Q Q1 ) (1.08 10) (10 7.31).08.69 0.61 Q Q (1.08 7.31) 4.77 4.77 3 3 1 0.13 Q 3 Itervals f F 1 1-3 4-6 7 9 7-9 13 10-1 18 40 13-15 10 50 1. F. */k 3 x 50/4 3 x 1.5 37.5 3. L : 10 0.5 x 1 9.5 4. f 4 18 5. F 3 6. I 1.5 9.5 3 7. Q I * 3 L + F 9.5 + f k 18 ( 37.5 ) 9.5 +.58 1. 08 3 14 71
17/03/015 3. Iterquartle asymmetry Q Itervals f F 1 1-3 4-6 7 9 7-9 13 10-1 18 40 13-15 10 50 1. F. */k x 50/4 x 1.5 5 3. L : 10 0.5 x 1 9.5 4. f 4 18 5. F 3 6. I 1.5 9.5 3 7. Q I * 3 L + F 9.5 + f k 18 ( 5 ) 9.5 + 0.5 10 143 3. Iterquartle asymmetry Q 1 Itervals f F 1 1-3 4-6 7 9 7-9 13 10-1 18 40 13-15 10 50 1. F. */k 1 x 50/4 1 x 1.5 1.5 3. L : 7 0.5 x 1 6.5 4. f 3 13 5. F 9 6. I 9.5 6.5 3 7. Q I * 3 L + F 6.5 + f k 13 ( 1.5 9) 6.5 + 0.81 7. 31 1 144 7
17/03/015 Group ad dvdual dexes Group dexes Idvdual dexes Cetral tedecy Varablty (Dsperso) Bas or Skewess (Asymmetry) Posto Decles (D ) Percetles (P ) Quartles (Q ) Raw scores ( ) Kurtoss Dfferetal scores (x ) Stadard scores (Z ) 145 Whch form s the dstrbuto? Is t flatteed or sharp? Kurtoss dex k r 146 73
17/03/015 Kurtoss dex 147 Itroducto K r < 0 Platykurtc K r K r 0 Mesokurtc (ormal dstrbuto) K r > 0 Leptokurtc 148 74
17/03/015 Kurtoss dex K r f ( 4 S ) 4 3 149 Kurtoss dex Example: wth the same example used before: Itervals f 1-3 4-6 7 7-9 13 10-1 18 13-15 10 1. Calculate the kurtoss dex.. Is the dstrbuto platykurtc, mesokurtc or Leptokurtc? 150 75
17/03/015 Kurtoss dex K r 4 4 Itervals f ( ) f( ) 1-3 -7.6 3371.47 674.94 4-6 5 7-4.6 455.58 3189.06 7-9 8 13-1.6 6.89 89.57 10-1 11 18 1.38 3.63 65.34 13-15 14 10 4.38 368.04 3680.4 f ( 50 13767.31 S ) 4 4 3 13767.31/ 50 3.4 4 75.35 3 3.5 3 0.5 110. 9.6 S 3.4 151 Kurtoss dex K r.5 K < 0 0 r Platykurtc 15 76
17/03/015 Group ad dvdual dexes Group dexes Idvdual dexes Cetral tedecy Varablty (Dsperso) Bas or Skewess (Asymmetry) Posto Decles (D ) Percetles (P ) Quartles (Q ) Raw scores ( ) Kurtoss Dfferetal scores (x ) Stadard scores (Z ) 153 1. Itroducto Raw scores ( ): they are the scores gve drectly (example: puctuato a test). Dfferetal scores (x ): x Stadard scores (Z ): Z S 154 77
17/03/015 1. Itroducto The comparso usg raw scores ( ) ca lead us to msleadg coclusos. The soluto based o the dstaces to the mea (x ) s ot a etrely satsfactory soluto. The soluto s to use stadard scores. 155. Raw scores CASE 1: a partcpat, a varable. A partcpat obtas a score o a tellgece questoare: Iterpretato: A raw score (example: 5) does ot gve us more tha a umber. Is t too much? Is t eough? It depeds o two factors: Mea Varablty 5 Cocluso: raw scores are ot eough to compare. 156 78
17/03/015. Raw scores CASE : a partcpat, two varables. Joh weghs 75 kg. ad he s 1.80 m. tall. Hs weght, s t more or less tha hs heght? They are ot drectly comparable. 157. Raw scores CASE 3: a partcpat, two supposedly comparable varables. A studet has recetly bee examed two dfferet subjects: Motvato ad Methods. If the scores obtaed were respectvely 30 ad 15, ca we say that the studet has doe better Motvato tha Methods? It depeds o hs group of referece. 158 79
17/03/015 3. Dfferetal scores Scores of Devato Dsperso Errors or bas x - 159 3. Dfferetal scores They tell us f a raw score s hgher, smaller or equal to the mea. Ths formato s ferred from the sg of the dfferetal score (postve, egatve or zero respectvely). 160 80
17/03/015 3. Dfferetal scores 7 9 10 11 13 1. Calculate the mea, the varace ad the stadard devato.. Calculate the dfferetal scores. 3. Calculate the mea, the varace ad the stadard devato of these dfferetal scores. 161 3. Dfferetal scores 7 49 9 81 10 100 11 11 13 169 S S S 50 10 5 50 10 104 100 4 5 4 50 50 16 81
17/03/015 3. Dfferetal scores x ( - ) x 7 7-10 -3 9 9 9-10 -1 1 10 10-10 0 0 11 11-10 1 1 13 13-10 3 9 50 0 0 0 5 S 0 0 4 0 4 5 S S 0 4 163 3 3 1 1 7 9 10 11 13 x -3-1 0 1 3 3 1 1 3 164 8
17/03/015 3. Dfferetal scores: coclusos RAW SCORES 10 DIFFERENTIAL SCORES x 0 S 4 S x 4 S S x 165 3. Dfferetal scores: example The tellectual level of two groups (A ad B) was measured: A: 97 10 107 11 117 B: 9 97 10 107 11 117 1 Supposg that studets, oe belogg to group A ad aother to group B, obtaed the same tellectual level of 117 pots, 1. Whch were the dfferetal scores for both studets?. Do both scores mply the same tellectual level? 166 83
17/03/015 3. Dfferetal scores: example A B A B 97 + 10 + 107 + 11 + 117 535 107 5 5 9 + 97 + 10 + 107 + 11 + 117 + 1 749 107 7 7 107 x - 117 107 + 10 167 3. Dfferetal scores: example Group A 97 10 107 11 117 Group B 9 97 10 107 11 117 1 168 84
17/03/015 3. Dfferetal scores: example Coclusos: Equalty dfferetal scores may be maskg dfferet stuatos. Group A was more homogeeous tha group B. Whle the score 117 group A represeted a extreme value (because t was the maxmum score), the same score group B was ot so extreme (there was a hgher score: 1). As a soluto to solve ths problem, stadard scores ca be used. 169 4. Stadard scores (Z ) They are also called typfed or stadardzed scores. Defto: The stadard, typfed or stadardzed score dcates the umber of stadard devatos that a partcular raw score s separated from ts mea. Z S 170 85
17/03/015 4. Stadard scores (Z ). Example 7 9 10 11 13 Calculate the dfferetal ad the stadard scores. 171 4. Stadard scores (Z ). Example x Z 7-3 -1.5 49 9-1 -0.5 81 10 0 0 100 11 1 0.5 11 13 3 1.5 169 Σ50 0 50 S 50 10 5 104 100 50 5 10 4 17 86
17/03/015 4. Stadard scores (Z) There s a shft to the left sde of the x-axs whe we covert raw scores to dfferetal: x [-3-1 0 1 3] [7 9 10 11 13] Ad there s a cocetrato of scores whe they become stadard: Z -1.5 0.5 0 0.5 1.5 173 4. Stadard scores (Z ). Propertes 1 The sum of stadard scores s 0: Z 0 Z 7-1.5 9-0.5 10 0 11 0.5 13 1.5 Σ50 174 87
17/03/015 4. Stadard scores (Z ). Propertes The mea of stadard scores s 0: Z 0 Z 7-1.5 9-0.5 10 0 11 0.5 13 1.5 Σ50 175 4. Stadard scores (Z ). Propertes The mea of stadard scores s 0: Z Z 0 7-1.5 0 9-0.5 Z Z 0 5 10 0 11 0.5 13 1.5 Σ50 Σ0 176 88
17/03/015 4. Stadard scores (Z ). Propertes 3 The sum of the stadard scores squared s : ΣZ Z 7-1.5 9-0.5 10 0 11 0.5 13 1.5 Σ50 Σ0 177 4. Stadard scores (Z ). Propertes 3 ΣZ The sum of the stadard scores squared s : Z Z 7-1.5.5 9-0.5 0.5 10 0 0 11 0.5 0.5 13 1.5.5 Σ50 Σ0 5 5 178 89
17/03/015 4. Stadard scores (Z ). Propertes 4 S S Z Z The stadard devato ad varace of stadard scores s equal to 1 : 1 1 Z Z 7-1.5.5 9-0.5 0.5 10 0 0 11 0.5 0.5 13 1.5.5 Σ50 Σ0 5 179 4. Stadard scores (Z ). Propertes 4 S S Z Z The stadard devato ad varace of stadard scores s equal to 1 : 1 1 Z Z 7-1.5.5 9-0.5 0.5 10 0 0 11 0.5 0.5 13 1.5.5 Σ50 Σ0 5 S S Z Z Z 5 Z 0 5 SZ 1 1 180 1 90