Numerical Summary Measures

Transcription

1 Eample: Brth weghts ( lb) of 5 babes bor from two groups of wome uder dfferet care programs. Group : 7, 6, 8, 7, 7 Group : 3, 4, 8, 9, Numercal Summary Measures Descrbe Dstrbuto wth Numbers Measure of Ceter Measure of Varato Measure of Posto Measure of Cetral Tedecy Mea: the average value of the data. Eample: Brth weghts ( lb) of 5 babes bor from a group of wome uder certa det. 7, 6, 8, 7, 7 If observatos are deoted by,,...,, ther (sample) mea s... Sol: mea [ear the ceter of the data set] 3 4 Meda: of a data set s the data value eactly the mddle of ts ordered lst f the umber of peces of data s odd, the mea of the two mddle data values ts ordered lst f the umber of peces of data s eve. [meda s ot flueced by outlers ad s best for o-symmetrc dstrbuto] Eample: (umber of hysterectomes performed by 5 doctors) 7, 50, 33, 5, 86, 5, 85, 3, 37, 44, 0, 36, 59, 34, 8 ordered lst => 0, 5, 5, 7, 8, 3, 33, 34, 36, 37, 44, 50, 59, 85, 86 meda = Descrptve Stat -

2 Eample: (Brth weghts for 6 fats.) 5, 7, 6, 8, 5, 9 ordered lst => 5, 5, 6, 7, 8, 9 Mode: of a data set s the observato that occurs most frequetly. meda = (6+7) / = Eample : (umber of tmes vsted class webste by 5 studets) 7, 50, 33, 5, 86, 5, 85, 3, 37, 44, 0, 36, 59, 34, 8 ordered lst => 0, 5, 5, 7, 8, 3, 33, 34, 36, 37, 44, 50, 59, 85, 86 Mode = 5 Eample : (Blood type of 5 studets) A, B, A, A, O, AB, A, A, B, B, O, O, A, A, A Mode = A A 8 B 3 O 3 Mea? Meda? Mode? Skewed to the Rght AB 9 0 Proporto Estmato Measure of Dsperso (Varablty) Rage = largest data value smallest data value Parameter: Populato Proporto p (or p) (Percetage of people has o health surace) Statstc: Sample Proporto pˆ s umber of successes s sample sze Data:, 0,, 0, 0 pˆ. 4 pˆ Sample from group I (det program I): 7, 6, 8, 7, 7 => mea = ( ) / 5 = 35/5 = 7 Sample from group II (det program II): 3, 4, 8, 9, => mea = ( ) / 5 = 35/5 = 7 Does the mother s det program affect the brth weghts of babes? Descrptve Stat -

3 Is there ay dfferece betwee the two samples? rage of sample I = 8-6 = rage of sample II = - 3 = 8 Varace ad Stadard Devato Measure the spread of the data aroud the ceter of the data. 3 4 Eample: Brth weghts ( lb) of 5 babes bor from a group of wome uder det program II. 3, 4, 8, 9, mea = = 7 Data Value Total Devato from mea Squared Dev. ( ) 3 7 = = = 9 7 = 4 7 = s A Short Cut formula: Data, Sample Varace = 46/4 =.5 lb, Sample Stadard Devato = 46/ 4 = 3.39 lb. 5 6 What s the stadard devato of the weghts of babes from the sample of mothers who receved det program I? Data: 7, 6, 8, 7, 7 s = ( )/(5-) = ½ s = 0.7 Does the mother s det program affect the brth weghts of babes? Det I: mea = 7, s = 0.7 Det II: mea = 7, s = If observatos are deoted by,,...,, ther varace ad stadard devato are ( ) Sample Varace: s (ubased estmator for varace of a fte populato.) Sample Mea: Sample Stadard Devato: s ( )... 8 Descrptve Stat - 3

4 Populato Parameters If N observatos are deoted by,,...,, are all the observato a fte populato, ther mea,, varace, ad stadard devato,, are... Populato Mea: N N ( ) Populato Varace: Populato Stadard Devato: N ( ) N 9 About s (sample stadard devato) : s measures the spread aroud the mea. the larger s s, the more spread out the data are. f s = 0, the all the observatos must be equal. s s strogly flueced by outlers. 0 The Use of Mea ad Stadard Devato Descrbe dstrbuto Uderstad the ceter ad the spread of the dstrbuto Ut: mg/ml Boe Desty Data Mea, Stadard Devato, s Female 0. 5 Male May dstrbutos ca be descrbed by a mathematcal fucto wth specfc parameters, such as mea ad stadard devato. Eample: Normal Dstrbuto (Bell-shaped) Emprcal Rule Propertes of a symmetrc ad bell-shaped (Normal) dstrbuto: The dstrbuto s symmetrc about t mea (), 68% of the area betwee ad, 95% of the area betwee ad, 99.7% of the area betwee 3 ad Descrptve Stat - 4

5 Heart rates for a certa populato at a certa codto follow a bell shape symmetrc dstrbuto wth mea 70 ad stadard devato. What percetage of people ths populato wll have heart rates betwee 66 ad 74? 95%?% Chebychev s Rule Chebychev s equalty There s at least (/k ) of the data a data set le wth k stadard devato of ther mea Eample: Heart rates for asthmatc patets a state of respratory arrest has a mea of beats per mute ad a stadard devato of 35.5 beats per mute. What percetage of the populato of ths type of patets have heart rates le betwee two stadard devatos of the mea a state of respratory arrest? It wll be at least 75%, because k =, ad (/ ) = ¾ = 75%. Heart rates eample: mea=44, s.d.=35.5 k = 75% = (/ ) At least 75% = = 8 What about wth three stadard devatos? Heart rates eample: mea=44, s.d.=35.5 At At least 89%?% k = 3?% 89% (/3 ) ) Whch of the followg data has relatvely lower varablty? Aalyst A: (Slde A) 3, 4, 8, 33, 6,, 9 Aalyst B: (Slde B) 9, 0, 3, 0,, 3, = = Descrptve Stat - 5

6 Coeffcet of Varato (C.V.): s the stadard devato epressed as a percetage of the mea. It s a ut-free measure of dsperso. It provdes a measuremet for comparg relatve varablty of data sets from dfferet scales. s C.V. = 00% Eample: Oe wshes to compare the qualty of works from two blood cell cout aalysts. The average from repeated couts o slde A used by aalyst A was 6.43 lb wth a s.d.= 3.87, ad average from aalyst B for slde B s.4 wth a s.d.=.57. C.V. (Aalyst A) = (3.87/6.43)00% = 3.06% C.V. (Aalyst B) = (.57/.4) 00% = 4.% Aalyst A has lower varablty! 3 3 Measure of Posto Stadard Score, Percetle, Quartle Z-score (Stadard Score) If s a observato from a dstrbuto that has mea, ad stadard devato, the stadardzed value of s, z-score of : Populato z-score mea z stadard devato 3 has a z-score 3, sce t s 3 s.d. from mea If a dstrbuto has a mea 0 ad a s.d., the value 7 has a z-score.5. z-score = (7 0)/ =.5. Sample z-score z s Eample: If the mea of a radom sample s 5 ad the stadard devato s, what would be the sample z-score of the value 6? 5, s, 6.5 s.d z Descrptve Stat - 6

7 Eample: Boe Meral Desty The WHO Workg Group defes osteopoross accordg to measuremets of boe meral desty (BMD) usg dualeergy X-ray absorptometry (DEXA). Thus osteopoross s defed as a boe desty T score at or below.5 stadard devatos (T score) below ormal peak values for youg adults. DEXA BMD Values T score > -.0 S.D T score betwee.0 ad.5 SD T score < -.5 SD T score < -.5 SD wth or more fraglty fractures Defto Normal boe meral desty Osteopaea Osteopoross Severe osteopoross These crtera were tally establshed for the assessmet of osteopoross Caucasa wome. BMD reports may clude a T score whch s the umber of stadard devatos by whch the subject of terest dffers from the mea for ther age Percetle (Measure of posto) Fd a Data Value Correspodg to a Gve Percetle Step : Sort the data. Step : Compute posto de c c = p / 00 = total umber of values p = percetle (If for 90 th percetle, p = 90.) Step 3 (fd posto): ) If c s ot whole umber, roud up c to the et whole umber. ) If c s a whole umber, the percetle s at the posto that s halfway betwee c ad c Eample: A sample of umber of tmes vsted class webste by 5 studets s the followg: 7, 50, 33, 5, 86, 5, 85, 3, 37, 44, 0, 36, 59, 34, 8. Fd the 90 th percetle of the data ths sample. Sol: = 5, p = 90. Ordered data: 0, 5, 5, 7, 8, 3, 33, 34, 36, 37, 44, 50, 59, 85, 86 c = p/00 = 5 90 / 00 = 3.5 Roud c to 4. The 4 th umber the ordered lst s the 90 th percetle ad that s 85. Quartles: (Measure of Posto) The frst quartle, Q, or 5 th percetle, s the meda of the lower half of the lst of ordered observatos. (Lower quartle) The thrd quartle, Q 3, or 75 th percetle, s the meda of the upper half of the lst of ordered observatos. (Upper quartle) Eample: [odd umber of data values] ( = ),6,63,64,64,65,65,65,66,67,69,7,7,7,7,7,7,7,73,74,75 Q =? 64.5 Meda = 69 Q 3 =? 7 Measure of spread: Iterquartle rage (IQR) = Q 3 Q IQR = = Descrptve Stat - 7

8 Eample: [eve umber of data] ( = ) 6,,6,63,64,64,65,65,65,66,67,69,7,7,7,7,7,7,7,73,74,75 Q = 64? Meda = 68? Q 3 =? 7 Measure of spread: Iterquartle rage (IQR) = Q 3 Q The fve-umber summary.mmum value.q.meda.q 3.Mamum value IQR = 7-64 = Eample: (data sheet wthout outler 6 ),6,63,64,64,65,65,65,66,67,69,7,7,7,7,7,7,7,73,74,75 M =, Q = 64.5, Meda = 69, Q 3 = 7, Ma = 75. Wth 6 the data: 6,,6,63,64,64,65,65,65,66,67,69,7,7,7,7,7,7,7,73,74,75 Q = 64 Meda = 68 Q 3 = 7 IQR = 7-64 = HEIGHT HEIGHT 46 Ier ad outer feces for outlers IQR = 7 64 = 8; Q = 64; Q 3 = 7 The er feces are located at a dstace of.5 IQR below Q (lower er fece = Q -.5 IQR ) ad at a dstace of.5 IQR above Q 3 (upper er fece = Q IQR ). The outer feces are located at a dstace of 3 IQR below Q (lower outer fece = Q 3 IQR ) ad at a dstace of 3 IQR above Q 3 (upper outer fece = Q IQR ). The er feces are located at a dstace of.5 IQR below Q (lower er fece = = 5 ) ad at a dstace of.5 IQR above Q 3 (upper er fece = = 84). The outer feces are located at a dstace of 3 IQR below Q (lower outer fece = = ) ad at a dstace of 3 IQR above Q 3 (upper outer fece = = 96) Descrptve Stat - 8

9 HEIGHT Descrptve Measures 84 UIF: = 84 Ier fece 96 UOF: = 96 Outer fece Ier fece IQR IQR 5 LIF: = 5 Ier fece LOF: = Ier fece Outer fece 0 Q = 64; Q 3 = 7; IQR = 7 64 = HEIGHT HEIGHT Mld ad Etreme outlers Sde-by-sde Bo Plot Data values fallg betwee the er ad outer feces are cosdered mld outlers. Data values fallg outsde the outer feces are cosdered etreme outlers Whe outlers est, the whsker eteded to the smallest ad largest data values wth the er fece Female Male 5 se 5 Remarks: If the dstrbuto of the data s symmetrc, the the mea ad meda wll be about the same. The fve-umber summary s best for o-symmetrc data. The meda, quartles, ter-quartle rage are ot flueced by outlers. The mea ad stadard devato are most approprate to use oly f the data are symmetrc because both of these measures are easly flueced by outlers. Boplot For the followg data: Fd the fve-umber-summary & IRQ Make a boplot Fd the th percetle Descrptve Stat - 9