The variance and standard deviation from ungrouped data

Transcription

1 BIOL 443 مقاييس التغير (التشتت ( (dsperso). Measures of Varato Just as measures of cetral tedecy locate the ceter of data, measures of varato measure ts spread. Whe the varato s small, ths meas that the values are close together (but ot the same). The fgure below shows the frequecy curves for two populatos that have equal meas but dfferet amouts of varato. Populato Populato Fgure. µ Two frequecy dstrbutos wth equal meas but dfferet amouts of varato. The most commoly used measures of data varato are the rage, the varace, stadard devato, Coeffcet of Varato, ad the Iterquartle rage. المدي Rage The rage s defed as the dfferece value betwee the hghest (maxmum) ad lowest (mmum) observato: Rage = x max - x m The rage ca be computed qucly, but t s ot very useful sce s cosders oly he extremes ad does ot tae to cosderato the bul of the observatos. The varace ad stadard devato from ugrouped data التباين والا نحراف المعياري للبيانات الغير مبوبة Oe way of measurg the spread of the data s to determe the extet to whch each observato devates from the arthmetc mea. Clearly, the larger the devatos, the greater the varablty of the observatos. However, we caot use the mea of these devatos as a measure of spread because the postve dffereces exactly cacel out the egatve dffereces. We overcome ths problem by squarg each devato, ad fdg the mea of these squared devatos; we call ths the varace. The Varace s less whe all values are close to the mea whle t s more whe the values are spread out from the mea. The sample varace, or s, s computed by ether formula where: x ( x x) = x = = s = s = OR s =sample varace x = sample mea 3

2 BIOL 443 = total umber of observatos the sample We ca see that ths s ot qute the same as the arthmetc mea of the squared devatos because we have dvded by - stead of. The reaso for ths s that we almost always rely o sample data our vestgatos. It ca be show theoretcally that we obta a better sample estmate of the populato varace f we dvde by -. The uts of the varace are the square of the uts of the orgal observatos, e.g. f the varable s weght measured g, the uts of the varace are g. The stadard devato s represeted by the symbol s. It s the square root of the varace.e. s = s It expresses exactly the same formato as the varace, but re-scaled to be the same uts as the orgal measuremet (raw data) ad the mea. The populato varace of the populato of the observatos x s defed the formula where: σ N ( x µ ) = σ = N (sgma squared) = populato varace x = the tem or observato µ = populato mea N = total umber of observatos the populato. The stadard devato of a populato s equal to the square root of the varace ( x µ ) = σ = σ = N Sce most populatos are large, the computato of σ s rarely performed. I practce, the populato varace (or stadard devato) s usually estmated by tag a sample from the populato ad usg s as a estmate of σ. N Example A pedatrc regstrar a dstrct geeral hosptal s vestgatg the amout of lead the ure of chldre from a earby housg estate. I a partcular street there are 5 chldre whose ages rage from year to uder 6, ad a prelmary study the regstrar has foud the amouts gve Table of urary lead (µmol/4hr), Table Urary cocetrato of lead 5 chldre from housg estate (µmol/4hr) 0.6,.6, 0.,., 0.4,.0, 0.8,.3,.,.5, 3.,.7,.9,.9,. What s the varace ad stadard devato? 3

3 BIOL 443 Soluto The calculato of the varace s llustrated ext Table wth the 5 readgs the prelmary study of urary lead cocetratos. The readgs are set out colum (). I colum () the dfferece betwee each readg ad the mea s recorded. I colum (3) the dffereces are squared, ad the sum of those squares s gve at the bottom of the colum. Calculato of stadard devato () Lead cocetrato () Dffereces from mea (3) Dffereces squared (4) Observatos col. () squared x Total x =.5 ( x x) = 0 ( ) x x =9.96 x = 43.7 = 5, x = l.5 The sum of the squares of the dffereces (or devatos) from the mea, 9.96, s ow dvded by the total umber of observato mus oe, to gve the varace. Thus, s = = ( x x) 33

4 BIOL I ths case we fd: s = = 0.74 (µmol/4hr) 4 Fally, the square root of the varace provdes the stadard devato: s = s = =0.843 µmol/(4hr) Meas ad varace from grouped data الوسط الحسابي و الا نحراف القياسي للبيانات المبوبة (الجداول التكرارية) More ofte tha ot, data are preseted grouped form. That s, the data are part summarzed ad grouped a frequecy table. I actualty, ths smplfes the hadlg of data -t's easer to wor wth the value of 8 serum cholesterol readgs of 30.5 tha to lst the value 30.5 separately 8 tmes. Meas ad stadard devatos may be computed from grouped data, but the equatos are a bt dfferet. Formulas for calculatg the mea ad the varace for grouped data: f x f x = f x = = x =, s = where x = mea of the data set, s = varace of the data set x = mdpot of the th class, f = frequecy of the th class, = total umber of observatos. Example Gve below are the frequecy dstrbutos for the heghts ( cetmeters) of a sample of 00 studets the Islamc Uversty, fd the approxmate value for the stadard devato for studets. Table 4.3 Frequecy of heghts of a sample of 00 studets the Islamc Uversty Class terval x x f fx fx ,04 9,368 07, ,649 3,454 54, ,44 3 5,0 83, , , , ,584 3,36 384, , ,39 Total = 00 6,65,649,035 For the grouped data, we obta x = = f x 6,65 = = 6.65cm 00 34

5 BIOL 443 f x = f x =,649,035,645,50.5 s = = = =5.97 cm 99 Note that there s some dfferece betwee results from computatos ugrouped ad grouped data. The sze of the dscrepacy depeds o wdth of the class terval ad o the umber of observatos wth a terval. Wth short class tervals ad large samples, the dscrepacy s eglgble. معامل الا ختلاف Coeffcet of Varato A dsadvatage of the stadard devato as a comparatve measure of varato s that t ca ot be used to compare varablty two dfferet ds of varables. For ths reaso, statstcas have defed the coeffcet of varato whch helps comparg the relatve varablty amog dfferet varables. Also, whle the stadard devato depeds o the uts of measuremets, the coeffcet of varato cv s utless or dmesoless, sce both stadard devato ad the mea are expressed same uts. Therefore, t s useful comparg scatter of varables measured dfferet uts. It also possble to use the coeffcet of varato to compare the relatve varato of eve urelated quattes or that are vastly dfferet scale or magtude of uts; elephat weght versus mouse weght. Stadard devato cv = Coeffcet of varato = 00 % x For example, we ca say that, the stadard devato (sd) of 5 mmhg case of systolc blood pressure (BP) readgs s small but a stadard devato of eve 3 g/dl case of hemoglob (Hb) level s large. Ths s because the stadard devato has to be assessed relato to ther mea. If the mea systolc BP level of the subject uder study s 00 mmhg the a sd = 5 mmhg s oly 5% of mea. If the mea Hb level s 0 g/dl the a sd = 3 g/dl s 30% of mea. Ths sd s surely hgher. The ext Table cotas CV of varous hematologcal parameters of chldre of megaloblastc aema the age group of 3½ moths to years. Note that the varablty mea corpuscular volume (stated femtolters = 0-5 lters) s much lower tha TLC (total leucocyte cout). It s relatvely more cosstet betwee patets. Such comparso caot be doe o the bass of sd. Hematologcal Data o 9 Chldre of Megaloblastc Aema Betwee 3½ Moths ad Years of Age Varable Mea (sd) Rage CV (%) Hb (g/dl) 5.3 (.88) TLC (cu mm) 8.39 (5.65) Platelet cout (0 9 /l) 0.83 (56.8) Mea corpuscular volume (fl) 0.3 (0.6) Source: Gomber et al. Note: We have computed the CV colum. Ths was ot gve by the authors. 35

6 BIOL 443 Measures of Postos I cases where our data dstrbuto are heavly sewed, we ofte get a better summary of the dstrbuto by utlzg relatve posto of data rather tha exact value. Measures of posto are used to descrbe the locato of a partcular observato relato to the rest of the data set. The meda s a example of measure computed by usg relatve posto of the data. If we are told that 8 s the meda score o bostatstcs test, we ow that after the data have bee ordered, 50% of the data fall below the meda value of 8. Percetles, Decles, ad Quartles المي ينات Percetles Percetles are values of x that dvde the ordered set to 00 equally szed groups. The pth percetle of a data set s a value such that p percet of the observatos the ordered set lyg below t ad (00 - p) percet of the observatos lyg above ths value. The frst percetle for example s the value of x that has % of the observatos the ordered set lyg below t (ad 99% of the observatos lyg above t). The value of x that has 4% of the observatos lyg below t s called the 4 th percetle, ad so o. The frst, secod 99 th percetles are expressed as P, P, P 99 respectvely. العشيرات Decles % % % % % % % % Lowest st d 3rd 4th 5th 98th 99th Hghest The values of x that dvde the ordered set to 0 equally szed groups are called decles. The frst decle s the value of x that has 0% of the observatos the ordered set lyg below t (ad 90% of the observatos lyg above t). The value of x that has 40% of the observatos lyg below t s called the 4 th decle, ad so o. The frst, secod 9 th decles are expressed as D, D,., D 9 respectvely. 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% Lowest st d 3rd 4th 5th 6th 7th 8th 9th Hghest الربيعات Quartles The values of x that dvde the ordered set to four equally szed groups. The frst quartle s the value of x that has 5% of the observatos the ordered set lyg below t (ad 75% of the observatos lyg above t). The value of x that has 50% of the observatos lyg below t s called the d quartle, ad the value of x that has 75% of the observatos the ordered set lyg below t s called the 3 rd quartle. The frst, secod thrd quartles are expressed as Q, Q, ad Q 3. 36

7 BIOL 443 Lowest Q Q Q3 Hghest Meda 50 th percetle Importat Note: All the quartles ad decles are percetles. For example, the 7 th decle s the 70 th percetle ad the st quartle s the 5 th percetle. The 50th percetle s the 5 th decle, the secod quartle ad the meda of the ordered set. Cosequetly, decles ad quartles are ofte stated as percetles. Procedure to compute the percetle of x observato The techques for fdg the varous measures of posto wll be llustrated by usg the data the ext Table whch cotas the aortc dameters measured cetmeters for 45 patets. Notce that the data the Table are already ordered. Raw data eed to be ordered pror to fdg measures of posto The percetle for observato x s foud by dvdg the umber of observatos less tha x by the total umber of observatos ad the multplyg ths quatty by 00. Ths percet s the rouded to the earest whole umber. Example Fd the percetle for the 5.5 observato. The umber of observatos the Table less tha 5.5 s..00 = 4.4% 45 5% 5% 5% 5% Ths percet rouds to 4. The dameter 5.5 s the 4th percetle ad we express ths as P 4 = 5.5. Example Fd the percetle for the 5 observato. The umber of observatos less tha 5.0 s = 0%.e. P 0 = Example Fd the percetle for the 0.0 observato. The umber of observatos less tha 0.0 s = 86.7% 87.e. P 87 =

8 BIOL 443 Procedure to compute the pth percetle The pth percetle for a raed data set cosstg of observatos s foud by a twostep procedure. ( p)( ). The frst step s to compute dex =. 00. If s ot a teger, the ext teger greater tha locates the posto of the pth percetle the raed data set. If s a teger, the pth percetle s the average of the observatos postos ad + the raed data set. Example Fd the 0 th percetle for the data. (0)(45) = = The ext teger greater tha 4.5 s 5. The observato the 5 th posto the Table s 3.6. Therefore, the 0 th percetle or P0 = 3.6. Example Fd the 40 th percetle for the data the Table. (40)(45) = = The forteth percetle s the average of the observatos the 8th ad 9th postos the raed data set. The observato the 8th posto s 6.0 ad the observato the 9th posto s 6.. (6.0)(6.) Therefore P40 = = = 6.. Decles ad quartles are determed the same maer as percetles, sce they may be expressed as percetles. Percetles are most commoly used for chld growth motorg purposes.e.for the motorg of physcal progress (weght ad heght) of fats ad chldre. Here, the same percetles, say the 90th, of weght or heght of groups of dfferet ages are joed by a curve. If the 95th percetle of weght of -year old boys s 4.8 g the t meas that 95% of such chldre have weght 4.8 g or less. The other 5% have a hgher weght. The growth curve s ot lear but rather sgmodal. The perods of rapd growth occur durg the frst moths of age. Growth ormally starts to slow dow at about to 5 moths of age, whch s reflected the growth chart. The growth chart s a essetal tool to dagose falure to thrve (FTT) or growth falure. Although there are o uversal crtera for FTT, most cosder the dagoss f the chld's weght s below the 5th percetle or drops more tha two major percetle les. Whe curves are outsde the 5th ad 95th percetles, t s useful to meto the age at whch the growth parameter s at ts meda value (50th percetle). For example, f a moth old baby weghs 8 g, ths weght s below the 5th percetle for a oe year old; ad, t s at the 50th percetle for a 6 moth old. Oe could state that the weght age s 6 moths, whch s a better quattatve descrpto of the growth abormalty. 38

9 BIOL 443 Fgure (a) Weght ad (b) heght curves. Alteratve procedure to compute quartles. Order the data from smallest to largest.. Fd the meda. Ths s the secod quartle. 3. The frst quartle Q s the the meda of the lower half of the data; that s, t s the meda of the data fallg below the Q posto (ad ot cludg Q ). 4. The thrd quartle Q 3 s the meda of the upper half of the data; that s, t s the meda of the data fallg above the Q posto (ad ot cludg Q ). Example.Eve umber Fd the meda, ad upper ad lower quartles of ths set: Soluto:, 9, 7, 3, 38, 5, 3, 6 Frst step, order the data: 9,, 5, 6, 7, 3, 3, 38 So, there are eght umbers, the meda s the average of the fourth ad ffth umbers. Meda (secod quartle) = (6+7)/ = 6.5 The lower quartle s the meda of the frst four umbers, Lower Quartle (frst quartle) = (+5)/ = 3.5 ad the upper quartle s the meda of the last four umbers. Upper Quartle (thrd quartle) = (3+3)/ = 3 39

10 BIOL 443 Iterquartle Rage (IQR) We ca obta a measure of spread that s ot flueced by outlers by excludg the extreme values the data set, ad determg the rage of the remag observatos. The terquartle rage s the dfferece betwee the frst ad the thrd quartles,.e. betwee the 5th ad 75th percetles. It cotas the cetral 50% of the observatos the ordered set, wth 5% of the observatos lyg below ts lower lmt, ad 5% of them lyg above ts upper lmt. The terquartle rage tells us the spread of the mddle half (50%) of the data ad s ot affected by extremes the data set. Outlers Iterquartle rage =Upper Quartle - Lower Quartle IQR = Q 3 Q A outler s a umber that s so far above the data set or below most of the data set as to be cosdered abormal ad therefore of questoable accuracy. Outlers may be from data collecto errors, data etry errors, or smply vald but uusual data values. Regardless of the reaso, t s mportat to detfy the outlers the data set ad exame outlers carefully to determe f they are a error. For may purposes a outler s defed to be ay data pot that s.5 IQRs below the lower quartle or above the upper quartle. Example 8, 55, 57, 58, 6, 6, 63, 65, 83 UQ = (65+63)/ = 64 LQ = (55+57)/ = 56 IQR = = 8 So ay umber below LQ.5(IQR) = 56.5(8) = 44 or ay umber above UQ +.5(IQR) = (8) = 78 s a outler. Therefore the outlers of ths data set are 8 & 83. Box-ad Whser Plots (Boxplot) The quartles together wth the low ad hgh data values gve us a very useful fve umber summary of the data ad ther spread. These Fve-umber summary clude; Lowest value, Q, meda, Q 3, ad hghest value. 5% 5% 5% 5% Q Q Q3 Lowest value Meda Hghest value These fve umbers ca be used to create sech of the data called a box-ad-whser plot. Box-ad-Whser plots provde aother useful techque for descrbg data. 40

11 BIOL 443 To mae Box-ad-Whser plot. Draw a vertcal scale to clude the lowest ad hghest data values.. To the rght of the scale draw a box from Q to Q Iclude a sold le through the box at the meda level. 4. Draw sold les, called whsers, from Q to the lowest value ad from Q 3 to the hghest value. Hghest value Q 3 Meda Q Lowest value Example Use the followg ordered set of data to mae a box-ad-whser plot 55, 58, 60, 6, 64, 64, 7 Meda = 6 Lower quartle Q= 58 Upper quartle Q3= 64 Lowest or mmum value = 55 Hghest or maxmum value = 7 Mmum value Meda Maxmum value Lower quartle Upper quartle 4