Descriptive Statistics

Transcription

1 Math 3 Lecture I Descrptve tatstcs Descrptve statstcs are graphcal or umercal methods utlsed to summarze data such a way that mportat features of the sample ca be depcted. tatstcs: tatstcs s cocered wth scetfc methods for collectg, orgazg, summarzg, presetg ad aalyzg data, drawg vald coclusos ad mag reasoable decsos of such aalyss. The term stasstcs s used to deote the data themselves or umbers derved from data, such as averages. As a example Employmet statstcs, Accdet statstcs ca be gve. Populato: I collectg data t s ofte mpossble or mpractcal to observe the etre for example sads o the beach, umber of defectve bolts produced a factory a gve day, all possble outcomes successve tosses of a far co, etc.. Therefore stead of examg the etre group called the Populato (uverse), oe exames a small part of t whch represets the group, called ample. A populato ca be fte or fte. Data: A collecto of values to be used for statstcal aalyss. Raw Data: Collected data whch does ot eed to be umercal..e. weghts of certa set of studets, days of the wee, etc. Array: Arragemet of raw umercal data ascedg or descedg order. Rage: Maxmum data mmum data. Class Iterval: A class terval s a dvso of data for use Hstogram(a type of Bar graph). For stace, t s possble to partto scores o a 00 pot test to class tervals of -5, 6-49, ad The ed umbers are called class lmts; the smaller umbers are Lower Class Lmts (LCL) ad the larger umbers are the Upper Class Lmts (UCL). The umbers , , , are called class boudares. For example 0.5 s a lower class boudary ad 5.5 s a upper class boudary of the frst class. Class Iterval ze (wdthessc): Upper class boudary lower class boudary. Class Frequecy: Number of dvduals belogg to each class. Class Mar(CM): The mdpot of the class terval. LCL + UCL LCB + UCB CM or CM. Frequecy Tables (Frequecy Dstrbutos): The frst step drawg a frequecy dstrbuto s to costruct a frequecy table. A frequecy table s a way of orgazg the data by lstg every possble score (cludg those ot actually obtaed the sample) as a colum of umbers ad the frequecy of occurrece of each score as aother. mply a frequecy dstrbuto s a arragemet of data by classes together wth the correspodg class frequecy. ouç Zorlu

2 Computg the frequecy of a score s smply a matter of coutg the umber of tmes that score appears the set of data. It s ecessary to clude scores wth zero frequecy order to draw the frequecy polygos correctly. Geeral Rules for formg Frequecy Dstrbutos:. step: Fd the rage. step: Dvde the rage to a coveet umber of class tervals havg the same sze (possble umber of class tervals are chagg betwee 5 ad 0). 3.step: fd the class frequeces. Hstograms: A hstogram s draw by plottg the scores (mdpots) o the -axs ad the frequeces o the Y-axs. A bar s draw for each score value, the wdth of the bar correspodg to the real lmts of the terval ad the heght correspodg to the frequecy of the occurrece of the score value Frequecy Polygos: A frequecy polygo s draw exactly le a hstogram except that pots are draw rather tha bars. The -axs begs wth the mdpot of the terval mmedately lower tha the lowest terval, ad eds wth the terval mmedately hgher tha the hghest terval. Relatve Frequecy of a class: It s a percetaeg whch s obtaed by dvdg the frequecy of the class to the total frequecy of all classes. Relatve Frequecy Dstrbuto: Arragemet of data by classes together wth the correspodg relatve frequeces. Cumulatve Frequecy: The total frequecy of all values less tha the upper class boudary of a gve class terval s called the cumulatve frequecy upto ad cludg that class terval. Plottg scores o the -axs ad the cumulatve frequecy o the Y-axs draws the Ogve (cumulatve frequecy polygo). The pots are plotted at the tersecto of the upper class boudary of the terval ad the cumulatve frequecy. Relatve Cumulatve Frequecy: Ths s also called the percetage cumulatve frequecy whch s obtaed by dvdg the cumulatve frequecy to the total frequecy. Drawg the -axs as before ad the relatve cumulatve frequecy o the Y-axs draws the Percetage Ogve (relatve cumulatve frequecy polygo). Ogve: The graph showg the cumulatve frequecy less tha ay upper class boudary s called a cumulatve-frequecy polygo or Ogve. ouç Zorlu

3 Example : The frequecy dstrbuto of the ages of sample of 400 dabetcs obtaed by a research physca are gve below. Age (years) No. of Class mars dabetcs Costruct (a) a Frequecy Hstogram ad a Frequecy Polygo. (b) a Relatve Frequecy Dstrbuto. (c) a Cumulatve Frequecy Dstrbuto ad a Ogve. (d) a Relatve Cumulatve Frequecy Dstrbuto ad a Percetage Ogve. (e) Estmate the percetage of dabetcs whose age s uder 4. Measures of Locato The ample Mea: Oe obvous ad very useful measure s the ample Mea. The mea s smply a umercal average. uppose that observatos a sample are,,...,. The sample mea s The ample Meda: The sample meda s ( ) f s odd + / ( / ) + ( / ) + f s eve ouç Zorlu 3

4 Measures of Varablty(Dsperso) Dsperso: The degree to whch mercal data ted to spread about a average value s called the dsperso or varato. The most commo measures of dspersos are rage,varace ad stadard devato. max ample Rage: The smplest measure of varablty (dsperso) s the sample rage. m ample Varace: Let,,..., deote sample values, the quatty ( ) or s called the sample varace. ample tadard Devato: The sample stadard devato s ( ). Example : A maufacturer of electroc compoets s terested determg the lfetme of a certa type of battery. A sample, hour of lfe, s as follows: 3, 6,, 0, 75, 6, 5,, 8, 7 (a) Fd the sample mea ad meda. 43 The sample mea s Arrage the data a Array as: 0,,6,7,8,,3,5,6,75. ( ) + 5 ( 6) 8 + ce 0 s eve, the meda s 0. (b) Fd the sample varace, stadard devato ad the rage ( ) The sample varace s ( ) ouç Zorlu 4

5 The sample stadard devato s ( ) The rage s max m Mea ad Varace computed from Grouped Data: Mea: If,,..., occur f, f,..., f tmes, respectvely (.e. occur wth frequeces f, f,..., f ), the arthmetc mea s where N s the total frequecy ad f N represets the class amr of the th class. Varace: If,,..., occur wth frequeces f, f,..., f respectvely, the varace ca be wrtte as f ( ) N or N f f N( N ). Example 3: Use the gve frequecy dstrbuto of the weghts of the 00 studets at YZ Uversty to fd the mea, varace ad the stadard devato. Weght(g) Frequecy( f ) Class mars( ) f ( ) f ( ) N f 00 f Mea f g N 00 ouç Zorlu 5

6 Varace f ( ) g N 99 f ( ) tadard Devato N g Example 4: The followg data represets the age of the buldgs ( years) of a gve area Lefosa (a) Costruct the frequecy table usg the followg classes: -7, 8-3, 4-9, 0-5, 6-3, (b) Draw the relatve cumulatve frequecy Hstogram ad the Percetage Ogve. (c) Estmate the percetage of houses whose age s uder 5 years. (a) Class Tally Freq.( f ) Cum Rel. Rel. cum boudary Freq( cf ) f Freq( N ) cf freq( N ) /30 6/ /30 6/ /30 3/ /30 7/ /30 9/ /30 30/30 6 f 30 ouç Zorlu 6

7 Example 5: Thrty AA Batteres were tested to determe how log they would last. The results, to the earest mute, were recorded as follows Costruct (a) a Frequecy Dstrbuto. (b) a Cumulatve Frequecy Dstrbuto. Example 6: I a -weely study of the productvty of worers, the followg data were obtaed o the total umber of acceptable peces whch 40 worers produced (a) Costruct a frequecy dstrbuto usg 6 classes. (b) Draw a frequecy hstogram. (c) Costruct a relatve frequecy hstogram. (d) Draw the percetage ogve. Example 7: The followg table shows a frequecy dstrbuto of the weely wages of 65 employees at the P&R Compay. Wages No. of employees $ $ $ $ $ $ $ N65 (a) Costruct a frequecy hstogram (b) Costruct a cumulatve-frequecy dstrbuto (c) Costruct a ogve (d) Evaluate the umber of employees earg () less tha $88.00 per wee () at least $63.00 per wee but less tha $75.00 per wee (e) Compute ad for grouped data. ouç Zorlu 7