Descriptive Measures (Measures of Skewness, Kurtosis, Central Tendency ,Dispersion), Study notes of Descriptive statistics

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DOLEEEL EE DY q J SHSFEDLEEEEEEYY Pb h Desc ip ve Measures: © Quantitative data- Tp a mass exhibit certain gene! characteristics oF they differ from each othe, in the folloising ways 1] Centra) tendenc) oy GIG oe They show a tendengy 40 concentrate at certain Values , usuarh) Somewhex in the Cente of the dietribution Measures of this tendeng) are callél measures of cemya) tendentyy of averages . 2] Measures of Variation or dispersion - The data vary about a measure Sp isainae Al) tendeny and these measures of deviation are calla] measures of varidHon oF dispersion We use measures of dispersion +e Know how Scatteral oy spread out the data is. @] Measures of skewness ~ The dota ina Frquenty distribution may fat) into symmetrical oF asymmericay poiterns. The measures of the direction and degree of asymmetry axe coNed measures of skewness ness GJ Measure of Ruriosis - Polynon of freq os Flamess of the frequengy curves ave called MeasUTES 0 + Frequeng 4able- A table Showing the distabutHons oF different classes is callel a Frequency tabee. 4 Grouped Frauency distribution - The manner in distributed over the Class intenals is called the grouped prgcng) Variabe tt Inclusive classes: The classes C\$—20, 21-24, 25 -26} in which both the upper and joer limi are included are catied ‘inclusive classess’ +tevmedt as inclusive type distribution ar Frequency densities « cer] digtribution exhibit Flotness oy peo iF Kuy+0S)s - the prequencies in the ic ve hich the class Frequencies o distyibution of the FrequerY _ rp one class imtenal has a Die ASIAe OS) ye class Width seman yange, the data is move dense in that interval. +t Inderminat classes. open end casses like lese tan 2 or grea tha B for approximate j=] + 3.322 a ; Where N is the total Frequeng oe The nitude of the closs Inierwal-. oneal difyererce yerween the apelass and the smaiiest observation ag frevence between the uppey ltoret — The width ox size of tHe class - that Is the at and \ower limit of +the class. | Magnitude = Upper Limit — Lower Limit | * Broad class ipteruas Cie less number of Classes) wil) yield ony rough esti mats * aml class interwals (i large number of classes) wi jie'd uy Hepes oF accuraagy Tassel = Lirias thal ence and laygest values that can bebo) to a class wrrva\ (roures Tequency table. 0 lowes a ieee Oss LiMit 3 The mayest value in the class. 5 20 4 Upper class Limits 3 The as Value in the class 335 Sys = aio) * The clase limits shourd be chosen in such a way thot the mid- value of the Class interway and aciuay AVeVae Of the observations In that class interwol are as NEO 40) each “Oey OS toss nie 4 continuous Payweng distribution T Not in interval jipe \5- * = Applicoble in B~\0 ;\0-)5 (ceceas We 41m) of the Frequeng distributor with * clases ig Rnown as Frequency distri bution © Exclusive type classipication- amy * Upper limit of each class are excluded from the TespeCtive classes and are included in the immediate next class are known as 19) 20-24 WN hich 14-20 noe tectdded com nuous TExdlusive classes’ and the classification is sewed as sexclusive type classi ficersion / ae Grophie tepresemation Of a Frequene distyi bution - T+ is oPem useful io reprecemt & Frequeng distribution by means of ~a Ataqyam which motes. the unaieldy clara Inieligible and the comparasion of two oF more Poqueng distr bution ie q Histogram ~The eee of consinuols rectangle ‘scaled histogram . If net consinuols — then make clase comHnudous * Note: ‘ eee bounderies: The upper arc) lower class himi+ of the new exclusive type Classes OTE Known as class boundaries . abw gl je thy aor between the Upper WMi+ oF any class and the lower limit Ff pe eeciRA class , the class boundaries for any class are then given by I Upper class bounday = 10pperoelaiss Mpiboted lower clase houndayy = Lower | Class limit = de He Praqyvency peugee: 1. Find Midpoi ns of each class. Midpoint = Lower Jimit + Upper limit Zz 2. Soin the points with ag Wes, NR — ae poe Oy CoMinuous Frequency distribution. , di = (1; -A) where Ais AP aybitray poim and P's the \coryran magni h OF class Interva), here we hare hai = Cx;~A) pa n 1s Asin Z Cpe N =! Brample 72 this : A=23) Bho? eee) [rid he [eying [ae] a, Oey | hy Pliner i pean Saeieoage ar i Desiree} Seas ee hs hill log = SNC Sl ie ie 5 | = =e iam Bogie eS Lie. 2h-32 | : Sn 23 2u | rs) (7) | 2—lo 36 |) | = ae | hy of feds a 1 | Tota E25 Me Pas = -solug Til 7 Properties Of Arithmetic Mean Property Me Bipgpiare sum of the deviation of a set OF pales Tig their arithmetic mean 35 Zcr0 TE GNF Fe 1yae.. gin 38 the Frequency, disribuien ther nH . . ee h mt h Me h Proog . 2 MiG =i) = Ze < xe Fis ait) een Le = = j2) =) n ae ENG pes Also 2 a : s, 2 Fi oNx p= a a ) ECG Ha NN HO, ip) Se SsSOeeseeSOeaeeoeaneanmeeannnannannunenr........ Property oles, We sum of the Syuares %) js minimum when +tayen alot mean. ec OD Proop. Xf; MM) ea ts -2 ae Me z= 2 tray 1=) par 4 - Qn 70 We hove ;, Prove that 92 4s minimum when A= “ 0) [ere | : Z is minimum j¢ OZ 2 an) apy 2. Soar etity-A)=o ey Sills eitva teal : EAI oe Ne H Bs ee = =) - =X a3 az Di ena =ENEne Foz 2 a Se R= Ni so Ae Giclee Property ey eG) =1,2, + KJ are the means of 4 CUVNIREN Beanies Gs Sz2s i} 7 CS Nae ky Yespectivey , then the mean Y oy the ) a: Wie eee Ge ) Re ee Saar ) S = 2s dia which is farhenvie equal of c 5 by Frain d+ an overage 2b] You can tave o srip cahich emails travelling $00 km. speed of 60 Km. per hour, 3009 km. by boat at an averose speed of 25 Kon ph | speds equa) = oo xm by plane at 350 Km. per hour and finally 15 km “by tai ot 25K. poh i nomic mean = Whod is the average spect for the entive distane - Jotal distance = 900 +3000 +h00 +350 +15 = L358 5 CaaS Hine GEREN Ge Un ete, A OO SE) rakes SS 60 2 3gs0 25 ed. led s es Bi Bin Gusiee speed = 3\.55 fics by the St Selection of An eS 4 distance Ayithmehic mean ig best of. all averages . e te Partition values . These ave the values which divides the Series imto a numbey ans OF equal parts: sé Go OGuariiles - te three points which divide the series into pour equa) ports ave called | rf quaranes. | OF ating a. @, = the value which exceed 25 4 \ + may i we Fi Gy = whith coincides With median | g 4 Ga~= whith coined exceed 15 % Thar cere obyained for | Seat z AYFEYEOY Values of Y the Number of heads,ave shown in the Follocin | a= ; sable de, | , median ; quartiles, U4 th decil E 47th percentile ; i J as Pa ‘Sa ee bya eaoes B coins wer fessied 2h K F he ; Be A= Ka OR ets 2 ey ea! ee se we 6h eee | 1 a ZO WoW BDSG Is als 25¢. 28.6 re FCs) Zi us\ 728/8.2 va | a ’ : 4 _af¢ \ty 290 249 2° dis EAA a gl S n® 285 256 | 250 = 254 - it) 24" 2 joo ag (TOP OG 95 Se 947 3) gy =4 9955 Dus Fo rps NZ a Eiample > Pious} ae in statistic. M's the digribution of mares obtained by 509 candidates FSTOR apecivil Services examination . Re ene eg m 1 © oF Candidates . uid Soa aera Calculate 4), eo eee Find 4), * l8ey Quartile fein : . Mina STF 70% of the candidates poss in the paper mM a OS Momene dents ; : ATS more 44g, |. eS ay @ pass leandidates . ae | ands, | Sa a 2 Ue ee __ class r cr ee ee ae Tae 29 Eo Abo lo-20° 60 100 Sa San Avo 20-30 300 ous cel Sele CEN 100 hoo aie \oo | ; | ho-So 70 4710 So } | —__ 30° | Bo -above . _ 30 500 ‘ | n= G08 y Q\= rake 125 ae C\0ss = 20-36 oS ~ \oo = eCXOMs Bi 2204+ (15-2) re cee 20 4 = 125 gailed the exams. Sabet. 225.5) 25h do - 30 =>2o+ 8 ore = Pome lor (25- Ao = 3045 200 4 O16 fared. UN S35 0 Da) We RO a rae: fai) 2560 é 25 — f| Sin aye oo ass = 115 . fai) = 28 12.5 i) poy (oe 8 half = 20 + ee ee race 25) a I pees 20850 Uso PES Remar : = 4 : ‘a The median can aise be cated ag Follows ., ) . from the poiny Intersection OF “less than! ogive <& “move than! y=) é aah perpendicaity jy OTIS Avsckine op te Point oC brained gives median. oe AN poision Values, viz, deciles & perce miles can be srmilary octed From ee Aissoved Review aa on Measures of cemtya) tendengy ‘ ( 2 uM (What ove Gree ont wngrouped Frequency ietributions? What ave thelr uses . . i TE. uen What axe the consideration that has 4o beor in mind while aed ude es distribution 2 G & 3 Sour 6 i ouped Frequengy distribution. il Je ividud! value along ¢ — This Shows dato in i+6 oes OY Yaw poe Si LCD INARI Xs g ¢ With irs Requeny ¢ how often i+ occurs) _ ¢ ®@ Use this When: ¢ WThe yrange of data is mal) ' 21 Each individuol vatue is important . , © Grouped Frequeng) distibusior { ( ( ( ‘ ( { ( Oise ie Gccd When the data is layge or spread out. Instead of fistin indlividuay values, dota is divided into class torervals, and the Fraquengy of each Class ji, shou —=Usz, EO emis larye io mance individdany 2 You wary to analyze the ata More eaci | © consider aon Uahile oe a Pequenc Distoibuston . 1) “aye of Pata °Find the difference between +he Meee %& lowest volues to decade how wide Jour data is, ZJ Number of clase, —- : Usual) between B40 18 fos aye: data Tos Few classes bide BELANS $55 ing BY confusir ae 31 class Width Csize of jnrerval)— PY) imtervats shoud ideaiy be of : : formula = Highes Va(We — Lowest value J que) ordth, Number of — c\accey 4] Moray) ©xclusire asses — No Each date poink shoud par | Overs \apc. 5 Exhaustive covevane — FY) data shouli] be included - no Value shouid be leh ow | 9) Tally Be Hequercy coum | 1) Class limits & boundaries — Be clan wheather youve Using lactustic BY EX clr | 3) dara +ype - disode oy continuo - N+0 ONN one Class, , 2) Explain the method of construct Rishon zs Frequency Polygar Which | oH OF these tuo, 15 beter -epresentdti¥e of Frequencies of ee ee ee AS4b 5304 0364 203/ >ofor o porscatar Geek HeenenD is better - eaciey +o read the exact frequenc © For whole cl compayison , Frequeny poyen is beHer - \+ clearly Shows the Overall pattern or trend in the data. “Nal SE con bars . 2) X-axis class intervais, Y- axis : Frequencies 3) Bars axe joined with no aps. * Frequency potijen Wade by plowiin mid points of classes and Spinks With Ines, Ystartke and end& at the X-akts Y particatoy eal: ~ Histogram - shows exact Frequency early D Whole group - Frequeng polygon - shows overal) tend better 3] What OIE the princhles gern the choice of : 1) Number OF class intervals. ii) The length of the class interval. iii) The aie of th class intewal. > 1) numbey of ¢loss Tntervals” Yshould not be +00 many or 400 fer . 2) Idea\ tame 5 40 16 intervals . re needed Use Formuia>. RK =1l +3. 322104 ,,0N) R=number of jmtervals . N= number of observations Dlength of class intervals, snany ; : ‘ unless specified (ayia) clase size helps in YAN imtervais should be of equal width , fect alts P. comparison ) a 2) Lae =CMaximum va)jue - Minimum value) = Number op classes 3] Mid pois of class Intervals * J Midpoint = Lrower lierit + upper limit J +2 cepya) value of the class and ic usec! in mean <& |PYeqUenty T+ yepresents the eee calculahions | ul Write Short mores on. Peete 4 4 “ie 8 distribution - A Frequeng distribUROM 15 a CIO) +40 ie eae 2) showing how oPten each value or qr? of values occurs. Tt heips 4° understand patters and is dsad 40 Create qrophs Nine Wee QZ frequeng polygons. dt When +o use each: ) DM Type Use ghen: AYN ==) AY valles are equarly imponont © no ettreme outliers, GM = Rates or pete ate involved . Hen => You ave Serr) rates |ike speed or time b] compare mean, Median & mode : Featurs ele Median mode Def = SYM/number D> Widdle value > Value that occurs / oF values When dota js the mos + Ordered € of : Thverage => Mathmial 2 positiond) = positiones ) Affected by No . Extveme value 28 ? Me A uses ANdaa DS xs => No => po, Easy to understand > yes A ys =) Jes € calculate Algebraic use D> Jes 3) 0) Dip vnigness > +e 2» yes > Ne & s i) ae) Discrete ] Frequeng Aater coi Ha Gest Fox > symrmeimta| — > SHEL ees distri puten qe when 4o use: Mean > When data is norma) | symmetrical and has no oupiers eq g pycsaie Salary OF simila?- level employes . Median > When doia is srecsed or has extreme values. eq : Income distribution ina populations Mode > Wher the mos Prequert value is important . eq: Most commen Sloe size sold. : = SD Bplain 9p = 4, - median , quart raphic we the wolues of uae 01] Medion e oe PPC method oF Jocas ah, aS =| Wey Nels igane Median GF G3 Steps: MSE CumUlative Frequeng Curve Cs oe ‘ dR, pore Cs Yass cumulative fy, juency table. 2) per ne ie a eh =| . Y= cumulative Prequencies . 3 We Ch= class) Vimks7 t= D “ the Wiese sl a YN z ' soe Foy Median On eae 3) an ; r “a @y Dia . ial a hor zone Wines from these poins 4o Inversect the carve Drop Nersiay Whes From, the pois of Intersection 40 the X-axis . 6 5i| TVhe X Valuec Whee Ahey +he axis ave Median, @, & a, Tay Mode — Osjn, Histogsany tallies Saw ( Mean ~ Cannot he Find Graphically. Can aie al foc D) eneme deviations about median ic least. Values | => Coa Un roupedl Data) : aaa let's consider ordered data~ i ee &] sum OF absolute Gh let's define a Function: n Fea) = |¥--a\ es case!: odd number of observations: le+ n= 2m4) 150 the median is nn , We WIN show +hat the funcion Fray i, Minimize at a=