Table of contents. Statistical analysis. Measures of statistical central tendencies. Measures of variability

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1 Table of contents Statistical analysis Meases of statistical cental tendencies Meases of vaiability Aleatoy ncetainties Epistemic ncetainties Meases of statistical dispesion o deviation The ange Mean diffeence The vaiance The standad deviation Coefficient of vaiation Meases of ncetainty Systems of events Entopy Random stochastic vaiables Discontinos discete andom vaiables Moments of discete andom vaiables Pobability distibtions of discete andom vaiables Binomial distibtion Poisson distibtion Continos andom vaiables Pobability Density Fnction Cmlative Distibtion Fnction Pobability distibtions of continos andom vaiables Unifom distibtion Simpson s tiangla distibtion Nomal distibtion Lognomal distibtion Shifted eponential distibtion Gamma distibtion Shifted Rayleigh distibtion Type I Lagest vale Gmbel distibtion Type III Smallest vales fo ε 0 it is nown as the Weibll distibtion Beta distibtion Type I Smallest vales distibtion Combinations of andom vaiables

2 Meases of statistical cental tendencies Meases of cental tendency of a set of data,,..., N locate only the cente of a distibtion of meases. Othe meases often ae needed to descibe data. The mean is often sed to descibe cental tendencies. Mean has two elated meanings in statistics: the aithmetic mean the epected vale of a andom vaiable. In mathematics and statistics, the aithmetic mean, often efeed to as simply the mean o aveage. The tem "aithmetic mean" is pefeed in mathematics and statistics. The aithmetic mean is analytically defined on a data set,,..., N as it follows: N N i i

3 Meases of vaiability Statistics ses smmay meases to descibe the amont of vaiability o spead in a set of data,,..., N. The vaiability applies to the etent to which data points in a statistical distibtion o data set divege fom the aveage o mean vale. Vaiability also efes to the etent to which these data points diffe fom each othe. Thee ae seveal commonly sed meases of vaiability: ange, mean diffeence, vaiance and standad deviation as well as the combined mease of vaiability defined as coefficient of vaiation with espect to the mean vale. Uncetainty epesents a state of having limited nowledge whee it is impossible to eactly descibe the eisting state, a fte otcome, o moe than one possible otcome. The ncetainty dobt in statistics and pobability theoy epesents the estimated amont o pecentage by which an obseved o calclated vale may diffe fom the te vale. Uncetainties can be distingished as being eithe aleatoy o epistemic. Aleatoy ncetainties Objective o etenal o iedcible ncetainty aises becase of natal, npedictable vaiability of the wave and wind climate o of ship opeations. The inheent andomness nomally cannot be edced althogh the nowledge of the phenomena may help in qantifying the ncetainty. Epistemic ncetainties Uncetainty is de to a lac of nowledge abot the climate popeties. The epistemic o sbjective o intenal o modelling ncetainty can be edced with sfficient stdy, bette measement facilities, moe obsevations o impoved modelling and, theefoe, epet jdgments may be sefl in its edction.

4 The ange A mease of statistical dispesion o deviation is a eal nmbe that is zeo if all the data ae identical, and inceases as the data becomes moe divese. It cannot be less than zeo. Most meases of dispesion have the same scale as the qantity being meased. In othe wods, if the measements have nits, sch as metes o seconds, the mease of dispesion has the same nits. Basic meases of dispesion inclde: Range Mean diffeence Vaiance Additional meases ae: Standad deviation the sqae oot of the vaiance Coefficient of vaiation the standad deviation divided by the mean vale See Ecel eample: GadationRateofNavalAchitecteinZageb The eample pesents the statistical popeties of the inpt and otpt ates of nmbes of stdents naval achitecte at the Faclty of Mechanical Engineeing and Naval Achitecte at the Univesity of Zageb. Upisano/diplomialo Stdij bodogadnje Upisano Diplomialo Godina

5 The ange In desciptive statistics, the ange is the length of the smallest inteval which contains all the data of a dataset,,..., N. The ange is calclated by sbtacting the smallest obsevation sample minimm S min fom the geatest sample maimm S ma and indicates of statistical dispesion RS ma -S min. The ange, in the sense of the diffeence between the highest and lowest scoes, is also called the cde ange. The midange point, i.e. the point halfway between the two etemes, is an indicato of the cental tendency of the data. It is not appopiate fo small samples. The mean diffeence In pobability theoy and statistics, the mean diffeence is sed as a mease of how fa a set of nmbes of a dataset,,..., N ae spead ot fom each othe. It is one of seveal desciptos of a pobability distibtion, descibing how fa the nmbes lie fom the mean epected vale. Fo andom vaiable X,,..., N with mean vale the mean diffeence of X is: N MD MD abs i RMD N i o the elative mean diffeence is then The vaiance In pobability theoy and statistics, the vaiance is anothe indicato sed as a mease of how fa a set of nmbes ae spead ot fom each othe. Fo andom vaiable X with epected vale mean E[X], the vaiance of X is: N N Va σ i i N i N i Poof: N N N i Va σ i i + N i N N N N The standad deviation i i i The widely sed mease of vaiability o divesity in statistics and pobability theoy is the standad deviation. It shows how mch vaiation o "dispesion" thee is fom the "aveage". The standad deviation is the sqae oot of its vaiance: σ X Va X The standad deviation, nlie vaiance, is epessed in the same nits as the data. The coefficient of vaiation Othe meases of dispesion ae dimensionless scale-fee. In They have no nits even if the vaiable itself has nits. In widest se is the coefficient of vaiation defined as follows: σ X COV X X Fo measements with pecentage as nit, the coefficient of vaiation and the standad deviation will have pecentage points as nit.

6 Meases of ncetainty Systems of events Random events ae in geneal consideed as abstact concepts and the elations among events ae chaacteized aiomatically. The algebaic stcte of the set of events tns ot to be Boolean algeba. The disjoined andom events E j with pobabilities p i p E i, i,,, N confige a system S N in a fom of an N-element finite scheme: E E Ej EN S N p p E p p E pj p Ej pn p EN The pobability of a system of events S N is then in geneal N p S p. Fo a complete distibtion is p PN o p S N. A system of N events: E, E,..., EN is called a complete system of events if the following aioms hold: E,,, N a E E fo j b j E + E + + EN I c The " " in a and b means an impossible event and "I" in c denotes a se event. The fact that Ej and E ae eclsive is epessed in b. The c denotes that at least one of the events E,,,..., N, occs. N i i Entopy Uncetainty of a single stochastic event E with nown pobability ppe 0 plays a fndamental ole in infomation theoy. To each pobability can be assigned the eqivalent nmbe of pobabilities o events ν E / p E. The entopy of a single stochastic event E can be intepeted accoding to Wiene 948 eithe as a mease of the infomation yielded by the event o how nepected the event was and can be defined as the logaithm of the eqivalent nmbe of events ν E as follows: [ ] H E log ν E log / p E log p E The nit of nepectedness H / epesses how nepected is fo eample to get a tail when flipping a coin. Moe impotant than nepectedness of a single stochastic event ae the ncetainties of systems of N events. The ncetainty of a complete system S of N events can be epessed as the weighted sm of nepectedness of all events by the Shannon s entopy Shannon and Weave, 949, as it follows: N N N H S p logυ p log/ p p logp N j j j j j j j j j The ncetainty of an incomplete system of N events S can be defined as the limiting case of the Renyi s entopy 970 of ode, as it is shown: N R HN S pj log pj S p j

7 The definition of the nit of ncetainty accoding to Renyi 970 is not moe and not less abitay than the choice of the nit of some physical qantity. E.g., if the logaithm applied is of base two, the nit of entopy is denoted as one "bit". One bit is the ncetainty of a system of two eqally pobable events. If the natal logaithm is applied, the nit is denoted as one "nit". Otcomes with 0 pobabilities do not change the ncetainty. By convention, 0 log 0 0. Some chaacteistics of the pobabilistic ncetainty meases and popeties of the entopy ae smmaized net. The entopy H N S is eqal to zeo when the state of the system S can be sely pedicted, i.e., no ncetainty eists at all. This occs when one of the pobabilities of events pi, i,,...,n is eqal to one, let s say p and all othe pobabilities ae eqal to zeo, p j 0, j. The entopy is maimal when all events ae eqally pobable and the pobability of faile is eqal to pi /N, fo i,,..., N, and it amonts to H N S ma log N, that is the Hatley's entopy 98. Hatley s entopy 98 coesponds to the Renyi s entopy of ode The entopy inceases as the nmbe of events incease. The entopy does not depend on the seqence of events: H n p,p,...,p N H n p,p,...,p N, whee is an abitay pemtation on,,...,n. The niqeness theoem by Khinchin 957 states that the entopy is the only fnction that meases the pobabilistic ncetainty of systems of events in ageement with hman epeience of ncetainty. see Ecel eample U-EntopyDieCoin

8 Random stochastic vaiables Deteministic vaiables ae nomally descibed by thei popeties: N - Nominal vale o the eact vale And possibly with toleances T - Toleance tt/n - Relative toleance Desciption of chaacteistics of andom vaiables: ON+N Mean vale O/N- Mean deviation of the nominal vale bias Vaσ Vaiance σva / Standad deviation COVσ/ Coefficient of vaiation F Pobability distibtion: PDF pobability density fnction CDF cmlative distibtion fnction Empiically, it is possible fo pactical pposes to elate the toleance of the deteministic vaiables to the standad deviation of andom vaiables: Tn σ Fo eample, spposing a nomal pobability distibtion, fo n3 is less than 7 samples ot of 0000 epected to be otside the toleable magins. In othe wods, the confidence inteval is 99.73% that a andom sample will be within the pescibed toleance inteval of ±3σ. See Ecel eample: MSpopety-plating-statistics The eample pesents the statistical analysis of mechanical popeties mild shipbilding steels MS fo olled plates and pofiles obtained by tensile testing in the Laboatoy fo epeimental mechanics at the Faclty of Mechanical Engineeing and Naval Achitecte at the Univesity of Zageb. See Ecel eample: MSpopety-pofils-statistics

9 Discontinos discete andom vaiables Definition: The vales of discontinos discete andom vaiables,,... ae pobabilities p, p,... with the popety p i. i Moments of discete andom vaiables m p i i i M p i i i Epectation Vaiance σ p i i i i i V p i

10 Pobability distibtions of discete andom vaiables Binomial distibtion n n P pq Mean np Sigma npq see Ecel eample DD-DistibtionBinomial.00 Binomial distibtion CDF P n5 p Mean.5 Sigma.5 PDF Binomial distibtion 0.5 n 0.4 P n5 n0 0. n0 n50 0 p0.5 p0.50 p

11 Poisson distibtion m m P e! m np> 0 Mean m Vaiance m Sigma m COV / see Ecel eample DD-DistibtionPoisson Poisson distibtion m Poisson disibtion Cmlative density fnction 0.6 P

12 Continos andom vaiables A andom vaiable is called continos if it can assme all possible vales in the possible ange of the andom vaiable. In continos andom vaiable the vale of the vaiable is neve an eact point. It is always in the fom of an inteval, the inteval may be vey small. Pobability Density Fnction PDF The pobability fnction of the continos andom vaiable is called pobability density fnction of biefly p.d.f. It is denoted by f and epesents the pobability that the andom vaiable X taes the vale between and +Δ whee Δ is a vey small change in X. Cmlative Distibtion Fnction CDF In tems of pobability density fnction the cmlative distibtion fnction is defined as: CDF f d see Ecel eample DC3-DistibtionNomal CDF PDF Mean Sigma Moments of continos andom vaiables m + f d + M f d Epectation Vaiance + f d + σ V f d

13 Pobability distibtions of continos andom vaiables Unifom distibtion f b a σ 3 b+ a a + b b a σ 3 see Ecel eample DC6-DistibtionUnifom F a a b a σ 3 b+ a

14 Simpson s tiangla distibtion a a f f b f b b a b+ a σ 3 f b b a a + b b a σ 3 see Ecel eample DC7-DistibtionSimpson Simpson's tiangla distibtion CDF a F f b b a PDF

15 Nomal distibtion σ f e F Φ πσ σ Standad nomal cmlative pobability ϕ e Φ π e σ > 0 π σ see Ecel eample DC3-DistibtionNomal CDF PDF Mean Sigma

16 Lognomal distibtion ln y σ y ln y f e F Φ σ y π σ y y ln / σ + σ y ln + σ / σ y y + y y e σ e e see Ecel eample DC4-DistibtionLogNomal CDF y σ + σ ln y σ y Mean 8 Sigma 5 PDF Mean y.9 Sigma y see Ecel eample DC4-DistibtionLogNomal-MildSteel CDF Lognomal distibtion of yield stess of mild shipbilding steel Mean 68 Sigma 30.5 Mean y5.58 Sigma y0.4 PDF

17 Shifted eponential [ ] o e f λ λ [ ] o e F λ > 0 λ o o λ λ + λ σ

18 Gamma distibtion λ λ f Γ Gamma fnction e λ e d o Γ, λ F λ > 0 > 0 Γ Γ Γ! Γ Incomplete gamma fnction o σ λ λ, e d λ λσ f e θ Γ / θ F, γ Γ θ θ σ θ σ θ σ see Ecel eample DC9-DistibtionGamma Gamma distibtion CDF PDF

19 Shifted Rayleigh α α o o e f α o e F π α + o v π α σ

20 Type I Lagest vale Gmbel αn o e f αn e e α n o αn o e F e α n > π n + σ αn α n π n αn αn σ 6 see Ecel eample DC5-DistibtionGmbel 0.05 CDF 0.04 CDF Mean 00 Sigma 0 n 09 an PDF

21 Type III Smallest vales fo 0 ε it is nown as the Weibll distibtion f e ε ε ε F e ε 0 > ε Γ + σ ε ε Γ + Γ + Type III Smallest vales fo 0 ε it is nown as the Weibll distibtion e f ε ε ε ε ε e F ε ε 0 > Γ + + ε ε + Γ Γ + ε σ

22 Beta distibtion, q q a b q B b a f 0 > q 0 > Beta fnction, q q q B + Γ Γ Γ q a b q a q q q a b σ

23 Type I Smallest vales distibtion o o e e f α α α e e F α > 0 α n α α π σ Type II Lagest vale o o o e f o e F 0 > Γ n Γ Γ n σ

24 Combinations of andom vaiables Fo linea combinations y ax + ax ax of andom vaiables X+ X X,,..., With given aithmetic means σ, σ,..., σ And standad deviations Theoem : The mean vale of the linea combination of andom vaiables is the sm of the mean vales of components: E a X + a X a X a E X + a E X a E X a+ a a. Theoem : The vaiance of the linea combination of andom vaiables is the sm of the vaiances of components: σ aσ + aσ aσ