Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca affect them. It s therefore essetal to take to accout these varabltes to use statstcal methods to terpret the results obtaed through a expermet. 6.. Itroducto A example of the use of statstcal aalyss of expermetal data s to use a represetato uder the form of a hstogram. Let us cosder the followg data represetg the measuremet of a temperature. umber of readgs Temperature ( C) 089 09 094 4 095 8 098 9 00 04 4 05 5 07 5 08 4 0 3 5 The data are frst arraged to groups called bs. Here the sze of a b s 5 C. The bs have to satsfy a couple of codtos: - The bs usually have the same sze ad cover the etre rage of the data wthout overlap. Fgure 6.. Hstogram. Istrumetato ad Measuremets \ LK\ 009 47
Chapter VI Statstcal Aalyss of Expermetal Data The above bell shaped curve of the hstogram s typcal of expermetal data (although ths s ot a rule, see fgure 6. for other types of hstograms). Fgure 6.. Dfferet dstrbutos of data. a) symmetrc; b) skewed; c) j-shaped; d) bmodale; e) uform. - Dscrete ad radom varables: Cotuous radom varables: It s a varable that ca take ay real value a certa doma. Dscrete varables: It a varable that ca take a lmted umber of values. 6.. Geeral cocepts ad deftos Populato: The populato comprses the etre collecto of objects, measuremets, observatos, ad so o whose propertes are uder cosderato ad about whch some geeralzatos are to be made. Sample: A sample s a represetatve subset of a populato o whch a expermet s performed ad umercal data are obtaed. Istrumetato ad Measuremets \ LK\ 009 48
Chapter VI Statstcal Aalyss of Expermetal Data Sample space: The set of all possble outcomes of a expermet s called the sample space. Is ca be a dscrete sample space of a cotuous sample space. Radom varable: It s a varable that wll chage o matter how you try to precsely repeat the expermet. A radom varable ca be dscrete or cotuous. Dstrbuto fucto: It s a mathematcal relatoshp used to represet the values of the radom varable. Parameter: It s a attrbute of the etre populato (exp. average, meda, ) Evet: It s the outcome of a radom expermet. Statstc: It s a attrbute of the sample (exp. average, meda, ) Probablty: It s the chace of occurrece of the evet a expermet. 6... Measures of cetral tedecy - Mea: x = = x x Ad for a fte umber of elemets: μ = = - Meda: t s the value at the ceter of a set, arraged ascedg or descedg order. If the sze of the set s eve, the meda represets the average of the two cetral peaks. - Mode: It represets the value of the varable that correspods to the peak value of the probablty of occurrece of a evet. 6..3. Measures of dsperso - devato: d = x x d - mea devato: d = = - stadard devato (for a populato wth a fte umber of elemets): ( x μ) σ = = - Sample stadard devato: Istrumetato ad Measuremets \ LK\ 009 49
Chapter VI Statstcal Aalyss of Expermetal Data devato. S = = ( x x) t s used to estmate the populato stadard - The varace: σ varace = S for populato for sample 6.3. Probablty Probablty s a umercal value expressg the lkelhood of occurrece of a evet relatve to all possbltes a sample space. The probablty of occurrece of a evet A s defed as the umber of successful occurreces (m) dvded by the total umber of possble outcomes () a sample space, evaluated for >>. probablt y of evet A = The evet ca be represeted by: ) a cotuous radom varable (the probablty s expressed as P(x)); ) a dscrete radom varable (the probablty s expressed as P(x )). Here are some propertes relatve to probablty: a- 0 P ( x or x ) b- If evet A s the complemet of evet A, the: P( = P( c- It the evets A ad B are mutually exclusve (A ad B ca ot occur smultaeously): P ( A + B) = P( + P( B) d- It the evets A ad B are depedet, the probablty that both A ad B wll occur tghter s: P ( AB) = P( P( B) e- The probablty of occurrece of A or B or both s: P( A B) = P( + P( B) P( AB) Example A dstrbutor clams that the chace that ay of the three major compoets of a computer (CPU, motor, ad keyboard) s defectve s 3%. Calculate the chace that all three wll be defectve a sgle computer? 6.3.. Probablty dstrbuto fuctos A mportat fucto of statstcs s to use formato from a sample to predct the behavor of a populato. For partcular stuatos, experece has show that the dstrbuto of the radom varable follow a certa mathematcal patter (fucto). The, f the parameters of ths fucto ca be determed usg the sample data, t wll be possble to predct the propertes of the paret populato. Such fuctos are called: probablty mass m Istrumetato ad Measuremets \ LK\ 009 50
Chapter VI Statstcal Aalyss of Expermetal Data fuctos for dscrete radom varables. For cotuous radom varables, these fuctos are called probablty desty fuctos. - Probablty mass fucto: = P( ) = ; x The mea of the populato for a dscrete radom varable (also called the expected value): μ = x P( ) = x The varace of the populato s gve by: σ = ( x μ) P( x ) - Probablty desty fucto: P ( x x x + dx) f ( x )dx = Ad the, to fd the probablty of x to occur betwee a ad b values: P = ( a x b) f ( x) + The mea of the populato s: μ x f ( x) b = = dx The varace of the populato s: σ ( x μ) f ( x) + a dx = dx Example Cosder the followg probablty dstrbuto fucto for a cotuous radom varable: 3x < x < 3 f ( x) = 35 0 elsewhere a- Show that ths fucto satsfes the requremets of a probablty dstrbuto fucto. b- Calculate the expected mea value of x. c- Calculate the varace ad the stadard devato of x. - Cumulatve dstrbuto fucto: Istrumetato ad Measuremets \ LK\ 009 5
Chapter VI Statstcal Aalyss of Expermetal Data Ths type of dstrbutos s used whe you wat to kow the probablty of evet to be lower that a certa value (x). x ( rv x) = F( x) = F f ( x) dx = P( rv x) For dscrete radom varable: F( rv x ) = P( x j j= Cumulatve dstrbuto fucto has the two followg propertes: P( a < x b) = F( b) F( a) P( x > a) = F( a) ) Istrumetato ad Measuremets \ LK\ 009 5