, which yields. where z1. and z2
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1 The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin Gaussian r nrmal PDF The Gaussian prbability density functin (als called the nrmal prbability density functin r simply the nrmal PDF) is the vertically nrmalied PDF that is prduced frm a signal r measurement that has purely randm errrs The nrmal prbability density functin is f x Here are sme f the prperties f this special distributin: It is symmetric abut the mean The mean and median are bth equal t, f(x) the expected value (at the peak f the distributin) [The mde is undefined fr a smth, cntinuus distributin] Its plt is cmmnly called a bell curve because f its shape The actual shape depends n the magnitude f the standard deviatin Namely, if is small, the bell will be tall and skinny, while if is large, the bell will be shrt and fat, as sketched x x exp 1 1 e Small Large x Standard nrmal density functin All f the Gaussian PDF cases, fr any mean value and fr any standard deviatin, can be cllapsed int ne nrmalied curve called the standard nrmal density functin This nrmaliatin is accmplished thrugh the variable transfrmatins intrduced previusly, ie, x and f f x, which yields 1 / 1 exp f f x e / This standard nrmal density functin is valid fr any signal measurement, with any mean, and with any standard deviatin, prvided that the errrs (deviatins) are purely randm A plt f the standard nrmal (Gaussian) density functin was generated in Excel, using the abve equatin fr f() It is shwn t the right It turns ut that the prbability that variable x lies between sme range x 1 and x is the same as the prbability that the transfrmed variable lies between the crrespnding range 1 and, where is the transfrmed variable defined abve In ther wrds, x1 x Px1 x x P1 where 1 and Nte that is dimensinless, s there are n units t wrry abut, s lng as the mean and the standard deviatin are expressed in the same units x Furthermre, since Px x x f xdx, it fllws that 1 x1 P x x x f d 1 We define A() as the area under the curve between and, ie, the special case where 1 = in the abve integral, and is simply In ther wrds, A() is the prbability that a measurement lies between and, r A f d, as illustrated n the graph belw 1
2 The Gaussian r Nrmal PDF, Page Fr cnvenience, integral A() is tabulated in statistics bks, but it can be easily calculated t avid the rund-ff errr assciated with lking up and interplating values in a table 1 f() A erf A() where erf() is the errr functin, defined as erf exp d Mathematically, it can be shwn that Belw is a table f A(), prduced using Excel, which has a built-in errr functin, ERF(value) Excel has anther A NORMSDIST ABS( ) 5 functin that can be used t calculate A(), namely T read the value f A() at a particular value f, G dwn t the rw representing the first tw digits f G acrss t the clumn representing the third digit f Read the value f A() frm the table Example: At = 54, A() = A(5 + 4) = These values are highlighted in the abve table as an example Since the nrmal PDF is symmetric, A() = A(), s there is n need t tabulate negative values f
3 The Gaussian r Nrmal PDF, Page 3 Linear interplatin: By nw in yur academic career, yu shuld be able t linearly A() interplate frm tables like the abve As a quick example, let s estimate A() at = The simplest way t interplate, which wrks fr bth increasing and decreasing values, is t always wrk frm tp t bttm, equating the fractinal values f the knwn and desired variables A() =? We m in n the apprpriate regin f the table, straddling the value f interest, and set up fr interplatin see sketch The rati f the red difference t the blue difference is the same fr either A clumn Thus, keeping the clr cde, we set up ur equatin as Slving fr A() at = 546 yields A Special cases: If =, bviusly the integral A() = This means physically that there is er prbability that x will exactly equal the mean! (T be exactly equal wuld require equality ut t an infinite number f decimal places, which will never happen) If =, A() = 1/ since f() is symmetric This means that there is a 5% prbability that x is greater than the mean value In ther wrds, = represents the median value f x Likewise, if =, A() = 1/ There is a 5% prbability that x is less than the mean value 1 If = 1, it turns ut that A 1 f d 3413 t fur significant digits This is a special case, since by definitin x / Therefre, = 1 represents a value f x exactly ne standard deviatin greater than the mean A similar situatin ccurs fr = 1 since f() is f() 1 symmetric, and A1 f d 3413 t fur significant digits Thus, = 1 represents a value f x exactly ne standard deviatin less than the mean Because f this symmetry, we cnclude that the prbability that lies between 1 and 1 is (3413) = 686 r 686% In ther wrds, there is a 686% 1 1 prbability that fr sme measurement, the transfrmed variable lies within ne standard deviatin frm the mean (which is er fr this pdf) P x In ther wrds, the Translated back t the riginal measured variable x, 686% prbability that a measurement lies within ne standard deviatin frm the mean is 686% Cnfidence level The abve illustratin leads t an imprtant cncept called cnfidence level Fr the abve case, we are 686% cnfident that any randm measurement f x will lie within ne standard deviatin frm the mean value I wuld nt bet my life savings n smething with a 68% cnfidence level A higher cnfidence level is btained by chsing a larger value Fr example, fr = (tw standard deviatins away frm the mean), it turns ut that A f d 477 t fur significant digits Again, due t symmetry, multiplicatin by tw yields the prbability that x lies within tw standard deviatins frm the mean value, either t the right r t the left Since (477) = 9544, we are 9544% cnfident that x lies within tw standard deviatins f the mean Since 9544 is clse t 95, mst engineers and statisticians ignre the last tw digits and state simply that there is abut a 95% cnfidence level that x lies within tw standard deviatins frm the mean This is in fact the engineering standard, called the tw sigma cnfidence level r the 95% cnfidence level Fr example, when a manufacturer reprts the value f a prperty, like resistance, the reprt may state R = 1 9 (hms) with 95% cnfidence This means that the mean value f resistance is 1, and that 9 hms represents tw standard deviatins frm the mean
4 The Gaussian r Nrmal PDF, Page 4 In fact, the wrds with 95% cnfidence are ften nt even written explicitly, but are implied In this example, by the way, yu can easily calculate the standard deviatin Namely, since 95% cnfidence level is abut the same as sigma cnfidence, 9, r 45 Fr mre stringent standards, the cnfidence level is smetimes raised t three sigma Fr = 3 (three 3 standard deviatins away frm the mean), it turns ut that A 3 f d 4987 t fur significant digits Multiplicatin by tw (because f symmetry) yields the prbability that x lies within three standard deviatins frm the mean value Since (4987) = 9974, we are 9974% cnfident that x lies within three standard deviatins frm the mean Mst engineers and statisticians rund dwn and state simply that there is abut a 997% cnfidence level that x lies within three standard deviatins frm the mean This is in fact a stricter engineering standard, called the three sigma cnfidence level r the 997% cnfidence level Summary f cnfidence levels: The empirical rule states that fr any nrmal r Gaussian PDF, Apprximately 68% f the values fall within 1 standard deviatin frm the mean in either directin Apprximately 95% f the values fall within standard deviatins frm the mean in either directin [This ne is the standard tw sigma engineering cnfidence level fr mst measurements] Apprximately 997% f the values fall within 3 standard deviatins frm the mean in either directin [This ne is the stricter three sigma engineering cnfidence level fr mre precise measurements] Mre recently, many manufacturers are striving fr six sigma cnfidence levels Example: Given: The same 1 temperature measurements used in a previus example fr generating a histgram and a PDF The data are prvided in an Excel spreadsheet (Temperature_data_analysisxls) T d: (a) Cmpare the nrmalied PDF f these data t the nrmal (Gaussian) PDF Are the measurement errrs in this sample purely randm? (b) Predict hw many f the temperature measurements are greater than 33 C, and cmpare with the actual number Slutin: (a) We plt the experimentally generated PDF (blue circles) and the theretical nrmal PDF (red curve) n the same plt The agreement is excellent, indicating that the errrs are very nearly randm Of curse, the agreement is nt perfect this is because n is finite If n were t increase, we wuld expect the agreement t get better (less scatter and difference between the experimental and theretical PDFs) (b) Fr this data set, we had calculated the sample mean t be x = 319 and sample standard deviatin t be S = 1488 Since n = 1, the sample sie is large enugh t assume that expected value is nearly equal t x, and standard deviatin is nearly equal t S At the given value f temperature (set x = 33 C), we nrmalie t btain, namely, x x x = C A() = C S (ntice that is nndimensinal) We calculate area A(), either by interplatin frm the abve table r by direct calculatin The table yields A() = 4955, 1 and the equatin yields A erf = erf = 4955 This means that 4955% f the measurements are predicted t lie between the mean (319 C) and the given value f 33 C (red Desired area
5 The Gaussian r Nrmal PDF, Page 5 area n the plt) The percentage f measurements greater than 33 C is 5% 4955% = 9448% (blue area n the plt) Since n = 1, we predict that = 9448 f the measurements exceed 33 C Runding t the nearest integer, we predict that 9 measurements are greater than 33 C Lking at the actual data, we cunt 81 temperature readings greater than 33 C Discussin: The percentage errr between actual and predicted number f measurements is arund 1% This errr wuld be expected t decrease if n were larger If we had asked fr the prbability that T lies between the mean value and 33 C, the result wuld have been 496 (t fur digits), as indicated by the red area in the abve plt Hwever, we are cncerned here with the prbability that T is greater than 33 C, which is represented by the blue area n the plt This is why we had t subtract frm 5% in the abve calculatin (5% f the measurements are greater than the mean), ie, the prbability that T is greater than 33 C is = 94 Excel s built-in NORMSDIST functin returns the cumulative area frm - t, the range-clred area in the plt t the right Thus, at = 1338, NORMSDIST() = 9955 This is the entire area n the left half f the Gaussian PDF (5) plus the area labeled A() in the abve plt The desired blue area is therefre equal t 1 - NORMSDIST() NORMSDIST() Desired area Cnfidence level and level f significance Cnfidence level, c, is defined as the prbability that a randm variable lies within a specified range f values The range f values itself is called the Area = c = 1 cnfidence interval Fr example, as discussed abve, we are 9544% cnfident that a purely randm variable lies within tw standard deviatins frm the mean We state this as a cnfidence level f c = 9544%, which we usually rund ff t 95% fr Area = / Area = / practical engineering statistical analysis Cnfidence Level f significance,, is defined as the prbability that a randm variable lies utside f a specified range f values In the abve example, we are = 456% cnfident that a purely randm variable lies either belw r abve tw standard deviatins frm the mean (We usually rund this ff t 5% fr practical engineering statistical analysis) interval Mathematically, cnfidence level and level f significance must add t 1 (r in terms f percentage, t 1%) since they are cmplementary, ie, c 1 r c 1 Cnfidence level is smetimes given the symbl c% when it is expressed as a percentage; eg, at 95% cnfidence level, c = 95, c% = 95%, and = 1 c = 5 Bth and cnfidence level c represent prbabilities, r areas under the PDF, as sketched abve fr the nrmal r Gaussian PDF The blue areas in the abve plt are called the tails There are tw tails, ne n the far left and ne n the far right The tw tails tgether represent all the data utside f the cnfidence interval, as sketched Cautin: The area f ne f the tails is nly /, nt This factr f tw has led t much grief, s be careful that yu d nt frget this!
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