Workshop on Understanding and Evaluating Radioanalytical Measurement Uncertainty November 2007

Transcription

1 Workshop on Understanding and Evalating Radioanalytical Measrement Uncertainty 5-16 November 007 Applied Statistics: Basic statistical terms and concepts Sabrina BARBIZZI APAT - Agenzia per la Protezione dell'ambiente e per Servizi Tecnici Servizio Metrologia Ambientale Via Castel Romano Rome ITALY

2 Workshop on Understanding and Evalating Radioanalytical Measrement Uncertainty Applied Statistics: Basic statistical terms and concepts Sabrina Barbizzi APAT- Environmental Protection Agency of Italy Trieste 8 november 007 1

3 Statistics for evalation of ncertainty: overwiev GUM approach Type A evalation Example Type B evalation Example Law of propagation of ncertainty Reporting reslts

4 GUM approach Define the otpt qantity, the qantity reqired to be measred Identify the inpt qantities pon which the otpt qantity depends Develop a model relating the otpt qantity to these inpt qantities On the basis of available knowledge assign probability density fnction normal, niform, etc. to the vales of the inpt qantities Estimate the ncertainties associated with the inpt qantities Propagate the vales of the inpt qantities and their associated ncertainties throght the model Obtain the estimate of the otpt qantity vale and its ncertainty. 3

5 GUM approach A measrement model is expressed by a fnctional relationship f Y f where Y is a single otpt qantity and represents the N inpt qantities 4

6 Qantify ncertainty Once identified the inpt qantities, the next step is to qantify the ncertainty arising from these qantities: evalating the ncertainty arising from each individal sorce and converting them to standard deviation determining directly the combined contribtion to the ncertainty on the reslt from some or all of these sorces sing method performance data. NOTE: Not all of the components make a significant contribtion to the combined ncertainty. 5

7 Convert to standard ncertainty The ncertainties associated with the inpt qantities may be groped into two categories according to the method sed to estimate them: A type A evalation of standard ncertainty may be based on any valid statistical method for treating data. A type B evalation of standard ncertainty is sally based on scientific jdgement sing all of the relevant information available. 6

8 Type A Examples are: calclating the standard deviation of the mean of a series of independent observations; sing the method of least sqares to fit a crve to data in order to estimate the parameters of the crve and their standard deviations; carrying ot an analysis of variance. 7

9 Type A Depending on the statistical distribtion of data, it is possible to estimate the corrisponding standard deviation: Normal Binomial Poisson Standard deviation s x n i 1 x i n n x 1 s np 1 p s λ 8

10 Normal Distribtion 9

11 Normal Distribtion For a set of n vales x i Average mean vale Arithmetic mean vale of a sample of n reslts n x 1 n x i 1 i 10

12 Normal Distribtion For a set of n vales x i Standard deviation: qantity characterizing the dispersion of the vales s x i n 1 1 n i 1 x i x 11

13 Normal Distribtion Variance V x i s x i Standard deviation of the mean Relative Standard deviation s x i s x n s x RSD s x i i or RSD% CV * 100 x x 1

14 Confidence Interval A confidence interval for some poplation parameter is an interval constrcted from a sample of n observations so that it will contain the parameter with some specified probability 1-α100% where α is some fraction between 0 and 1 sally α is less than 0.5. Confidence interval of the mean of a normal distribtion 95% CI t0.05, n - 1 * s n µ x ± 1 α % CI n 13

15 Binomial distribtion An elementary example is this: Roll a standard die ten times and cont the nmber of sixes. The distribtion of this random nmber is a binomial distribtion with n 10 and p 1/6. the expected vale of Y is ynp the standard deviation is s y np 1 p 14

16 Poisson distribtion The poisson distribtion is generally appropriate for conting experiments where the data represent the nmber of events observed per nit interval. It is important in the stdy of random processes sch as those associated with the radioactive decay of elementary particles or nclear states. the expected vale of Y is yλ the standard deviation is sy λ λ average nmber of events that occr within the given time period or area 15

17 Type B Relevant information available: previos measrement data experience with, or general knowledge of, the behavior and property of relevant materials and instrments manfactrer s specifications data provided in calibration and other reports ncertainties assigned to reference data taken from handbooks 16

18 Type B Depending on the information available, it is possible to estimate the standard ncertainty on the basis of assmed probability distribtions which the inpt qantities approximate. Probability distribtions: xi xi min xi max xi xi min xi max x min max i xi x 3 x 1 β x i min x i max 6 xi xi min xi max x i x min i max x 6 17

19 rectanglar distribtion a... a the inpt qantity vale is between the limits the expectation y x ± a estimated standard ncertainty s x a / 3 Little information are available abot the inpt qantity and all one can do is sppose that the inpt qantity is described by a niform distribtion with a constant probability for the vale to lie anywhere within the interval 1/a a±a 18

20 Example of rectanglar distribtion It is likely that the vale is somewhere in that range Rectanglar distribtion is sally described in terms of: the average vale and the range ±a certificates or other specification give limits where the vale cold be, withot specifying a level of confidence or degree of freedom. Examples: concentration of calibration standard is qoted as 1000 ±mg/l assming rectanglar distribtion the standard ncertainty is: x a / 3 / mg/l the prity of the cadmim is given on the certificate as ±0.01% assming rectanglar distribtion the standard ncertainty is: x a / / % 19

21 trianglar distribtion It is sed when the available information abot the inpt qantity is less limited than those for the rectanglar distribtion and it is sggested that vales near the centre of the range are more likely than near the extreme the expectation y x ± a a±a estimated standard ncertainty s x a / 6 1/a 0

22 Example of trianglar distribtion vales close to x are more likely than near the bondaries Examples volmetric glassware: The manifactrer qotes a volme for the flask of 100 ±0.1mL at a T0 C Nominal vale most probable! Assming trianglar distribtion the standard ncertainty is: x a / / ml In case of dobt, se the rectanglar distribtion 1

23 trapezoidal distribtion Distribtion sed when it is sggested that vales near the extremes of the range are more likely than near the centre the expectation y x ± a estimated standard ncertainty s x a 1 6 β a±a

24 Law of propagation of ncertainty For the calclation phase of ncertainty evalation, GUM applies the law of propagation of ncertainty 3

25 The law of propagation of ncertainty x 1, x 1 x, x x 3, x 3 Yf y, y 4

26 5 Following the law of propagation of ncertainty, the combined ncertainty can be calclated combining all the components expressed as standard deviations: where x i are the inpt components, while covx ij is the covariance between x i and x j n i n j ij j i n i i i c x x y x y x x y y cov Combined ncertainty

27 Combined ncertainty When there is no correlation between inpt qantities the covariance is zero and the combined standard ncertainty is evalated as the sqare root of the combined variance simplified: n dy y x c i i 1 dx i 6

28 7 Law of ncertainty propagation withot correlation...,, n f Y n n c x Y x Y x Y y 1 1 Y Y / 1 1 Y Y 1 Y c 1 1 y Y c

29 8 r y c y y C B A Y C B A Y hc B A Y D C B A Y C B A C B A C r B r A r C B A C h B A C h B A C r B r A r [ ] [ ] [ ] D C B A D A B C AB AC BC D 1 [ ] [ ] [ ] C B A AB AC BC C B A h A B C r B r A r D r C r B r A r The most common qantitative expression relating the vale of the measrand to the parameters on which it depends are: r y is the relative combined standard ncertainty Combined ncertainty y y c /

30 Reporting reslts A reslt is given with its ncertainty C Cs ±. Bq / kg bt what is.? Standard deviation Rectanglar interval Trianglar interval Confidence interval Combined ncertainty Expanded ncertainty with k? 9

31 THANK YOU 30

32 a±a 1/a 31

33 3

34 a±a 1/a 33

35 a±a 34