Computing Confidence Intervals for Sample Data

Transcription

1 Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios How may measuremets are eeded for desired error? 1

2 What are statistics? A brach of mathematics dealig with the collectio, aalysis, iterpretatio, ad presetatio of masses of umerical data. Merriam-Webster We are most iterested i aalysis ad iterpretatio here. Lies, dam lies, ad statistics! 3 What is a statistic? A quatity that is computed from a sample [of data]. Merriam-Webster A estimate of a populatio parameter 4

3 Statistical Iferece populatio statistics iferece parameters sample 5 Why do we eed statistics? A set of experimetal measuremets costitute a sample of the uderlyig process/system beig measured Use statistical techiques to ifer the true value of the metric Use statistical techiques to quatify the amout of imprecisio due to radom experimetal errors 6 3

4 Experimetal errors Errors oise i measured values Systematic errors Result of a experimetal mistake Typically produce costat or slowly varyig bias Cotrolled through skill of experimeter Examples Temperature chage causes clock drift Forget to clear cache before timig ru 7 Experimetal errors Radom errors Upredictable, o-determiistic Ubiased equal probability of icreasig or decreasig measured value Result of Limitatios of measurig tool Observer readig output of tool Radom processes withi system Typically caot be cotrolled Use statistical tools to characterize ad quatify 8 4

5 Example: Quatizatio Radom error 9 Quatizatio error Timer resolutio quatizatio error Repeated measuremets X ± Δ Completely upredictable 10 5

6 A Model of Errors Error Measured value Probability -E x E ½ +E x + E ½ 11 A Model of Errors Error 1 Error Measured value Probability -E -E x E ¼ -E +E x ¼ +E -E x ¼ +E +E x + E ¼ 1 6

7 A Model of Errors 13 Probability of Obtaiig a Specific Measured Value 14 7

8 A Model of Errors Pr(X=x i ) = Pr(measure x i ) = umber of paths from real value to x i Pr(X=x i ) ~ biomial distributio As umber of error sources becomes large, Biomial Gaussia (Normal) Thus, the bell curve 15 Frequecy of Measurig Specific Values Accuracy Precisio Mea of measured values Resolutio True value 16 8

9 Accuracy, Precisio, Resolutio Systematic errors accuracy How close mea of measured values is to true value Radom errors precisio Repeatability of measuremets Characteristics of tools resolutio Smallest icremet betwee measured values 17 Quatifyig Accuracy, Precisio, Resolutio Accuracy Hard to determie true accuracy Relative to a predefied stadard o E.g. defiitio of a secod Resolutio Depedet o tools Precisio Quatify amout of imprecisio usig statistical tools 18 9

10 Cofidece Iterval for the Mea 1-α α/ c1 c α/ 19 Statistical Iferece populatio statistics iferece parameters sample 0 10

11 Why do we eed statistics? A set of experimetal measuremets costitute a sample of the uderlyig process/system beig measured Use statistical techiques to ifer the true value of the metric Use statistical techiques to quatify the amout of imprecisio due to radom experimetal errors Assumptio: radom errors ormally distributed 1 Iterval Estimate a b The iterval estimate of the populatio parameter will have a specified cofidece or probability of correctly estimatig the populatio parameter. 11

12 Properties of Poit Estimators I statistics, poit estimatio ivolves the use of sample data to calculate a sigle value which is to serve as a "best guess" for a ukow (fixed or radom) populatio parameter. Example of poit estimator: sample mea. Properties: Ubiasedess: the expected value of all possible sample statistics (of give size ) is equal to the populatio parameter. E[ X ] = µ E [ s ] =! Efficiecy: precisio as estimator of the populatio parameter. Cosistecy: as the sample size icreases the sample statistic becomes a better estimator of the populatio parameter. 3 Ubiasedess of the Mea ' i= 1 X i X = & E$ $ E[ X ] = % ' i= # Xi!!" = ' i= 1 ' i= 1 µ 1 µ = = µ E[ X ] i = 4 1

13 Sample size= % of populatio Sample 1 Sample Sample E[sample] Populatio Error Sample Average % Sample Variace % Efficiecy (average) 18% 18% 80% Efficiecy (variace) 59% 1% 173% 5 Sample size = 87 10% of populatio Sample 1 Sample Sample Populatio % Rel. Error Sample Average % Sample Variace % Efficiecy (average) 7.5% 5.7% 4.5% Efficiecy (variace).9% 10.0% 0.1% 6 13

14 Cofidece Iterval Estimatio of the Mea Kow populatio stadard deviatio. Ukow populatio stadard deviatio: Large samples: sample stadard deviatio is a good estimate for populatio stadard deviatio. OK to use ormal distributio. Small samples ad origial variable is ormally distributed: use t distributio with -1 degrees of freedom. 7 Cetral Limit Theorem If the observatios i a sample are idepedet ad come from the same populatio that has mea µ ad stadard deviatio σ the the sample mea for large samples has a ormal distributio with mea µ ad stadard deviatio σ/ x ~ N( µ,! / ) The stadard deviatio of the sample mea is called the stadard error. 8 14

15 c =Q(1-α/) c 1 =Q(α/) µ N ( µ,! / ) 0 1-α α/ α/ c 1 c 1-α α/ α/ x 1 0 x c N(0,1) = c1 = µ + µ + " " z1#! / z! / = x =z 1-α/ x 1 =z α/ = - z 1-α/ µ # " z1#! / 9 Cofidece Iterval - large (>30) samples 100 (1-α)% cofidece iterval for the populatio mea: s ( x " z1 "! /, x + z1 "! / s ) x : sample mea s: sample stadard deviatio : sample size z : (1-α/)-quatile of a uit ormal variate ( N(0,1)). 1 "! / 30 15

16 Populatio Sample Average Sample Variace Efficiecy (average) 7.5% 5.7% 4.5% Efficiecy (variace).9% 10.0% 0.1% 95% iterval lower % iterval upper Mea i iterval YES YES YES 99% iterval lower % iterval upper Mea i iterval YES YES YES 90% iterval lower % iterval upper Mea i iterval YES YES YES I Excel: ½ iterval = CONFIDENCE(1-0.95,s,) iterval size α Note that the higher the cofidece level the larger the iterval 31 µ Sample Iterval iclude µ? 1 YES YES 3 NO 100 YES 100 (1 α) of the 100 samples iclude the populatio mea µ. 3 16

17 Cofidece Iterval Estimatio of the Mea Kow populatio stadard deviatio. Ukow populatio stadard deviatio: Large samples: sample stadard deviatio is a good estimate for populatio stadard deviatio. OK to use ormal distributio. Small samples ad origial variable is ormally distributed: use t distributio with -1 degrees of freedom. 33 Studet s t distributio t( v) ~ N(0,1)! ( v) / v v: umber of degrees of freedom. v! ( ) : chi-square distributio with v degrees of freedom. Equal to the sum of squares of v uit ormal variates. the pdf of a t-variate is similar to that of a N(0,1). for v > 30 a t distributio ca be approximated by N(0,1)

18 Cofidece Iterval (small samples) For samples from a ormal distributio N(µ,σ ), (X " µ) /(# / ) has a N(0,1) distributio ad ( "1)s /# has a chi-square distributio with -1 degrees of freedom Thus, (X " µ) / s / has a t distributio with -1 degrees of freedom 35 Cofidece Iterval (small samples, ormally distributed populatio) 100 (1-α)% cofidece iterval for the populatio mea: x s ( x! t[1!" / ;! 1], x + t[1!" / ;! 1] s ) : sample mea s: sample stadard deviatio : sample size t [ 1! " / ;! 1] : critical value of the t distributio with -1 degrees of freedom for a area of α/ for the upper tail

19 Usig the t Distributio. Sample size= E[sample] Populatio Error Sample Average % Sample Variace % Efficiecy (average) 18% 18% 80% Efficiecy (variace) 59% 1% 173% 95% iterval lower %,-1 critical value % iterval upper Mea i iteval YES YES YES I Excel: TINV(1-0.95,15-1) α 37 How may measuremets do we eed for a desired iterval width? Width of iterval iversely proportioal to Wat to miimize umber of measuremets Fid cofidece iterval for mea, such that: Pr(actual mea i iterval) = (1 α) [(1! e) x,(1 e x] ( 1 + c, c ) = ) 38 19

20 How may measuremets? z ( c 1' ( / 1, c) = (1 m e) = x m z s = xe & = $ % z 1' ( / 1' ( / xe x s #! " s 39 How may measuremets? But depeds o kowig mea ad stadard deviatio! Estimate s with small umber of measuremets Use this s to fid eeded for desired iterval width 40 0

21 How may measuremets? Mea = 7.94 s Stadard deviatio =.14 s Wat 90% cofidece mea is withi 7% of actual mea. 41 How may measuremets? Mea = 7.94 s Stadard deviatio =.14 s Wat 90% cofidece mea is withi 7% of actual mea. α = 0.90 (1-α/) = 0.95 Error = ± 3.5% e =

22 How may measuremets? & z1 '( / s # = = $ % xe! " & 1.895(.14) # = $! % 0.035(7.94) " measuremets 90% chace true mea is withi ± 3.5% iterval 43 Cofidece Iterval Estimates for Proportios

23 Cofidece Iterval for Proportios For categorical data: E.g. file types {html, html, gif, jpg, html, pdf, ps, html, pdf } If 1 of observatios are of type html, the the sample proportio of html files is p = 1 /. The populatio proportio is π. Goal: provide cofidece iterval for the populatio proportio π. 45 Cofidece Iterval for Proportios The samplig distributio of the proportio formed by computig p from all possible samples of size from a populatio of size N with replacemet teds to a ormal with mea π ad stadard error! ( 1#! ). " p = The ormal distributio is beig used to approximate the biomial. So, "!

24 Cofidece Iterval for Proportios The (1-α)% cofidece iterval for π is p(1! p), p + z ( p! z1! " / 1! " / p(1! p) ) p: sample proportio. : sample size z : (1-α/)-quatile of a uit ormal variate ( N(0,1)). 1 "! / 47 Example 1 Oe thousad etries are selected from a Web log. Six hudred ad fifty correspod to gif files. Fid 90% ad 95% cofidece itervals for the proportio of files that are gif files. Sample size () 1000 No. gif files i sample 650 Sample proportio (p) 0.65 *p 650 > 10 OK 90% cofidece iterval alpha alpha/ 0.95 z Lower boud 0.65 Upper boud % cofidece iterval alpha alpha/ z Lower boud 0.60 Upper boud I Excel: NORMSINV(1-0.1/) NORMSINV(1-0.05/) 48 4

25 Example How much time does processor sped i OS? Iterrupt every 10 ms Icremet couters = umber of iterrupts m = umber of iterrupts whe PC withi OS 49 Proportios How much time does processor sped i OS? Iterrupt every 10 ms Icremet couters = umber of iterrupts m = umber of iterrupts whe PC withi OS Ru for 1 miute = 6000 m =

26 Proportios p(1! p) ( c 1, c) = p m z1!" / (1! ) = m = (0.1018,0.1176) 95% cofidece iterval for proportio So 95% certai processor speds % of its time i OS 51 Number of measuremets for proportios (1! e) p = ep = = p! z z 1! " / z 1! " / 1! " / p(1! p(1! p) p(1! ( ep) p) p) 5 6

27 Number of measuremets for proportios How log to ru OS experimet? Wat 95% cofidece ± 0.5% 53 Number of measuremets for proportios How log to ru OS experimet? Wat 95% cofidece ± 0.5% e = p =

28 Number of measuremets for proportios = z 1"# / p (1" p ) (ep) = (1.960) (0.1097)(1" ) 0.005(0.1097) [ ] =1,47,10 10 ms iterrupts 3.46 hours 55 Cofidece Iterval Estimatio for Variaces 8

29 Cofidece Iterval for the Variace If the origial variable is ormally distributed the the chi-square distributio ca be used to develop a cofidece iterval estimate of the populatio variace. The (1-α)% cofidece iterval for! is ( # 1) s! U $ " $ ( # 1) s! L! L! U : lower critical value of : upper critical value of!! 57 Chi-square distributio Not symmetric! α/ Q(α/) 1-α α/ Q(1-α/) 58 9

30 95% cofidece iterval for the populatio variace for a sample of size 100 for a N(3,) populatio. 1-α/ average variace.1716 std deviatio lower critical value of chi-square for 95% upper critical value of chi-square for 95% lower boud for cofidece iterval for the variace upper boud for cofidece iterval for the variace I Excel: CHIINV (0.975, 99) CHIINV (0.05, 99) α/ The populatio variace (4 i this case) is i the iterval (3.6343, 6.36) with 95% cofidece. 59 Cofidece Iterval for the Variace If the populatio is ot ormally distributed, the cofidece iterval, especially for small samples, is ot very accurate

31 Key Assumptio Measuremet errors are Normally distributed. Is this true for most measuremets o real computer systems? 1-α α/ c1 c α/ 61 Key Assumptio Saved by the Cetral Limit Theorem Sum of a large umber of values from ay distributio will be Normally (Gaussia) distributed. What is a large umber? Typically assumed to be > 6 or

32 Normalizig data for cofidece itervals If the uderlyig distributio of the data beig measured is ot ormal, the the data must be ormalized Fid the arithmetic mea of four or more radomly selected measuremets Fid cofidece itervals for the meas of these average values o We ca o loger obtai a cofidece iterval for the idividual values o Variace for the aggregated evets teds to be smaller tha the variace of the idividual evets 63 Summary Use statistics to Deal with oisy measuremets Estimate the true value from sample data Errors i measuremets are due to: Accuracy, precisio, resolutio of tools Other sources of oise Systematic, radom errors 64 3

33 Summary (cot d): Model errors with bell curve Accuracy Precisio Mea of measured values Resolutio True value 65 Summary (cot d) Use cofidece itervals to quatify precisio Cofidece itervals for Mea of samples Proportios Variace Cofidece level Pr(populatio parameter withi computed iterval) Compute umber of measuremets eeded for desired iterval width 66 33