Two New Unbiased Point Estimates Of A Population Variance

Transcription

1 Journal of Modern Applied Statistical Methods Volue 5 Issue Article Two New Unbiased Point Estiates Of A Population Variance Matthew E. Ela The University of Alabaa, ela@baa.ua.edu Follow this and additional works at: Part of the Applied Statistics Coons, Social and Behavioral Sciences Coons, and the Statistical Theory Coons Recoended Citation Ela, Matthew E. (006 "Two New Unbiased Point Estiates Of A Population Variance," Journal of Modern Applied Statistical Methods: Vol. 5 : Iss., Article 7. DOI: 0.37/jas/ Available at: This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCoons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCoons@WayneState.

2 Journal of Modern Applied Statistical Methods Copyright 006 JMASM, Inc. May, 006, Vol. 5, No., /06/$95.00 Two New Unbiased Point Estiates Of A Population Variance Matthew E. Ela Departent of Industrial Engineering The University of Alabaa Two new unbiased point estiates of an unknown population variance are introduced. They are copared to three known estiates using the ean-square error (MSE. A coputer progra, which is available for download at is developed for perforing calculations for the estiates. Key words: Unbiased, point estiate, variance, range, standard deviation, oving range. Introduction The statistical analysis of saple data often involves deterining point estiates of unknown population paraeters. A desirable property for these point estiates is that they be unbiased. An unbiased point estiate has an expected value (or ean equal to the unknown population paraeter it is being used to estiate. For exaple, consider the ean x and variance v calculated fro a rando saple of size n (x, x,, x n obtained fro a population with unknown ean µ and variance σ. The equations for these two statistics are equations ( and (: n i= n x = x / n ( v = = i i ( x x /(n i ( It is well known that x and v are unbiased point estiates of µ and σ, respectively (e.g., see Theores 8.. and 8.., respectively, in Bain & Engelhardt, 99. This eans the expected value of the sapling distribution of x is equal to µ (i.e., E( x =µ and the expected value of Matthew E. Ela is Assistant Professor of Industrial Engineering at The University of Alabaa. He is a eber of the ASQ and IIE, and is an ASQ Certified Quality Engineer. Eail hi at ela@baa.ua.edu. the sapling distribution of v is equal to σ (i.e., E(v=σ. It is iportant to have a saple that is rando when calculating unbiased point estiates of unknown population paraeters. In a rando saple, each value coes fro the sae population distribution. If the values coe fro different population distributions (i.e., populations with different distributions, eans, and/or variances, then the point estiates they are used to calculate will be inaccurate. For exaple, if the values coe fro population distributions with different eans, then v calculated fro this saple using equation ( will be inflated. Many situations exist in which it is difficult to obtain a rando saple. One of these is when the population is not well-defined, as is the case when studying on-going processes. Ongoing processes are often encountered in anufacturing situations. An approach to obtain unbiased point estiates of unknown population paraeters fro these types of processes is to collect data as soe nuber of subgroups, each having size n. This is the procedure that is used when constructing control charts to onitor the centering and/or spread of a process. The idea is for the data within a subgroup to coe fro the sae process distribution. If any changes are to occur in the process distribution, it is desirable for the to show up between subgroups. An additional procedure in control chart construction, which ay be called a deleteand-revise (D&R procedure, is perfored as an additional safeguard to ensure data within subgroups has the sae distribution. 9

3 MATTHEW E. ELAM 95 Two new unbiased point estiates of an unknown population variance are introduced. They are derived assuing the saple data is drawn fro an on-going process as subgroups, each of size n. The Methodology section has an exaple showing how the control charting procedure works. Also, it presents the three known unbiased point estiates used in the situation considered in this article, it derives the two new unbiased point estiates, and it explains a Mathcad (999 coputer progra that perfors calculations for the unbiased point estiates. The Results section has ean-square error (MSE results for the unbiased point estiates. These are useful for the purpose of coparing the unbiased point estiates. The Conclusion section suarizes the interpretations of the analyses in the Results section. Methodology Control Charting Procedure. Consider the data in Table obtained fro a norally distributed process with µ=00.0 and σ=7.0 (the data was generated in Minitab (003 and a few changes were ade to siulate a process with a nonconstant ean. The true unknown variability for the process is estiated using within subgroup variability. A control chart for spread ay be used to deterine if data within a subgroup coes fro the sae process distribution. The control chart for spread used here is the range (R chart. It is constructed using equations (3a-(3c: UCL = D R (3a CL = R (3b LCL = D3 R (3c UCL, CL, and LCL are the upper control liit, center line, and lower control liit, respectively, for the R chart. Values for the control chart factors D and D 3 for various n are widely available in control chart factor tables (e.g., see Table M in the appendix of Duncan, 97. The value R (Rbar is the ean of the subgroup ranges. The subgroup ranges are calculated for each subgroup as the axiu value in the subgroup inus the iniu value in the subgroup (these calculations are in the "R" colun of Table. Equations (a-(c are the R chart control liit calculations for the data in Table : UCL = D R = = (a CL = R = 3.58 (b LCL = D3 R = = 0.0 (c Figure is the R control chart generated in Minitab (003. The delete-and-revise (D&R procedure involves identifying any subgroup ranges that are greater than the UCL or less than the LCL. The identified subgroups are then reoved fro the analysis as long as, in this case, each identified subgroup was an indication of a shift in the process ean. The R chart control liits are recalculated using the reaining subgroups. For the Table data, the range (R for subgroup seven is above the UCL (see the point arked with a "" in Figure. The new value for R calculated using the reaining =9 subgroups after subgroup seven is reoved is shown as the Revised R in Table. The revised control liits are calculated in equations (5a-(5c: UCL = D R =.8.60 = 8.76 (5a CL = R =.60 (5b LCL = D3 R = = 0.0 (5c Because all of the reaining subgroup ranges are between the revised control liits, the conclusion is that the data within each subgroup coes fro the sae process distribution.

4 96 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE Table. Data Collected as =0 Subgroups, Each of Size n= Subgroup X X X 3 X R R 3.58 Revised R.60 R Control Chart UCL= e g n 0 a R le p 5 a S 0 _ R= LCL= Saple Figure. R Control Chart for the Data in Table

5 MATTHEW E. ELAM 97 The next two subsections, Known Unbiased Point Estiates of σ and Two New Unbiased Point Estiates of σ, explain how data collected and cleaned in this anner is used to obtain an unbiased point estiate of an unknown process variance using the following statistics: v, the ean of the subgroup variances, where each subgroup variance is calculated using equation (. v c, the variance of the n data values grouped together as one saple. It is calculated using equation ( with n replaced by n and with x calculated using equation (, also with n replaced by n. It should be noted that v c cannot be used when cleaning subgrouped data using a deleteand-revise (D&R procedure as explained in this subsection. The reason is it would include between subgroup variability, which would inflate its value if the process fro which the data is collected is operating under ultiple distributions. R, as previously deonstrated. s, the ean of the subgroup standard deviations, where each subgroup standard deviation is calculated using the square root of equation (. MR, the ean of the oving ranges. When data is collected as individual values, - oving ranges ay be calculated as the absolute value of the difference between consecutive individual values. In this case, the subgroup size n is taken to be two. For exaple, if the first three individual values are 5., 5.3, and.8, the first two oving ranges are =0. and =0.5. Known Unbiased Point Estiates of σ The three known unbiased point estiates of σ calculated fro data collected as subgroups, each of size n, considered in this article are v, v c, and ( R d. The unbiasedness of v is shown in the Appendix of Ela and Case (003. Wheeler (995, in his Tables 3.6, 3.7, and., indicated the unbiasedness for v (listed as the pooled variance as well as for ( R d. The value d ay be called an unbiasing factor, as ( R by itself is a biased point estiate of σ. The value d is tabled for various and n (e.g., see Table D3 in the appendix of Duncan, 97. David (95 gave the equation for d (i.e., dstar as equation (6: dstar = d + d3 / (6 The value d (i.e., d is the ean of the distribution of the range W. Its values for various n are widely available in control chart factor tables. Assuing a noral population with ean µ and variance equal to one, Harter (960 gave the equation for d as equation (7 (with soe odifications in notation: d = n (n + W (F(x W 0 n F(x f (x + WdW] f (x dx (7 The function F(x is the cuulative distribution function (cdf of the standard noral probability density function (pdf f(x. The value d3 (i.e., d 3 is the standard deviation of the distribution of the range W. Its values for various n are widely available in control chart factor tables. It is calculated using equation (8: d3 = EW d (8 Harter (960 gave the equation for EW, the expected value of the second oent of the distribution of the range W for subgroups of size n sapled fro a noral population with ean µ and variance equal to one, as equation (9 (with soe odifications in notation: EW = n (n + W (F(x W 0 n F(x f (x + WdW] f (x dx (9 Equations (6-(9 are the fors used in the Mathcad (999 coputer progra explained in

6 98 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE the Mathcad (999 Coputer Progra subsection. Two New Unbiased Point Estiates of σ Ela and Case (005a, in their Appendix 7, derived the equation for the factor that allows for an unbiased point estiate of σ to be calculated usings. Ela and Case (005a denoted this factor as c (i.e., cstar and gave its equation as equation (0: cstar = c + c5 / (0 The fact that ( s is an unbiased point estiate of σ is shown in the Appendix. In equation (0, the value c (i.e., is the ean of the distribution of the standard deviation. Its values for various n are widely available in control chart factor tables. Mead (966 gave the equation for c as equation ( when σ=.0 (with soe odifications in notation: = /(n exp(galn(n / galn( (n / ( The equivalency of this for to that given by Mead (966 is shown in Appendix 3 of Ela and Case (005a. The function galn represents the natural logarith of the gaa (Γ function. The value c5 (i.e., c 5 is the standard deviation of the distribution of the standard deviation. Mead (966 also gave the equation for c5 as equation ( when σ=.0 (with soe odifications in notation: c 5 = [( /(n [exp(galn((n + / galn ((n / exp( (galn(n / ]] 0. 5 galn((n / ( The equivalency of this for to that given by Mead (966 is shown in Appendix of Ela and Case (005a. The value c5 is also equal to. Equations (0-( are the fors used in the Mathcad (999 coputer progra explained in the Mathcad (999 Coputer Progra subsection. Ela and Case (006a, in Appendix, derived the equation for the factor that allows for an unbiased point estiate of σ to be calculated using MR. Ela and Case (006a denoted this factor as d (MR (i.e., dstarmr and gave its equation as equation (3: dstarmr = dn + dn r (3 The fact that ( d (MR MR is an unbiased point estiate of σ is shown in the Appendix. In equation (3, the value dn is d when n is equal to two. Harter (960 gave the equation for dn as equation ( (with soe odifications in notation: d n = / π ( The value r is the ratio of the variance to the squared ean, both of the distribution of the ean oving range MR σ, an approxiation to which is derived in Ela and Case (006a. Pal and Wheeler (990 gave the equation for r as equation (5:.5 r = (( π ( π /(6 ( (5 Equations (3-(5 are the fors used in the Mathcad (999 coputer progra explained in the Mathcad (999 Coputer Progra subsection. Mathcad (999 Coputer Progra A coputer progra was coded in Mathcad (999 with the Nuerical Recipes Extension Pack (997 in order to calculate the unbiasing factors d, c, and d (MR in equations (6, (0, and (3, respectively, regardless of the nuber of subgroups and the subgroup size n. The progra is in the Appendix and is naed UEFactors.cd. It is on one page which is divided into seven sections. Download instructions for the progra are available at The first section of the progra is the data entry section. The progra requires the user

7 MATTHEW E. ELAM 99 to enter (nuber of subgroups and n (subgroup size. Before a value can be entered, the cursor ust be oved to the right side of the appropriate equal sign. This ay be done using the arrow keys on the keyboard or by oving the ouse arrow to the right side of the equal sign and clicking once with the left ouse button. The progra is activated by paging down once the last entry is ade. The user is allowed to iediately page down to the output section of the progra (explained later after the last entry is ade. In section. of the progra, the value TOL is the tolerance. The calculations that use this value will be accurate to ten places to the right of the decial. The functions dnor(x, 0, and pnor(x, 0, in Mathcad (999 are the pdf and cdf, respectively, of the standard noral distribution. Section. of the progra has the equations for d, d3, and EW given earlier as equations (7, (8, and (9, respectively. Section.3 of the progra has the equations for c and c5 given earlier as equations ( and (, respectively. The function galn is a nuerical recipe in the Nuerical Recipes Extension Pack (997. Using it in equations ( and ( allows for c and c5, respectively, to be calculated for large values of n. Section. of the progra has the equations for dn and r, given earlier as equations ( and (5, respectively. Section.5 of the progra has the equations for dstar, cstar, and dstarmr, given earlier as equations (6, (0, and (3, respectively. The last section of the progra has the output. The two values entered at the beginning of the progra are given. Accurate values for the unbiasing factors d, c, and d (MR are also given. The value for d (MR is always calculated for n=, regardless of the value for n entered at the beginning of the progra. To copy results into another software package (like Excel, follow the directions fro Mathcad s (999 help enu or highlight a value and copy and paste it into the other software package. When highlighting a value with the ouse arrow, place the arrow in the iddle of the value, depress the left ouse button, and drag the arrow to the right. This will ensure just the nuerical value of the result is copied and pasted. Results The two new unbiased point estiates of σ are copared to the three known unbiased point estiates of σ using the ean-square error (MSE calculation in equation (6, which is based on Luko s (996 equation (A3: MSE( ˆ The value ˆ ˆ σ = Var( σ + [E( σ σ ] (6 or ( MR d (MR ˆσ represents v, v c, ( d R, ( s,, and Var represents the variance as calculated in equation (. Because these five point estiates of σ are all unbiased, E( σ ˆ σ = 0. Therefore, calculating their MSEs is identical to calculating their variances. Better point estiates are those with saller MSEs. MSEs for v, v c, ( R, and ( s are calculated using the FORTRAN (99 coputer progra naed "siulate" in the Appendix. The progra siulates the rando sapling of subgroups (: -0, 5, 30, 50, 75, 00, 50, 00, 50, 300, each of size n (n: -8, 0, 5, 50, fro a standard noral distribution (unifor (0, rando variates are generated using the Marse-Roberts code (983. This process is repeated 5000 ties for each cobination of and n in order to generate 5000 values each of v, v c, ( R, and ( s so that their variances can be deterined. The necessary values for d and c are taken fro Table A in Appendix III: Tables of Ela and Case (00 and Table A. in Appendix II of Ela and Case (005b, respectively. d d MSEs for ( d (MR MR are calculated using the FORTRAN (99 coputer progra naed "siulate_mr" in the Appendix. The progra siulates the rando sapling of subgroups (: -0, 5, 30, 50, 75, 00, 50, 00, 50, 300 fro a standard noral distribution (unifor (0, rando variates are generated using the Marse-Roberts

8 00 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE code (983. This process is repeated 5000 ties for each in order to generate 5000 ( d (MR MR values so that the variance can be deterined. The necessary values for d (MR are taken fro Table A. in Appendix of Ela and Case (006b. The Appendix has the MSE results for v, v c, ( R, ( s, and ( d (MR d MR in its Tables A.-A.5, respectively. As increases for any n, or as n increases for any, the MSEs in Tables A.-A. decrease. As increases, the MSEs decrease in Table A.5. This is not surprising because as ore inforation about the process is at hand, the unbiased estiates should perfor better. Only the MSEs for v, v c, ( d R, and ( c copared to the MSE for ( d (MR s when n= and = can be MR when =. In this case, the oving range is interpreted to be the sae as the range. These results are the sae. Tables A.6-A.8 in the Appendix have the percent change in MSE ( v over MSE (v c, ( s c MSE ( R over ( s MSE over MSE ( v, and d MSE, respectively. The calculations in Tables A.6-A.8 were perfored using Excel s full accuracy. Because ost of the percentages in these tables are zero or positive, it can be stated that, in general, MSE (v c MSE ( v ( MSE ( R d MSE s. The following additional conclusions can be drawn fro Tables A.6-A.8: In Tables A.6 and A.7, the percent changes decrease as n increases for any. This eans the MSEs for v, v c, and ( s converge to each other as n increases for any. The MSEs for v, v c, and ( sae when =. s are the The MSE for ( R d when n= and = is alost identical to that for v, v c, and ( c (or any, the MSEs for ( s ; however, as n gets larger for = R grow. This is because of the well known fact that the range calculation loses efficiency as the size of the saple fro which it is calculated increases. d larger than those for v, v c, and ( s The MSEs for ( R and ( d s when n= are alost identical. This is because the range and standard deviation calculations differ by only a constant when n=. Conclusion Fro the analyses in the Results section, it ay be concluded that ( s is at least as good of an unbiased point estiate of σ as ( R d fact, as n increases for any, (. In s becoes a uch better unbiased point estiate of σ than ( R. Also, the perforance of ( s d approaches that of v and v c as n increases for any. Additionally, ( d (MR MR appears to be an adequate unbiased point estiate of σ, as indicated by its reasonably sall MSE values. This eans that, for the first tie, there is an alternative to equation ( for obtaining an unbiased point estiate of σ fro individual values.

9 MATTHEW E. ELAM 0

10 0 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE progra siulate iplicit none INTEGER, paraeter :: DOUBLE=SELECTED_REAL_KIND(p=5 real(kind=double :: ean, sd, pi, dstar, cstar, r, r, X, large, sall, v, s, R, vc real(kind=double :: suvc, suvc, suvbar, suvbar, susbar, susbar, surbar, surbar real(kind=double :: sux, sux, suv, sus, sur, suxsv, suxsv real(kind=double :: vbar, sbar, Rbar, varvc, varvbar, varsbar, varrbar INTEGER :: c, b, a, rep, i, j, seed = integer, diension(:9 :: integer, diension(:0 :: n open(unit=, file="siulate.txt" open(unit=, file="dstar.txt" open(unit=3, file="cstar.txt" ean = 0.0 sd =.0 pi = ACOS(-.0 = (/ (c, c =, 0, 5, 30, 50, 75, 00, 50, 00, 50, 300 / n = (/, 3,, 5, 6, 7, 8, 0, 5, 50 / write(, 5 "n", "", "cstar", "dstar", "varvc", "varvbar", "varsbar", "varrbar" 5 forat(x, A, 3X, A, X, A, X, A, 5X, A, 8X, A, 5X, A, 5X, A do b =, 0 n loop do a =, 9 loop suvc = 0.0 suvc = 0.0 suvbar = 0.0 suvbar = 0.0 susbar = 0.0 susbar = 0.0 surbar = 0.0 surbar = 0.0 read(, dstar read(3, cstar do rep =, 5000 replication loop sux = 0.0 sux = 0.0 suv = 0.0 sus = 0.0 sur = 0.0 do i =, (a suxsv = 0.0 suxsv = 0.0 new subgroup do j =, n(b call rando(r, seed call rando(r, seed X = ean + sd ((SQRT(-. LOG(r (COS(. pi r sux = sux + X sux = sux + X(.0 suxsv = suxsv + X suxsv = suxsv + X(.0 if (j == then large = X sall = X else if (X > large large = X if (X < sall sall = X end if end do

11 MATTHEW E. ELAM 03 v = (suxsv - ((suxsv(.0 / n(b / (n(b- s = v(0.5 R = large - sall suv = suv + v sus = sus + s sur = sur + R end do vc = (sux - ((sux(.0 / ((an(b / ((an(b-.0 vbar = suv / (a sbar = ((sus / (a/cstar Rbar = ((sur / (a/dstar suvc = suvc + vc suvc = suvc + vc(.0 suvbar = suvbar + vbar suvbar = suvbar + vbar(.0 susbar = susbar + sbar susbar = susbar + sbar(.0 surbar = surbar + Rbar surbar = surbar + Rbar(.0 replication loop end do varvc = (suvc - ((suvc(.0 / (rep -.0 / (rep -.0 varvbar = (suvbar - ((suvbar(.0 / (rep -.0 / (rep -.0 varsbar = (susbar - ((susbar(.0 / (rep -.0 / (rep -.0 varrbar = (surbar - ((surbar(.0 / (rep -.0 / (rep -.0 write(, 0 n(b, (a, cstar, dstar, varvc, varvbar, varsbar, varrbar 0 forat(x, I, X, I3, X, F7.5, X, F7.5, X, F.0, X, F.0, X, F.0, X, F.0 loop end do n loop end do stop contains subroutine rando(uniran, seed This subroutine generates Unifor (0, rando variates using the Marse-Roberts code iplicit none INTEGER, paraeter :: DOUBLE=SELECTED_REAL_KIND(p=5 REAL(KIND=DOUBLE, INTENT(OUT :: uniran INTEGER, INTENT(IN OUT :: seed INTEGER :: hi5, hi3, low5, lowprd, ovflow INTEGER, PARAMETER :: ult =, ult = 63, & be5 = 3768, be6 = 65536, & odlus = hi5 = seed / be6 lowprd = (seed - hi5 be6 ult low5 = lowprd / be6 hi3 = hi5 ult + low5 ovflow = hi3 / be5 seed = (((lowprd - low5 be6 - odlus + & (hi3 - ovflow be5 be6 + ovflow if (seed < 0 seed = seed + odlus

12 0 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE hi5 = seed / be6 lowprd = (seed - hi5 be6 ult low5 = lowprd / be6 hi3 = hi5 ult + low5 ovflow = hi3 / be5 seed = (((lowprd - low5 be6 - odlus + & (hi3 - ovflow be5 be6 + ovflow if (seed < 0 seed = seed + odlus uniran = ( (seed / 56 + / return end subroutine rando end progra siulate progra siulate_mr iplicit none INTEGER, paraeter :: DOUBLE=SELECTED_REAL_KIND(p=5 real(kind=double :: ean, sd, pi, dstarmr, r, r, X, first, second, MR real(kind=double :: sumrbar, sumrbar, sumr, MRbar, varmrbar INTEGER :: c, a, rep, i, seed = integer, diension(:8 :: open(unit=, file="siulate_mr.txt" open(unit=, file="dstarmr.txt" ean = 0.0 sd =.0 pi = ACOS(-.0 = (/ (c, c =, 0, 5, 30, 50, 75, 00, 50, 00, 50, 300 / write(, 5 "", "dstarmr", "varmrbar" 5 forat(3x, A, X, A, 3X, A do a =, 8 loop sumrbar = 0.0 sumrbar = 0.0 read(, dstarmr do rep =, 5000 replication loop sumr = 0.0 do i =, (a call rando(r, seed call rando(r, seed X = ean + sd ((SQRT(-. LOG(r (COS(. pi r if (i == then first = X else second = X MR = abs(first - second sumr = sumr + MR first = second end if end do

13 MATTHEW E. ELAM 05 MRbar = ((sumr / ((a - /dstarmr sumrbar = sumrbar + MRbar sumrbar = sumrbar + MRbar(.0 replication loop end do varmrbar = (sumrbar - ((sumrbar(.0 / (rep -.0 / (rep -.0 write(, 0 (a, dstarmr, varmrbar 0 forat(x, I3, X, F7.5, X, F.0 loop end do stop contains subroutine rando(uniran, seed This subroutine generates Unifor (0, rando variates using the Marse-Roberts code iplicit none INTEGER, paraeter :: DOUBLE=SELECTED_REAL_KIND(p=5 REAL(KIND=DOUBLE, INTENT(OUT :: uniran INTEGER, INTENT(IN OUT :: seed INTEGER :: hi5, hi3, low5, lowprd, ovflow INTEGER, PARAMETER :: ult =, ult = 63, & be5 = 3768, be6 = 65536, & odlus = hi5 = seed / be6 lowprd = (seed - hi5 be6 ult low5 = lowprd / be6 hi3 = hi5 ult + low5 ovflow = hi3 / be5 seed = (((lowprd - low5 be6 - odlus + & (hi3 - ovflow be5 be6 + ovflow if (seed < 0 seed = seed + odlus hi5 = seed / be6 lowprd = (seed - hi5 be6 ult low5 = lowprd / be6 hi3 = hi5 ult + low5 ovflow = hi3 / be5 seed = (((lowprd - low5 be6 - odlus + & (hi3 - ovflow be5 be6 + ovflow if (seed < 0 seed = seed + odlus uniran = ( (seed / 56 + / return end subroutine rando end progra siulate_mr

14 06 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE Table A.. MSE of v n

15 MATTHEW E. ELAM 07 Table A.. MSE of v c n

16 08 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE Table A.3. MSE of ( R n d

17 MATTHEW E. ELAM 09 Table A.. MSE of ( s n

18 0 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE Table A.5. MSE of ( MR d (MR MSE

19 MATTHEW E. ELAM Table A.6. Percent change in MSE( v (Table A. over MSE(v c (Table A. n

20 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE Table A.7. Percent change in ( MSE s (Table A. over MSE( v (Table A. n

21 MATTHEW E. ELAM 3 Table A.8. Percent change in MSE ( R (Table A.3 over ( s d MSE (Table A. n References Bain, L. J. & Engelhardt, M. (99. Introduction to probability and atheatical statistics (nd ed.. Belont, CA: Duxbury Press. David, H. A. (95. Further applications of range to the analysis of variance. Bioetrika, 38, Duncan, A. J. (97. Quality control and industrial statistics ( th ed.. Hoewood, IL: Richard D. Irwin, Inc. Ela, M. E. & Case, K. E. (00. A coputer progra to calculate two-stage shortrun control chart factors for ( X, R charts. Quality Engineering, (, Ela, M. E. & Case, K. E. (003. Twostage short-run ( X, v and ( X, v control charts. Quality Engineering, 5(3, -8. Ela, M. E. & Case, K. E. (005a. Two-stage short-run ( X, s control charts. Quality Engineering, 7(,

22 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE Ela, M. E. & Case, K. E. (005b. A coputer progra to calculate two-stage shortrun control chart factors for ( X, s charts. Quality Engineering, 7(, Ela, M. E. & Case, K. E. (006a. Two-stage short-run (X, MR control charts. Journal of Modern Applied Statistical Methods, 5(, in press. Ela, M. E. & Case, K. E. (006b. A coputer progra to calculate two-stage shortrun control chart factors for (X, MR charts. Journal of Statistical Software, 5(. FORTRAN PowerStation v.0 in Microsoft Developer Studio (99. Redond, WA: Microsoft Corporation. Harter, H. L. (960. Tables of range and studentized range. Annals of Matheatical Statistics, 3, -7. Luko, S. N. (996. Concerning the estiators R d and R d in estiating variability in a noral universe. Quality Engineering, 8(3, Marse, K. & Roberts, S. D. (983. Ipleenting a portable FORTRAN unifor (0, generator. Siulation, (, Mathcad 000 Professional (999. Cabridge, MA: MathSoft, Inc. Mead, R. (966. A quick ethod of estiating the standard deviation. Bioetrika, 53, Minitab Release. Statistical Software (003. State College, PA: Minitab Inc. Nuerical Recipes Extension Pack (997. Cabridge, MA: MathSoft, Inc. Pal, A. C. & Wheeler, D. J. (990, Unpublished anuscript. Equivalent degrees of freedo for estiates of the process standard deviation based on Shewhart control charts. Wheeler, D. J. (995. Advanced Topics in Statistical Process Control. Knoxville, TN: SPC Press, Inc. Show: ( of Appendix σ ; i.e., show E ( s = σ s is an unbiased point estiate E = E s ( c s ( c si i= = E ( c E s i i= = ( c = Var s i + E si ( c i= i= = Var ( si + E ( si ( c i= i= because the s i ' s are independent. ( c5 σ s i= ( c ( c σ i= E =, c + because Var(s = c 5 σ and E (s = σ (by definition, s Var = c5 σ Var(s = c5 Var(s = c5 σ ; σ by definition, s E = c E(s = c E(s = c σ. σ σ,

23 MATTHEW E. ELAM 5 E = s c ( c c5 = ( c + c σ σ = ( ( c c ( c ( c s c = σ E [ c σ + ( c σ ] 5 ( c σ + c σ, since c = σ 5 c 5 c + = σ ( c = c ( = σ Show: ( d (MR of c 5 + MR is an unbiased estiate σ ; i.e., show E ( MR d (MR = σ 0.5. One first needs to deterine the variance of the distribution of the ean oving range MR σ. MR Var = Var σ σ Fro Pal and Wheeler (990, where with Var ( MR d σ r =, b ( c r = ( π b = and π c = = Var σ MR d σ d r d r = σ Var MR σ = d Var r = E = E MR ( ( d MR ( d MR because Var ( ( E( MR Var MR = + ( d MR i i= = d r σ + E ( d ( σ = ( σ ( = d r Var Var MR d r Var MR = d r σ MR E d ( MR d r σ + = ( d E MR i ( i=, d r σ + = ( d ( E MR i ( i=.

24 6 TWO NEW UNBIASED POINT ESTIMATES OF A POPULATION VARIANCE d r σ + = ( d ( d σ ( i= because E (MR d σ (by definition, = MR E = d σ E(MR = d E(MR = d σ. σ MR E d ( MR d r σ + = ( d (( d σ ( = + ( d ( d ( d r σ d σ σ ( d d r = + = σ ( d because d ( d ( d d = σ (MR r E QED =. MR d (MR ( = σ