ISSN 1684-8403 Journal of Statstcs Volume 17, 010, pp. 3-3 Abstract X- Chart Usng ANOM Approach Gullapall Chakravarth 1 and Chaluvad Venkateswara Rao Control lmts for ndvdual measurements (X) chart are avalable n the lterature usng movng range as an estmate of process standard devaton. Ths paper presents new control lmts for X - chart usng analyss of means (ANOM) approach. In dervng the control lmts, sample standard devaton s used as an estmate of process standard devaton. The expected length of the nterval between exstng (old) control lmts s compared wth the expected length of the nterval between new control lmts for the same confdence coeffcent and recommendatons are made to the practtoner when to use the new control lmts. Keywords Analyss of means, X-chart, Control lmts 1. Introducton For testng the equalty of several populaton means, a graphcal procedure, namely, Analyss of Means (ANOM) was ntroduced by Ott (1967) as an alternatve to Analyss of Varance (ANOVA). In Ott s procedure, the results are summarzed n an ANOM chart. Ths chart s smlar n appearance to a control chart. Instead of control lmts, decson lnes are used n ANOM procedure. The man dfference between ANOM chart and control chart s that the value of k (number of samples) s usually as large as 0 or more n control charts, whereas k n an ANOM chart. When there are exactly means k the ANOM s smply a graphcal form of Student s t - test. In ANOM chart, the sample mean values are compared to the overall grand mean, about whch the upper and lower 1 Department of Statstcs, P.B. Sddhartha College of Arts & Scence, Vjayawada, Inda Emal: chakravarth_math@yahoo.co.n Department of Statstcs, Acharya Nagarjuna Unversty, Guntur - 5510, Inda Emal: cvrao1950@yahoo.co.n
4 Chakravarth and Rao decson lnes have been constructed. If a sample mean falls outsde these decson lnes, t s declared sgnfcantly dfferent from the grand mean. There are several advantages of ANOM plottng, e.g., () t provdes a comparson of the relatve mportance and magntude of the factors as well as ther statstcal sgnfcance, () t provdes a pn-pontng of sources of non-randomness, and () t encourages the translaton of conclusons nto scentfc acton and for takng manageral decsons. Hence, ANOM plots reveal the statstcal sgnfcance as well as practcal sgnfcance of samples beng compared. Several authors extended the ANOM technque for comparng several () proportons () counts () treatment effects (v) nteracton effects (v) Lnear contrasts (v) varances (v) correlaton coeffcents (v) Regresson coeffcents (x) ntercepts (x) autocorrelaton coeffcents (x) coeffcents of varaton. Recently, Nelson et al. (005) wrote a book exclusvely on ANOM graphcal method for comparng means, rates and proportons. Rao (005) revewed papers on analyss of means startng from the frst paper n 1967 upto 004. Kan et al. (008) developed new control lmts for X - chart wth unknown based on the t - dstrbuton. Wth k subgroups, each wth n observatons, the control lmts obtaned by them are: X t /, 1 S kn b k 1 / kn (1.1) where S k n X X k( n 1) b j 1 j1 k X X k, X X j 1 j1 n n 1/ X are random varables for 1,,..., k subgroups and j 1,,..., n j observatons n each subgroup and t s the crtcal value obtaned from /, kn1 t - dstrbuton wth level of sgnfcance and degrees of freedom kn ( 1).
5 X- Chart Usng ANOM Approach It may be remarked here that the dervaton of Shewhart control lmts for X - chart depends on: X X E S.E. (1.) / whereas the dervaton of Kan et al.(008) control lmts depends on: E X S.E. X X (1.3) / as n ANOM approach. Control lmts for ndvdual measurements(x) chart are avalable n the lterature usng movng range as an estmate of process standard devaton. Ths paper presents new control lmts for X - chart usng ANOM approach. The expected length of the nterval between exstng (old) control lmts s compared wth the expected length of the nterval between new control lmts for the same confdence coeffcent.. Revew of X-Chart and Exstng Control Lmts The purpose of the X- chart or ndvdual measurements chart s same as that of the X - chart to montor the process mean. The assumptons for an X - chart are the same as the assumptons for an X - chart - normalty and ndependence. The normalty assumpton s far more mportant when ndvdual observatons are plotted than t s when averages are plotted, snce there s no central lmt theorem- type effect wth ndvdual observatons. Juran and Godfrey (004, p. 45.11) suggest to collect 0 or more observatons for a tral study. There are many stuatons n whch the subgroup sze used for process montorng s 1, that s, the subgroup conssts of an ndvdual unt. Some examples of these stuatons are gven n Montgomery (001) Exstng (old) Control Lmts The Shewhart control lmts for X - chart wth known parameters are gven by /. (.1) When the parameters and are unknown, the control lmts for X- chart wth k ndvdual unts X 1,,..., k are gven by:
6 Chakravarth and Rao UCL CL X MR X / d (.) LCL MR X / d where k X MR ˆ k d 1 X, X, ˆ, MR max X mn X, for 1,,..., k (.3) In the chosen n successve observatons and n k, d s a functon of n successve observatons to obtan movng ranges. MR s the average of all the MR computed from the sample of sze k. Z s the crtcal value from Standard Normal Dstrbuton wth level of sgnfcance. 3. Dervaton of New Control Lmts for X- Chart In dervng the new control lmts we wll use the method of Kan et al.(008). X 1,,..., k denote the k ndvdual measurements drawn from a normal Let populaton wth mean and varance. Let X X X and X ~ N, k 1,,..., k defne: X X E X X Z S. E. X X On dervng E X X and S. E. X X k X X k be the mean of 1. Consderng X X as a varate, we. (3.1), we obtan:
7 X- Chart Usng ANOM Approach X 0 E X S. E. X X k 1 k. (3.) We know that: X X Z S. E. X X Substtutng (3.) n (3.3), we get: ~ N 0,1. (3.3) X X Z ~ N0,1. (3.4) k 1 k We know that: k1 s ~ k 1, (3.5) where k 1 s k 1 1 X X wth degrees of freedom k 1. We have: s s (3.6) Snce sample mean and sample varance are ndependently dstrbuted, Z and are ndependent random varables. Hence, Fsher s t s gven by: Z t ~ t. (3.7) Substtutng (3.4) and (3.6) n (3.7), we get: t s X X k 1 k ~ t k1. (3.8)
8 Chakravarth and Rao As a result, the new control lmts for X-chart are obtaned as: LCL X t, k 1s k 1 k CL X UCL X t, k 1s k 1 k where (3.9) P t t. (3.10), k1 1 Plot the ndvdual measurements X aganst the control lmts n (3.9) and f any pont falls outsde the control lmts, the process s sad to be out of control wth respect to the process mean. 4. Comparson of New Control Lmts wth Exstng Control Lmts The new control lmts of X - chart usng ANOM approach are gven n (3.9) and the length of the nterval between the control lmts s gven by: L t s k 1 k. (4.1) 1, k1 and the expected length s: E L t c k 1 k, (4.) 1, k1 4 E s snce 4 c, where c 4 s a functon of k ( sample sze) and s tabulated n several qualty control books for k 5. When k >5, c 4 ~ (see Montgomery, 001). The exstng (old) control lmts of X- chart usng movng ranges are gven n (.) and the length of the nterval between the control lmts s gven by:
9 X- Chart Usng ANOM Approach MR L Z (4.3) d and the expected length s: E L Z. (4.4) We wsh to compare the expected lengths EL 1 and EL confdence coeffcent 1 (4.4), we wll compare 1 E L for the same. Snce s common n the two equatons (4.) and E L and EL effectvely, where: 1 t,k-1c4 k 1 k (4.5) and E L Z. (4.6) We have compared the expressons n (4.5) and (4.6) by computng the dfference E L1 E L for varous values of k and 0.01,0.05,0.10 and s compled n Tables 1-3 untl the dfference becomes zero. Even though the computaton of expected lengths s made for many values of k not appearng n the Tables, the results presented only for few values of k are to show the trend of dfference between expected lengths. When the dfference s zero, the X- chart wth new control lmts and the X- chart wth old control lmts have equal expected lengths for the same confdence coeffcent.
30 Chakravarth and Rao Table 1: Comparson of Expected Lengths at α = 0.01 k d.f. t-value C4 E(L1)/σ E(L)/σ E(L1)/σ - E(L)/σ 0 19.861 0.987.75.576 0.176 100 99.66 0.997.607.576 0.031 00 199.601 0.999.591.576 0.015 500 499.586 0.999.58.576 0.006 1500 1499.579 1.000.578.576 0.00 000 1999.578 1.000.577.576 0.001 3000 999.577 1.000.577.576 0.001 4000 3999.577 1.000.577.576 0.001 5000 4999.577 1.000.576.576 0.000 6000 5999.577 1.000.576.576 0.000 Table : Comparson of Expected Lengths at α = 0.05 k d.f. t-value C4 E(L1)/σ E(L)/σ E(L1)/σ - E(L)/σ 0 19.093 0.987.014 1.960 0.054 50 49.010 0.995 1.979 1.960 0.019 100 99 1.984 0.997 1.969 1.960 0.009 00 199 1.97 0.999 1.965 1.960 0.005 300 99 1.968 0.999 1.963 1.960 0.003 500 499 1.965 0.999 1.96 1.960 0.00 1000 999 1.96 1.000 1.961 1.960 0.001 1500 1499 1.96 1.000 1.961 1.960 0.001 000 1999 1.961 1.000 1.960 1.960 0.000 500 499 1.961 1.000 1.960 1.960 0.000
X- Chart Usng ANOM Approach Table 3: Comparson of Expected Lengths at α = 0.10 31 k d.f. t-value C4 E(L1)/σ E(L)/σ E(L1)/σ - E(L)/σ 0 19 1.79 0.987 1.663 1.645 0.018 50 49 1.677 0.995 1.651 1.645 0.006 100 99 1.660 0.997 1.648 1.645 0.003 00 199 1.653 0.999 1.646 1.645 0.001 300 99 1.650 0.999 1.646 1.645 0.001 400 399 1.649 0.999 1.646 1.645 0.001 500 499 1.648 0.999 1.645 1.645 0.000 1000 999 1.646 1.000 1.645 1.645 0.000 E L E L1 From the Tables we observe that n general, and as k ncreases E L1 E L the dfference reduces and becomes zero. 5. Recommendatons and Concludng Remarks I. In the X- chart wth new control lmts, standard devaton ( s ) s computed from the sample only once, whereas n the X- chart wth old control lmts several MR are to be computed fnally to get MR. Hence, one can save some tme n computng the new control lmts whch s the advantage n usng these lmts. Moreover, the concept of movng range enters along wth the concept of dependence of subgroups but ndependence of subgroups s essental n the development of any control chart. Calculaton of s nstead of MR overcomes ths objecton n the present X- chart. Based on the results n the Tables 1 3, the practtoner s advsed to use the new control lmts n the followng stuatons: () 0.01 and k 5000 () 0.05 and k 000 () 0.10 and k 500
3 Chakravarth and Rao II. For degrees of freedom k 1, z t, k-1 whatever may be. Hence the expected length EL 1 between new control lmts s always less than the expected length EL between old control lmts snce c4 1 and k1 k 1. In ths stuaton, the X chart wth new control lmts s always preferred over the X chart wth old control lmts. Acknowledgements The authors wsh to thank the referees for ther careful readng and valuable suggestons that mproved the presentaton of the paper. References 1. Juran, J. M. and Godfrey, A. B. (004). Juran s Qualty Handbook, Ffth Edton. McGraw -Hll, New York.. Kan,M., Panaretos, J. and Psaraks, S. (008). A new procedure to montor the mean of a qualty characterstc. Communcatons n Statstcs - Smulaton and Computaton, 37, 1870-1880. 3. Montgomery, D. C. (001). Introducton to Statstcal Qualty Control, Thrd Edton, John Wley & Sons, Inc., New York. 4. Nelson, P.R., Wludyka, P.S. and Copeland, K.A.F.(005). The Analyss of Means: A Graphcal Method for Comparng Means, Rates and Proportons. ASA- SIAM Seres on Statstcs and Appled Probablty, Phladelpha. 5. Ott, E. R. (1967). Analyss of means a graphcal procedure. Industral Qualty Control, 4, 101 109. 6. Rao, C.V. (005). Analyss of means - a revew. Journal of Qualty Technology, 37, 308-315.