Technical Note: An Improved Range Chart for Normal and Long-Tailed Symmetrical Distributions

Technical Note: An Improved Range Chart for Normal and Long-Tailed Symmetrical Distributions Pandu Tadikamalla, 1 Mihai Banciu, 1 Dana Popescu 2 1 Joseph M. Katz Graduate School of Business, University of Pittsburgh, Pittsburgh, Pennsylvania 2 Stern School of Business, New York University, New York, New York Received 19 February 2007; revised 7 September 2007; accepted 7 October 2007 DOI 10.1002/nav.20265 Published online 26 November 2007 in Wiley InterScience (www.interscience.wiley.com). Abstract: The distribution of the range of a sample, even in the case of a normal distribution, is not symmetric. Shewhart s control chart for range and other approximations for range from skewed distributions and long-tailed (leptokurtic) symmetrical distributions assume the distribution of range as symmetric and provide ±3 sigma control limits. We provide accurate approximations for the R-chart control limits for the leptokurtic symmetrical distributions, using a range quantile approximation (RQA) method and illustrate the use of the RQA method with a numerical example. As special cases, we provide constants for the R-chart for the normal, logistic, and Laplace distributions. 2007 Wiley Periodicals, Inc. Naval Research Logistics 55: 91 99, 2008 Keywords: range control charts; long-tailed distributions; normal distribution; logistic distribution; Laplace distribution; Shewhart method; statistical quality control 1. INTRODUCTION Since Shewhart [8] introduced his control charts in 1920, they have become extremely popular among practitioners largely because of their ease of use. Although recent advances in statistics and computational mathematics make possible significantly more precise methods, traditional control charts remain the most widely taught and used tool in quality control. With a sufficiently large sample size, the Shewhart control charts are accurate [1]. Most of the time, however, using a large sample size is impractical and prohibitively expensive. When the sample size is small, on the other hand, it is more likely that the sample mean of the quality characteristic is not normally distributed, and thus the Shewhart charts become less reliable. Several authors have tried to address these issues and construct improved methods for skewed distributions [4] and for long-tailed (leptokurtic 1 ) symmetrical distributions of the quality characteristics [3, 12]. The Correspondence to: P.R. Tadikamalla (pandu@pitt.edu) 1 We define the kurtosis for a normal distribution as equal to zero and distributions with positive kurtosis are termed leptokurtic distributions. In some text books and research papers, the kurtosis for normal distribution is defined as three. restrictive normality assumption of the quality characteristic is not the only drawback in the case of small sample sizes. As shown in Montgomery [7], the use of the ±3 sigma limits for the range (R) chart produces a higher type I error than for the X chart. This means that even with a normally distributed quality characteristic the use of the ±3 sigma for the R-chart is not appropriate, as it does not guarantee the desired type I error of 0.27%. Another problem with the ±3 sigma limits is that the distribution of R is skewed and because of this, small sample sizes yield negative numbers for the lower control limit (LCL). Since the range is bounded below by zero, the LCL in those cases is truncated to zero. In the Shewhart control chart, for sample sizes less than 7, we always have the LCL for the R-chart equal to 0, although in reality, the exact LCL is higher. By treating very low sample ranges as in control when they are not, we are more likely to overlook improvements in the process as a result of reduced variability, and consequently fail to investigate the causes of these improvements. Khoo and Lim [6] address this shortcoming of the traditional range control charts and show a way to derive the exact upper control limit (UCL) and LCL for the range of a normally distributed quality characteristic. They do not, however, tabulate the constants needed by practitioners to construct these charts. In this study, we derive a range quantile 2007 Wiley Periodicals, Inc.

92 Naval Research Logistics, Vol. 55 (2008) approximation (RQA) method to compute more precise limits for the case of long-tailed symmetrically distributed quality characteristics. We then tabulate the constants needed to construct the range control charts. Finally, we include the exact control limits for the normal, logistic, and Laplace distributions. The rest of this study is organized as follows: Section 2 explains the rationale behind our approach and shows how to compute the control chart constants for the leptokurtic distributions. Section 2 also illustrates the use of these constants with a numerical example. Section 3 presents the range control chart constants for the normal distribution. Section 4 compares the performance of our charts against the traditional Shewhart charts for the case where the quality characteristic has a logistic distribution or a Laplace distribution. Finally, Section 5 presents our conclusions. 2. THE IMPROVED R-CHART FOR LEPTOKURTIC DISTRIBUTIONS 2.1. Description of the RQA Method Let (X1 i, Xi 2,..., Xi n ) i=1,2,...,r be r subgroups of size n from a process distribution with mean µ, standard deviation σ, skewness 0, and kurtosis k 4. Let f(x) be the probability density function (pdf) of the process distribution and let F(x) be the corresponding cumulative density function (cdf). Also, let X(1) i and Xi (n) be respectively the smallest and the largest observations in the subgroup i. We denote by R i = X(n) i Xi (1) the range of the subgroup i. IfF(x) is known, the exact distribution function of R i is given by [9]: G(R) = n [F(x + R) F(x)] n 1 df (x) (1) Sometimes the cdf cannot be expressed in closed form. In that case, we can rewrite (1) in terms of the pdf, f(x): G(R) = n x+r x f(x)dx n 1 f(x)dx (2) To find the UCL and the LCL of the R-chart, we need to solve the following two equations: G(UCL R ) = 1 α/2 (3) G(LCL R ) = α/2 (4) where UCL R and LCL R are the UCL and LCL for R and α is the desired type I error. We define d 2 as the mean of R/σ. The formula for d 2 is as follows (see, for example, [13]): d 2 = [1 (1 F(x)) n (F (x)) n ]dx (5) The control chart constants will be given by: D 3 = LCL R d 2 D 4 = UCL R d 2 and the control chart limits will be: UCL R = RD 4 CL R = R (7) LCL R = RD 3 where R is the average of sample range values. (6) Table 1. Values of LCL constant D3 for the RQA method. n Kurtosis 0.5 1 1.2 1.5 2 2.5 3 4 5 6 2 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 3 0.040 0.039 0.039 0.039 0.038 0.038 0.038 0.037 0.037 0.036 4 0.104 0.101 0.100 0.100 0.098 0.097 0.096 0.094 0.093 0.092 5 0.164 0.160 0.158 0.157 0.154 0.152 0.150 0.147 0.145 0.144 6 0.216 0.209 0.207 0.205 0.201 0.198 0.196 0.192 0.189 0.186 7 0.258 0.250 0.248 0.244 0.240 0.236 0.233 0.228 0.224 0.221 8 0.293 0.284 0.281 0.277 0.269 0.267 0.264 0.258 0.253 0.250 9 0.323 0.313 0.309 0.305 0.298 0.294 0.289 0.283 0.278 0.274 10 0.349 0.337 0.333 0.328 0.321 0.316 0.311 0.304 0.299 0.294 12 0.390 0.376 0.371 0.366 0.358 0.351 0.346 0.338 0.332 0.327 15 0.435 0.419 0.414 0.407 0.398 0.390 0.384 0.375 0.368 0.362 20 0.486 0.467 0.461 0.453 0.442 0.434 0.427 0.416 0.408 0.401

Tadikamalla, Banciu, and Popescu: Range Charts for Normal and Long-Tailed Symmetrical Distributions 93 Table 1a. Values of D 3 for Student t and Johnson S u distributions. n Kurtosis 0.5 1 1.2 1.5 2 2.5 3 4 5 6 2 0.0021 a 0.0021 0.0020 0.0020 0.0020 0.0019 0.0019 0.0019 0.0020 0.0020 0.0021 b 0.0021 0.0020 0.0020 0.0020 0.0019 0.0019 0.0020 0.0019 0.0019 3 0.0403 0.0396 0.0394 0.0390 0.0386 0.0383 0.0381 0.0377 0.0374 0.0372 0.0402 0.0394 0.0391 0.0387 0.0381 0.0376 0.0372 0.0365 0.0358 0.0354 4 0.1037 0.1015 0.1008 0.0999 0.0987 0.0978 0.0970 0.0959 0.0951 0.0945 0.1036 0.1009 0.1001 0.0992 0.0971 0.0957 0.0944 0.0923 0.0907 0.0892 5 0.1644 0.1604 0.1591 0.1576 0.1554 0.1538 0.1525 0.1505 0.1492 0.1482 0.1641 0.1594 0.1578 0.1562 0.1527 0.1501 0.1478 0.1438 0.1415 0.1389 6 0.2158 0.2101 0.2083 0.2060 0.2030 0.2007 0.1989 0.1962 0.1943 0.1928 0.2153 0.2086 0.2064 0.2035 0.1991 0.1956 0.1925 0.1874 0.1833 0.1800 7 0.2584 0.2511 0.2488 0.2460 0.2422 0.2393 0.2370 0.2336 0.2311 0.2294 0.2578 0.2493 0.2465 0.2429 0.2373 0.2328 0.2290 0.2233 0.2175 0.2133 8 0.2938 0.2851 0.2825 0.2790 0.2745 0.2711 0.2684 0.2643 0.2615 0.2594 0.2932 0.2830 0.2797 0.2756 0.2640 0.2636 0.2586 0.2513 0.2454 0.2410 9 0.3236 0.3137 0.3106 0.3068 0.3016 0.2977 0.2946 0.2900 0.2868 0.2844 0.3229 0.3114 0.3075 0.3024 0.2953 0.2893 0.2842 0.2757 0.2692 0.2637 10 0.3490 0.3380 0.3346 0.3303 0.3246 0.3203 0.3168 0.3118 0.3082 0.3055 0.3482 0.3355 0.3313 0.3257 0.3177 0.3111 0.3055 0.2969 0.2891 0.2827 12 0.3901 0.3772 0.3732 0.3682 0.3616 0.3565 0.3525 0.3466 0.3424 0.3393 0.3892 0.3744 0.3695 0.3634 0.3538 0.3462 0.3398 0.3291 0.3213 0.3142 15 0.4354 0.4202 0.4156 0.4097 0.4020 0.3960 0.3914 0.3845 0.3796 0.3760 0.4344 0.4172 0.4115 0.4040 0.3935 0.3847 0.3772 0.3657 0.3559 0.3481 20 0.4862 0.4684 0.4630 0.4561 0.4470 0.4400 0.4345 0.4264 0.4207 0.4165 0.4852 0.4652 0.4587 0.4499 0.4379 0.4279 0.4195 0.4058 0.3953 0.3863 a First entry in cell corresponds to the Student t distribution. b Second entry in cell corresponds to the S u distribution. For most known distributions, Eqs. (3) and (4) can be solved numerically. In general, however, the underlying process distribution is unknown and thus it is not possible to solve Eqs. (3) and (4) to get the exact UCL and LCL. Even for known f(x) and/or F(x), the use of the control charts can be enhanced with the readily available control chart constants. We propose a method where we assume that a symmetric system of distributions such as the Student t or Johnson s S u [5] can effectively approximate any leptokurtic symmetrical distribution by matching the mean, variance, and kurtosis. Although the degrees of freedom parameter of the t distribution is treated as an integer in practice, the t distribution is defined for all real values of the degrees of freedom. This gives the family of t distributions the flexibility to have any kurtosis greater than 0. 2.2. Derivation of the Control Charts Constants Using Eqs. (3) and (4), we calculated D 3 and D 4 values for two different families of symmetric, leptokurtic Table 2. Values of UCL constant D4 for the RQA method. n Kurtosis 0.5 1 1.2 1.5 2 2.5 3 4 5 6 2 4.340 4.619 4.719 4.854 5.063 5.242 5.312 5.663 5.874 6.069 3 3.207 3.446 3.530 3.648 3.820 3.968 4.099 4.314 4.488 4.637 4 2.790 3.012 3.091 3.199 3.358 3.494 3.613 3.808 3.972 4.111 5 2.566 2.778 2.853 2.956 3.107 3.237 3.350 3.536 3.687 3.814 6 2.423 2.629 2.701 2.801 2.947 3.072 3.180 3.361 3.505 3.625 7 2.322 2.523 2.594 2.691 2.833 2.955 3.061 3.231 3.376 3.494 8 2.247 2.444 2.513 2.609 2.748 2.867 2.969 3.141 3.280 3.394 9 2.188 2.382 2.449 2.544 2.681 2.798 2.900 3.067 3.205 3.314 10 2.140 2.332 2.399 2.491 2.626 2.742 2.842 3.008 3.139 3.250 12 2.067 2.255 2.321 2.410 2.543 2.656 2.753 2.914 3.044 3.151 15 1.990 2.174 2.238 2.327 2.455 2.565 2.659 2.817 2.940 3.047 20 1.908 2.087 2.150 2.236 2.361 2.468 2.560 2.711 2.832 2.846

94 Naval Research Logistics, Vol. 55 (2008) Table 2a. Values of D 4 for Student t and Johnson S u distributions. n Kurtosis 0.5 1 1.2 1.5 2 2.5 3 4 5 6 2 4.3407 a 4.6174 4.7154 4.8490 5.0407 5.2010 5.3360 5.5501 5.7105 5.8356 4.3391 b 4.6198 4.7222 4.8589 5.0860 5.2828 5.2884 5.7758 6.0379 6.3022 3 3.2082 3.4475 3.5312 3.6452 3.8085 3.9441 4.0585 4.2385 4.3737 4.4785 3.2050 3.4435 3.5295 3.6505 3.8311 3.9925 4.1396 4.3899 4.6018 4.7963 4 2.7921 3.0161 3.0943 3.2008 3.3529 3.4791 3.5855 3.7530 3.8786 3.9762 2.7877 3.0081 3.0869 3.1977 3.3622 3.5087 3.6411 3.8632 4.0650 4.2462 5 2.5685 2.7838 2.8591 2.9612 3.1074 3.2288 3.3307 3.4916 3.6122 3.7059 2.5631 2.7725 2.8472 2.9502 3.1069 3.2446 3.3693 3.5803 3.7616 3.9222 6 2.4259 2.6357 2.7088 2.8083 2.9507 3.0691 3.1683 3.3250 3.4425 3.5338 2.4196 2.6216 2.6934 2.7931 2.9424 3.0741 3.1915 3.3978 3.5671 3.7162 7 2.3258 2.5315 2.6032 2.7010 2.8405 2.9567 3.0542 3.2080 3.3235 3.4130 2.3186 2.5150 2.5847 2.6808 2.8257 2.9530 3.0669 3.2531 3.4292 3.5738 8 2.2506 2.4534 2.5241 2.6205 2.7580 2.8726 2.9687 3.1205 3.2344 3.3228 2.2429 2.4349 2.5029 2.5979 2.7376 2.8614 2.9695 3.1617 3.3261 3.4660 9 2.1918 2.3922 2.4622 2.5574 2.6935 2.8067 2.9018 3.0521 3.1648 3.2524 2.1836 2.3720 2.4349 2.5313 2.6682 2.7890 2.8980 3.0812 3.2452 3.3765 10 2.1443 2.3427 2.4121 2.5065 2.6412 2.7536 2.8479 2.9968 3.1087 3.1955 2.1355 2.3209 2.3863 2.4764 2.6117 2.7301 2.8355 3.0184 3.1695 3.3046 12 2.0714 2.2670 2.3352 2.4283 2.5616 2.6722 2.7654 2.9124 3.0229 3.1087 2.0617 2.2423 2.3058 2.3920 2.5243 2.6388 2.7413 2.9161 3.0657 3.1930 15 1.9955 2.1879 2.2552 2.3469 2.4783 2.5876 2.6795 2.8248 2.9340 3.0189 1.9846 2.1598 2.2214 2.3071 2.4323 2.5425 2.6391 2.8089 2.9456 3.0748 20 1.9143 2.1035 2.1697 2.2602 2.3896 2.4975 2.5884 2.7319 2.8398 2.9237 1.9019 2.0710 2.1302 2.2126 2.3323 2.4376 2.5318 2.6903 2.8235 2.7686 a First entry in cell corresponds to the Student t distribution. b Second entry in cell corresponds to the S u distribution. distributions: Student t and Johnson S u, for sample sizes of n = 2ton = 25, α = 0.0027, and kurtosis ranging from 0.5 to 6. As we can see from Tables 1a and 2a, the corresponding quantiles of these two distributions are very close to one another. We experimented with other symmetric leptokurtic distributions from the Tadikamalla-Johnson s L U family of distributions [11], the Burr [2] distribution, and the Exponential power distribution [10]. The D 3 and D 4 values from these distributions are very similar to the values in Tables 1a and 2a. This led us to infer that distributions with identical first four moments closely approximate one another. It is then reasonable to assume that an average of the values of D 3 for Student t and Johnson S u, would give a good estimate of the actual LCL for any symmetrical leptokurtic distribution with the same mean, standard deviation, and kurtosis. Similarly, averaging the values of D 4 for the two distributions would give a good approximation for the UCL. We define D 3 and D 4 as follows: D 3 = D 3t dist + D 3S u dist 2 D 4 = D 4t dist + D 4S u dist 2 (8) Tables 1 and 2 give the D3 and D 4 values for different kurtosis and sample size values. 2.3. Numerical Example Here we illustrate the use of the proposed method with a numerical example using the same data from Tadikamalla and Popescu [12]. The data comes from a technology company that engages in the development, manufacture, and marketing of materials and derivative products for precision use in industrial, medical, military, surgical, and aerospace applications. The quality characteristic is the center thickness of an optical lens. Table 3 shows the data for 40 subgroups of size n = 5 from a process that is known to be in control. A look at the summary statistics of the data revealed symmetry (skewness of 0.24) and a heavy tailed distribution (kurtosis = 2.9). Following the procedure described in Tadikamalla and Popescu [12], statistical tests indicate that the skewness is not statistically significant (P-value = 0.156), and the kurtosis is highly significant (P-value 0). Using the constants in Tables 1 and 2, we calculated the control chart limits for the R-chart using the proposed method

Tadikamalla, Banciu, and Popescu: Range Charts for Normal and Long-Tailed Symmetrical Distributions 95 Table 3. Center thickness of the optical lenses. Group 1 2 3 4 5 X R 1 66.98 60.21 58.10 58.30 58.26 60.37 8.88 2 60.18 60.42 55.96 59.04 60.06 59.13 4.46 3 60.50 60.10 60.67 64.08 55.26 60.12 8.82 4 59.93 64.72 60.77 60.60 58.03 60.81 6.69 5 58.59 60.38 63.23 61.27 64.14 61.52 5.55 6 60.25 59.81 60.86 61.50 58.05 60.09 3.45 7 62.57 64.27 60.14 63.34 58.67 61.80 5.60 8 60.88 60.91 58.66 60.07 60.06 60.12 2.25 9 58.21 61.92 58.44 61.70 59.90 60.04 3.72 10 62.70 57.70 49.03 62.08 61.75 58.65 13.67 11 63.04 61.53 63.09 62.64 60.38 62.14 2.71 12 60.91 60.15 56.50 62.58 59.15 59.86 6.07 13 58.59 56.63 56.02 56.91 62.00 58.03 5.98 14 58.89 60.44 61.74 62.36 59.88 60.66 3.47 15 65.18 59.72 59.88 60.04 58.90 60.74 6.28 16 55.18 61.50 60.77 61.79 58.66 59.58 6.60 17 60.22 61.47 58.27 59.30 59.50 59.75 3.20 18 67.13 63.26 58.16 58.72 60.60 61.57 8.97 19 60.58 63.24 60.14 58.49 60.24 60.54 4.76 20 61.30 59.22 59.34 58.47 60.38 59.74 2.84 21 57.35 59.52 60.05 60.00 59.78 59.34 2.71 22 63.89 62.15 61.45 61.07 60.89 61.89 3.00 23 60.05 58.39 59.54 57.66 58.55 58.84 2.39 24 58.26 62.11 62.25 59.47 58.81 60.18 3.99 25 60.07 57.43 56.89 59.25 63.54 59.43 6.65 26 62.32 59.34 62.99 59.64 61.67 61.19 3.65 27 60.53 57.56 58.96 61.16 59.60 59.56 3.60 28 64.75 59.92 58.63 57.38 57.38 59.61 7.37 29 59.95 59.76 58.62 59.86 58.87 59.41 1.33 30 59.76 59.87 61.41 64.19 60.79 61.21 4.43 31 58.27 60.37 60.57 61.62 54.63 59.09 7.00 32 61.62 59.34 59.96 58.81 58.81 59.71 2.81 33 58.16 56.06 59.06 61.50 62.17 59.39 6.11 34 59.60 61.93 62.02 63.42 60.98 61.59 3.82 35 58.83 58.28 61.28 58.40 59.75 59.31 3.00 36 61.58 63.01 63.98 64.78 63.68 63.41 3.20 37 60.33 60.43 59.13 59.09 63.73 60.54 4.64 38 60.86 59.46 59.71 58.69 57.54 59.25 3.32 39 59.28 58.15 60.62 60.60 59.23 59.58 2.47 40 62.69 60.73 55.28 60.32 60.93 59.99 7.40 X = 60.195, R = 4.921 RQA, as follows (we use the constants corresponding to kurtosis = 3.0) 2 : CL R = R = 4.921 UCL R = D4 R = (3.35) 4.921 = 16.485 LCL R = D3 R = (0.15) 4.921 = 0.738 2 In general, for intermediate kurtosis values, a linear interpolation seems to be quite satisfactory. For a given value of n, we ran simple linear regressions for the D3 and D 4 values as the dependent variables and the kurtosis values as the independent variable. The R 2 values for these regressions range from 0.923 to 0.976. Figure 1 shows the R-chart with the control limits calculated from our proposed RQA method, KC method (Tadikamalla and Popescu [12]), and Shewhart s method. Note that both the RQA and the KC method indicate that the process under investigation is in statistical control, as is determined by the engineers, while the traditional Shewhart s method sends an out of control signal. (Tadikamalla and Popescu [12]) give the corresponding X chart, which shows that the process as being out of control using the Shewhart chart and to be in control using the KC method). 3. THE NORMAL DISTRIBUTION 3.1. Exact Control Limits for a Normal Process Distribution Consider the case of a standard normal process distribution: f(x)= 1 e x2 2 (9) 2π Substituting (9) in (2), we can compute D3 and D 4 as follows: LCL R = G 1 (α/2) ( ) 1 n n 2π = α/2 UCL R = G 1 (1 α/2) ( ) 1 n n 2π = 1 α/2 x+lcl R x x+ucl R x e x2 2 dx e x2 2 dx n 1 n 1 Then the control chart constants will be given by: where d 2 = D 3 = LCL R d 2 D 4 = UCL R d 2 e x2 2 dx e x2 2 dx (10) (11) [1 (1 (x)) n ( (x)) n ]dx and (x) is the cumulative density function of the standard normal distribution. Equation (10) can be solved numerically for different values of n and α = 0.0027 to yield the corresponding control limits. Table 4 below presents the R-chart constants for a process with an underlying normal distribution computed using our

96 Naval Research Logistics, Vol. 55 (2008) Figure 1. R-chart for the quality characteristic showing the RQA, KC and Shewhart upper and lower control limits. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.] approach as compared with the traditional Shewhart constants derived using the 3-sigma limits. 3.2. The Importance of the LCL Simulated Example We have always wondered about the use of an accurate approximation to the LCL of the R-chart. The purpose of an R-chart, in general, is to monitor the process variance. A typical out of control signal (a point outside the 3 sigma UCL) warns that the process variation may have increased, which in turn warrants an investigation for an assignable cause. In Shewhart control charts, the LCL is zero for n<7, and an out of control signal on the LCL side is impossible for small sample sizes. As in the case of the p-chart (proportion defective), an out of control signal on the LCL side of the R-chart could be a good thing. Such a signal may lead us to an assignable cause, which could result in an unusually low variance in the process quality characteristic. An accurate (nonzero) approximation to the LCL of the R-chart may provide such an opportunity. We simulated several data sets from a normal distribution (µ = 50, σ = 2, n = 5, 6, and r = 40). During the Figure 2. R-chart for the simulated quality characteristic showing the RQA, KC and Shewhart upper and lower control limits. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Tadikamalla, Banciu, and Popescu: Range Charts for Normal and Long-Tailed Symmetrical Distributions 97 Table 4. R-chart constants for a normal process distribution. Exact method Shewhart s method n D 3 D 4 D 3 D 4 2 0.002 4.017 0 3.267 3 0.041 2.925 0 2.575 4 0.107 2.525 0 2.282 5 0.170 2.312 0 2.114 6 0.224 2.176 0 2.004 7 0.269 2.081 0.076 1.924 8 0.307 2.009 0.136 1.864 9 0.339 1.954 0.184 1.816 10 0.366 1.910 0.223 1.777 12 0.410 1.840 0.283 1.717 15 0.459 1.768 0.347 1.653 20 0.513 1.691 0.415 1.585 simulation (in between the simulation of the subgroups), we randomly (with a probability of 0.03) induced an abnormal variation to the process by reducing the variance by half (σ = 1). Table 5 gives one such simulated data set. Figure 2 shows the corresponding range chart with control limits both from the Shewhart method and the proposed method (RQA). Note that the Shewhart method (and the KC method, not shown here) would have an LCL of zero and thus would not have detected an out of control situation where as the proposed method detected an out of control situation on the LCL side. 4. EVALUATION OF THE PERFORMANCE OF THE PROPOSED R-CHARTS 4.1. The Logistic Distribution Case The pdf and the cdf of the standard logistic distribution (with zero mean and unit variance) are given as: and f(x)= π 3 e πx 3 (1 + e πx 3 ) 2 (12) F(x) = (1 + e πx 3 ) 1 (13) Note that the logistic distribution is symmetric and has a kurtosis of 1.2. The control chart constants for the logistic distribution are calculated very similarly to the normal case described above. The constants D 3 and D 4 for the logistic distribution are given in Table 6. Table 6 also gives the corresponding constants for the Shewhart s method, the KC method [12], and the proposed RQA method. Figure 3 shows the exact LCL and UCL values for the R-chart for the standard logistic case and compares them to different approximations. We find that the LCL constant (D 3 ) is very closely approximated by RQA. The KC and the RQA approximations are comparable to the UCL constant (D 4 ). The performance of the Shewhart control limits is significantly worse. 4.2. The Laplace Distribution Case The Laplace distribution is a symmetric distribution with kurtosis of 3.0. The pdf and cdf of the standard Table 5. The importance of a nonzero LCL in an R-chart (Simulated data). Group 1 2 3 4 5 6 R 1 80.49 85.00 82.01 81.28 81.63 81.26 4.51 2 80.07 82.25 78.30 78.90 79.66 78.87 3.95 3 77.93 74.35 79.28 79.62 78.47 81.35 7.01 4 80.69 81.18 77.60 78.12 76.84 82.61 5.77 5 78.86 79.26 81.69 80.20 79.52 78.53 3.16 6 82.31 80.28 81.78 79.21 79.09 79.70 3.22 7 80.21 80.99 77.54 79.90 83.21 80.87 5.67 8 78.40 78.98 81.90 79.67 82.43 76.99 5.44 9 78.02 76.75 78.62 83.08 79.74 79.64 6.33 10 79.61 81.33 81.47 78.49 79.72 77.64 3.84 11 79.17 79.69 77.50 84.02 80.01 81.22 6.52 12 81.08 77.04 80.62 82.02 77.89 79.16 4.98 13 81.74 80.86 77.14 80.75 81.27 78.66 4.60 14 83.35 81.69 80.56 81.01 81.86 81.65 2.79 15 80.76 80.77 81.63 78.50 81.23 81.18 3.14 16 76.92 80.20 80.47 78.63 81.35 80.31 4.43 17 80.91 79.45 83.56 82.73 80.13 78.03 5.53 18 79.08 79.75 82.34 80.09 81.08 80.70 3.26 19 77.58 80.35 79.52 78.07 78.65 78.94 2.77 20 80.39 80.96 82.02 82.39 77.28 77.27 5.12 21 79.17 78.44 79.34 79.64 75.77 77.87 3.87 22 80.12 77.16 80.16 76.04 79.90 77.27 4.12 23 78.74 80.16 81.88 78.80 79.88 79.08 3.14 24 78.88 82.57 79.67 83.79 81.62 82.50 4.91 25 83.43 81.72 79.74 76.57 81.08 81.39 6.86 26 81.65 80.21 76.21 80.14 82.48 82.29 6.26 27 81.29 80.83 76.76 80.08 79.66 78.54 4.52 28 80.20 75.61 79.00 80.09 78.57 84.00 8.39 29 84.25 81.73 81.49 79.13 77.38 77.37 6.88 30 81.61 82.84 82.23 79.44 83.14 80.63 3.70 31 81.10 81.55 80.54 80.20 77.95 78.08 3.60 32 78.30 78.68 78.52 78.31 78.84 79.21 0.90 33 82.26 80.70 78.01 78.22 77.16 78.38 5.09 34 79.64 82.33 80.45 83.94 78.34 82.19 5.60 35 81.85 83.38 77.61 81.31 79.83 81.57 5.76 36 82.25 78.50 80.50 82.89 80.39 81.60 4.39 37 79.72 79.36 79.65 83.74 82.17 81.14 4.38 38 81.83 78.42 79.32 80.60 77.89 82.93 5.03 39 76.51 81.09 82.56 80.93 80.72 79.52 6.05 40 79.97 81.54 80.14 82.32 82.13 81.40 2.35 R = 4.7 Shewhart Limits: UCL = 9.4188; LCL = 0 Exact Limits: UCL = 10.2272; LCL = 1.0528

98 Naval Research Logistics, Vol. 55 (2008) Table 6. R-chart control limits for a logistic process distribution. Table 7. R-chart control limits for a Laplace process distribution. Exact Shewhart s KC RQA method method method method n D3 D4 D3 D4 D3 D4 D3 D4 2 0.002 4.717 0 3.267 0 4.481 0.002 4.719 3 0.039 3.515 0 2.575 0 3.363 0.039 3.530 4 0.099 3.066 0 2.282 0 2.957 0.100 3.091 5 0.156 2.821 0 2.114 0 2.738 0.158 2.853 6 0.204 2.663 0 2.004 0 2.598 0.207 2.701 7 0.243 2.550 0.076 1.924 0 2.498 0.248 2.594 8 0.276 2.465 0.136 1.864 0 2.422 0.281 2.513 9 0.303 2.398 0.184 1.816 0 2.363 0.309 2.449 10 0.326 2.343 0.223 1.777 0 2.314 0.333 2.399 12 0.364 2.259 0.283 1.717 0 2.239 0.371 2.321 15 0.406 2.169 0.347 1.653 0 2.159 0.414 2.238 20 0.453 2.072 0.415 1.585 0 2.071 0.461 2.150 Exact Shewhart s KC RQA method method method method n D3 D4 D3 D4 D3 D4 D3 D4 2 0.002 5.493 0 3.267 0 5.138 0.002 5.312 3 0.033 4.125 0 2.575 0 3.956 0.038 4.099 4 0.080 3.588 0 2.282 0 3.509 0.096 3.613 5 0.124 3.283 0 2.114 0 3.266 0.150 3.350 6 0.161 3.082 0 2.004 0 3.109 0.196 3.180 7 0.193 2.937 0.076 1.924 0 2.997 0.233 3.061 8 0.219 2.823 0.136 1.864 0 2.913 0.264 2.969 9 0.242 2.738 0.184 1.816 0 2.847 0.289 2.900 10 0.262 2.661 0.223 1.777 0 2.793 0.311 2.842 12 0.295 2.547 0.283 1.717 0 2.709 0.346 2.753 15 0.334 2.427 0.347 1.653 0 2.621 0.384 2.659 20 0.381 2.295 0.415 1.585 0 2.525 0.427 2.560 Laplace distribution (mean zero and unit variance) are given below: f(x)= 1 e 2 x (14) 2 and 1 F(x) = 2 e 2x, x 0 1 1 2 e 2x, x>0 (15) The control chart constants (D 3 and D 4 ) for the Laplace distribution can be calculated very similarly to the normal and logistic cases. The constants D 3 and D 4 for the Laplace distribution are given in Table 7. Table 7 also gives the corresponding constants for Shewhart s method, the KC method [12], and the RQA method. Exact values for the LCL and UCL of the Laplace distribution are shown in Figure 4 along with the other approximations. Once again, RQA outperforms the other approximations to the LCL and is comparable to the KC method for the UCL. 5. CONCLUSIONS This study presents a method to calculate more accurate limits for the R-chart under the assumption of a symmetric, leptokurtic distribution of the quality characteristic. Our Figure 3. Upper and lower control limits for the R-chart for the logistic distribution.

Tadikamalla, Banciu, and Popescu: Range Charts for Normal and Long-Tailed Symmetrical Distributions 99 Figure 4. Upper and lower control limits for the R-chart for the Laplace distribution. method computes the true quantiles directly for the distribution of R and thus overcomes the disadvantage of the Shewhart R-chart and other approximations that use ±3 sigma limits which result in a LCL of zero for smaller sample sizes. We tabulated the D3 and D 4 constants for different values of n and of the kurtosis and illustrate the implementation of our method with a numerical example. By providing these constants, the implementation of our method is identical with Shewhart s method and is just as simple. As a special case, we tabulated the exact constants of the range chart for the normal distribution. Finally, we compared the performance of our method for the case of the logistic and the Laplace distributions and found increased accuracy for the LCL relative to the Shewhart method and the KC method. REFERENCES [1] W. Albers and W.C.M. Kallenberg, Estimation in Shewhart control charts: Effects and corrections, Metrika 59 (2004), 207 234. [2] I.W. Burr, Cumulative frequency functions, Ann Math Stat 13 (1942), 215 232. [3] P. Castagliola, Control charts for data having a symmetrical distribution with a positive kurtosis, Recent Advances in Reliability and Quality Engineering, Hoang Pham (editor), World Scientific Publishers, 2001, pp. 1 16. [4] L.K. Chan and H.J. Cui, Skewness correction x-bar and R-charts for skewed distributions, Nav Res Logistics 50 (2003), 555 573. [5] N.L. Johnson, Systems of frequency curves generated by methods of translation, Biometrika 36 (1949), 149 176. [6] M.B.C. Khoo and E.G. Lim, An improved R (range) control chart for monitoring the process variance, Qual Reliab Eng Int 21 (2005), 43 50. [7] D.C. Montgomery, Introduction to statistical quality control, Wiley, New York, 2001. [8] W.A. Shewhart, Economic control of quality of manufacturing processes, American Society for Quality Control, Milwaukee, 1931. [9] A. Stuart and J.K. Ord, Advanced theory of statistics, Oxford University Press, New York, 1987. [10] P. Tadikamalla, Random sampling from the exponential power distributions, J Am Stat Assoc 75 (1980), 683 686. [11] P. Tadikamalla and N.L. Johnson, Tables to facilitate fitting Lu distributions, Commun Stat Simulation Computation 11 (1982), 249 271. [12] P. Tadikamalla and D. Popescu, Kurtosis correction method for x-bar and R control charts for long-tailed symmetrical distributions, Nav Res Logistics 54 (2007), 371 383. [13] L.H.C. Tippett, On the extreme individuals and the range of sample taken from a normal population, Biometrika 17 (1925), 364 387.