Control charts for X and R are maintained on the shear strength of spot welds. One hundred observations divided into subgroups of size five are used as a baseline to construct the charts, and estimates of and are computed from these observations.

Assume that the 100 observations are X1, X2, . . . , X100 and the ranges of the 20 subgroups are R1, R2, . . . , R20. From these baseline data the following quantities are computed:

Using this information, compute the values of three-sigma limits for both charts.

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100 20 = 97,500, R = 1,042. == i=1 j=1