Calculation of Control Limits and Standard Deviation

The data in the accompanying table illustrated length measurements (in.) taken on a machined workpiece. Sample size is five and the sample number is 10; therefore the total number of parts measured is 50. The quantity x  is the average of five measurements in each sample. We first calculate the average of averages x,

Since the sample size is five, we determine from Table , the following constants :

A₂ = 0.58, D₄ = 2.11, and D₃  =  0. We can now calculate the control limits using Eqs. Thus, for averages we have

UCL_x = 4.430 + (0.58) (0.103) = 4.489 in.,

LCL_x = 4.430 - (0.58) (0.103) = 4.370 in.,

We can calculate the standard deviation using Eq. σ =  R /d₂ and a value of d₂ = 2.326.

Thus,

σ = 0.103 /2.326

= 0.044 in.

Table  : Factors for Estimating the Standard Deviation (σ′) of a Parent Population from Sample Data for the Average Range ( R ) or the Average Sample Standard Deviation ( σ )