Calculation of X̄ & R Charts

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X̄ and R for each sub group:
Let X_ij, j= 1, 2, ..., n be the measurement on the ith sample (i = 1, 2,...,k). The mean X̄i, the range R and the standard deviation Si, for the ith sample are given by -

Next, we find X̄̄ (X-bar-bar), R̄ (R-bar) and  S̄ (S-bar), the average of sample means, sample ranges and standard deviation respectively as follows-

----(ii)

Setting the control Limits:

We know if σ is the process standard deviation, then the standard error of sample mean is σ/√n, where n is the sample size i.e.

S.E (X̄_i) = σ/√n, i = 1,2,..., k

Also from the sampling distribution of range, we know, 

E (R) = d₂. σ , where

d₂ is the constant depending on the sample size. 

Thus an estimate of σ can be obtained from R by the relation -

R̄ (R-bar) = d₂. σ

=> σ = R̄ /d₂ ---(iii)

Also, X̄̄ (X-bar-bar) gives an unbiased estimate of population mean since