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Let X

**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-

**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) = d

_{2}. σ , where
d

_{2}is the constant depending on the sample size.
Thus an estimate of σ can be obtained from R by the relation -

R̄ (R-bar) = d

_{2}. σ
=> σ = R̄ /d

_{2}---(iii)
Also, X̄̄ (X-bar-bar) gives an unbiased estimate of population mean since

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