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I recently came across quality control charts in my studies and found out that the estimator

$$\hat \sigma =\frac{s}{c_4}$$

is preferred for UCL and LCL calculations in an x-bar chart as it is considered unbiased ($s$ is the sample standard deviation). I am not used to all these new estimators although I already took 2 basic statistics courses and I am familiar with typical confidence intervals and basics estimators.

Now, say that I would like to build a simple x-bar chart and s-chart with both $\mu$ and $\sigma$ unknown for a list of 50 samples each of consisting of 5 observations.

According to what I understood from my study materials I would write something like this in R

s <- apply(sd,samples) # Calculate sd for each sample
s_bar <- mean(s)       # Center line for the s-chart
UCL_s <- s_bar + 3*s_bar/c4 * sqrt(1-c4^2)
LCL_s <- s_bar - 3*s_bar/c4 * sqrt(1-c4^2)

and for the x-bar chart

x <- apply(mean,samples) # Calculate mean for each sample
x_bar <- mean(s)         # Center line for the x-chart
UCL_x <- s_bar + 3*s_bar/(c4 * sqrt(n))
LCL_x <- s_bar - 3*s_bar/(c4 * sqrt(n))

Now my questions:

  1. Is the estimation process of $\sigma$ and its intervals correct and unbiased?
  2. How would I calculate the $c_4$ term? I could look it up on the tables but is there a more fast way to find it? Is there an R function that provides such coefficient?

  3. Say that the standard deviation is now given and equal to $\sigma$. The UCL and LCL should now be the following

    UCL_s <- c4 * sigma + 3 * sigma * sqrt(1-c4^2)

    LCL_s <- c4 * sigma + 3 * sigma * sqrt(1-c4^2)

Why do I need to multiply the known $\sigma$ for the coefficient $c_4$?

mdewey
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mickkk
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  • Although it's unclear what "$c4$" actually refers to, your reference to an unbiased estimator of a standard deviation suggests your question might already have an answer at http://stats.stackexchange.com/a/27984/919. Please let us know if it does not. – whuber Nov 28 '15 at 21:12
  • @whuber Yes it does answer my question. Should I delete this question? – mickkk Nov 28 '15 at 23:11
  • M.A. Mahmoud et al. go into depth on this topic in their paper "Estimating the Standard Deviation in Quality-Control Applications" published in the _Journal of Quality Technology_ in Vol. 42, No. 4, October 2010. $\sigma_s$ is a multiple of $\sigma$ in the form $\sigma_s = \sqrt{\left(1-c_4^2\right)}\sigma$ In that paper, $c_4=\frac{\sqrt{\frac{2}{n-1}\times \Gamma[n/2]}}{\Gamma[(n-1)/2]}$ and can be approximated by $c_4\approx\frac{4n-4}{4n-3}$. NIST defines $c_4$ as $c_4=\sqrt{\frac{2}{n-1}}\frac{\left ( \frac{n-2}{2} \right )!}{\left ( \frac{n-3}{2} \right )!}$ – Tavrock Dec 16 '16 at 17:54

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