# The 8 steps to creating a $\bar{X}$ and s control chart

Once you decide to monitor a process and after you determine using an $\bar{X}$ & s chart is appropriate, you have to construct the charts.

The $\bar{X}$ & R charts use the range as an approximation of the variation in the population. When feasible use the standard deviation, s, rather than the range, R for the improved efficiency in detecting meaningful changes in process variation.

Constructing the two charts is not difficult and by following the 8 steps below you will have a robust way to monitor the stability of your process.

## 1. Determine Sample Plan

Determine the sample size, n, and frequency of sampling. Consider cost of sampling, required resources, and balance with minimizing time (and produced units) between measurements. Of course more samples and more frequent measurements is better statistically.

## 2. Collect initial set of samples

Shewhart recommended 100 individual units in 25 samples of 4 each. Basically we want enough samples to create reasonable estimates of the mean values of $\bar{X}$ and s. Keep the data in time sequence following the time of the sample selection, which should be in the same order of manufacture.

## 3. Calculate $\bar{X}$

Calculate the average for each set of samples. This is the $\bar{X}$ for each sample.

## 4. Calculate s and $\sigma$ and ${\sigma _s}$

Calculate the standard deviation of each set of samples. This is the sample standard deviation, s, which is a biased estimate of the unknown population standard deviation. Handheld calculators and spreadsheets calculate this value.

$\displaystyle s = \sqrt {{{\sum\limits _{i = 1}^n {{{\left( {{x _i} - \bar x} \right)}^2}} } \over {n - 1}}}$

If the underlying distribution is normal, then we can correct for the bias with a constant, c4 , which is a function of sample size.

$\displaystyle {c _4} = \sqrt {{2 \over {n - 1}}} {{\left( {{n \over 2} - 1} \right)!} \over {\left( {{{n - 1} \over 2} - 1} \right)!}}$

The calculation uses a non-integer factorial. For n/2 we define it as follows

$\displaystyle \left( {{n \over 2}} \right)! = \left( {{n \over 2}} \right)\left( {{n \over 2} - 1} \right)\left( {{n \over 2} - 2} \right) \cdots \left( {{1 \over 2}} \right)\sqrt \pi$

The last term in the progression before the 1/2 squre root of pi is between zero and one. For example, for n = 7

$\displaystyle \left( {{7 \over 2}} \right)! = (3.5)! = \left( {3.5} \right)\left( {2.5} \right)\left( {1.5} \right)\left( {0.5} \right)\left( {1.77246} \right) = 11.632$

A Table of c4 Constants speeds up the calculations.

n c4
2 0.7979
3 0.8862
4 0.9213
5 0.9400
6 0.9515
7 0.9594
8 0.9650
9 0.9693
10 0.9727

The corrected (unbiased estimator of) standard deviation, ${\sigma}$ thus becomes

$\displaystyle \sigma = {s \over {{c _4}}}$

And, the standard deviation of the sample standard deviation, ${\sigma _s}$, is

$\displaystyle {\sigma _s} = \sigma \sqrt {1 - {c _4}}$

## 5. Calculate $\bar{\bar{X}}$

Calculate the average of the $\bar{X}$‘s. This is the centerline of the $\bar{X}$ control chart.

## 6. Calculate $\bar{s}$

Calculate the average of the s values. This is the centerline of the s control chart.

## 7. Calculate Control Limits

First calculate the s chart limits.

$\displaystyle UC{L _s} = \bar s + 3{{\bar s} \over {{c_4}}}\sqrt {1 - c _4^2}$

$\displaystyle LC{L _s} = \bar s - 3{{\bar s} \over {{c _4}}}\sqrt {1 - c_4^2}$

Be sure to plot the data on the s chart and if not in control, before continuing with building the control chart, work to bring the variability of the process under control.

For the $\bar{X}$ chart limits use

$\displaystyle UCL = \bar {\bar x} + 3{{\bar s} \over {{c _4}\sqrt n }}$

$\displaystyle LCL = \bar {\bar x} - 3{{\bar s} \over {{c _4}\sqrt n }}$

## 8. Plot the data

With the control limits in place, gather samples, and plot the data. Look for special or assignable causes and adjust the process as necessary to maintain a stable and in control process.

formulas from

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, Dec 21, 2013.