# 3 - σ Control Limits & Steps for X̄ & R chart

3 - σ Control Limits:
Consider the statistic: t = t (x1, x2, ..., xn) a function of the sample observations x1, x2, xn.
Let, E (t) = μt and Var (t) = σt2
If the statistic t is normally distributed, then from the fundamental property of normal distribution, we have,
P[ μt - 3σt < t < μt + 3σt ] = 0.9973
P[|t - μt| < 3σt] = 0.9973
P[|t - μt| > 3σt] = 0.0027
In other words, the probability that a random value of t goes outside the 3 - σ limit is 0.0027, which is very small.
Hence, if t is normally distributed, the limits of variation should be between μt + 3σt and μt - 3σt which are termed as upper control limit (UCL) and lower control limit (LCL) respectively.
If for the ith sample,
* The observed ti lies between UCL & LCL, the variation between samples is attributed to chance, in this case the process is in statistical control.
* The observed ti falls outside the control limits, it indicates that some assignable causes has crept in which must be identified and eliminated.

X̄ (X -bar) and R Charts:
The control limits in the X̄ (X-bar) & R charts are so placed that they reveal the presence or absence of assignable causes of variation in the -
i. average - mostly related to machine setting
ii. range - mostly related to negligence on the part of the operations.

Steps for X̄ (X -bar) & R charts:
1. Measurement:
The work of control chart stands first with measurement. Since the conclusion drawn from control chart are based on the variability in the measurement as well as the variability in the quality being measured. It is important that the mistakes in the reading measurements instruments or errors in recording data should be minimized so that valid conclusions can be drawn from control charts.
2. Selection of samples or subgroups:
It is essential to make the rational selection of the samples or subgroups in order to make the control chart analysis effective.
The choice of the sample size n and the frequency of sampling (i.e. the time between the selection of two groups) depend upon the process and no hard rules can be laid down for this purpose. Usually n is taken to 4 or 5 while the frequency of sampling depends on the state of control exercised.