Shewhart Control Charts, also known as process-behavior charts, are a statistical tool used to determine if a manufacturing or business pgrocess is in a state of control. They help in monitoring the stability of processes over time by distinguishing between common cause variation (inherent to the process) and special cause variation (due to external factors). The Central Limit Theorem (CLT) plays a crucial role in the formulation of Shewhart Control Charts, as it allows for the approximation of the distribution of sample means to a normal distribution, regardless of the shape of the population distribution, given a sufficiently large sample size.
The formula for the control limits in a Shewhart Control Chart is given by:
UCL: \( \bar{X} + 3\frac{\hat\sigma}{\sqrt{n}} \)
CL: \( \bar{X} \)
LCL: \( \bar{X} - 3\frac{\hat\sigma}{\sqrt{n}} \)
where \( \bar{X} \) is the average of the sample means, \( \hat\sigma \) is the estimated standard deviation of the process, and \( n \) is the sample size in the group.
The c4 function returns the expected value of the standard deviation of \( n \) independent, normally distributed random variables with the same mean and with standard deviation of 1. The value c4 can be expressed as:
\( c_4 = \frac{\Gamma\left(\frac{n}{2}\right) \sqrt{2/(n-1)}}{\Gamma\left(\frac{n-1}{2}\right)} \)
where \( \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt \) is the gamma function. As n grows, \( c_4 \) is asymptotically equal to \( (4n-4)/(4n-3) \).
The d2 function returns the expected value of the sample range of \( n \) independent, normally distributed random variables with the same mean and a standard deviation of 1. The value d2 can be expressed as:
\( d_2 = \int_{-\infty}^{\infty} \left[ 1 - \left( 1 - \Phi(x) \right)^n - \Phi(x)^n \right] \, dx \)
where \( \Phi(x) \) is the standard normal cumulative distribution function.
The d3 function returns the standard deviation of the range of \( n \) independent, normally distributed random variables with the same mean and with unit standard deviation. The value d3 can be expressed as:
\( d_3 = \sqrt{2 \int_{-\infty}^{\infty} \int_{-\infty}^{y} f(x, y) \, dx \, dy - d_2^2} \)
where
\( f(x, y) = 1 - \left( \Phi(y) \right)^n - \left( 1 - \Phi(x) \right)^n + \left( \Phi(y) - \Phi(x) \right)^n \)
where \( \Phi(\cdot) \) is the standard normal cumulative distribution function and \( d_2 \) is the expected range.