Understanding the Difference Between Long and Short Term Sigma

for postsWe have seen that the main difference between Cpk and the Ppk is the way in which the value of sigma (standard deviation) is being calculated.

In Cpk, the value of sigma comes from the control chart and usually given by the formula

Where  is the average of the absolute value of range (obtained as a difference of two consecutive points when, data is arranged in a time order). The term d2 is a statistical constant that depend on the sample size.

This sigma-short is affected by the time order to the data i.e. every time you change the time order, sigma-short would change.

Whereas, in Ppk the sigma is calculated using traditional formula and is also called as the overall sigma or sigma-long.

In this case, sigma-long is not affected by the time order of the data points. This is called as overall standard deviation.

Usually, sigma-short is less than sigma-long.

Let’s do a simulation in R to check whether sigma-short is really affected by the time order or not

 #setting the seed for reproducibility
 #load library QCC
 # Generate a normal sample of 50 data points
 # Generate a data set for storing output of the control chart, sigma-short and   sigma-long
 # Generate a blank matrix of 10 rows and 50 columns to store 10 10   random samples each having 50 data points.
sam<-matrix(nrow=10,ncol=50,byrow = TRUE)
 # Code for generating 10 random samples from the normal sample   generated as (d) above
for(i in 1:10){
sam[i,]<-sample(d,50,replace=FALSE) #generate ith sample and store in   the matrix sam.#generate I-MR chart of the ith sample.

#calculate sigma-short of the ith sample.

#calculate sigma-long of the ith sample.

#print data frame   containing sigma-short and sigma-long of all 10 sample.

Table-1: Short and long sigma generated from the same simulated data but with different time order.

sigma_short   sigma_long
1.1168596          1.09059
1.1462365          1.09059
1.1023853 1.09059
0.9902320 1.09059
1.1419678 1.09059
1.2173854 1.09059
0.9941954 1.09059
1.0408088 1.09059
1.1038588 1.09059
1.2275286 1.09059

It is evident from the simulation that sigma-short do get affected by the time order of the data. Therefore, the sigma or the standard deviation calculated from the control charts (short sigma) and the overall sigma are different.

for more on Cpk and Ppk see below links

Car Parking & Six-Sigma

What Taguchi Loss Function has to do with Cpm?

What do we mean by garage’s width = 12σ and car’s width = 6σ?