# Variable Control Chart Limits

The control limits for the control charts are calculated using the following formulas and Table 3.1 for control chart factors. The control chart factors were designated with variables such as A2, Ds, and D4 to calculate the control limits of X and R control charts. The factor d2 is important in linking the average range and hence the standard deviation of the sample (s), to the population standard deviation a. The control chart factors shown in Table 3.1 stop at the number 20 of observations of the subgroup. Control charts are based on taking samples to approximate a large production output. If the sample be, comes large enough, there is no advantage to using samples and their associated normal distributions to generate variable control charts.

Instead, 100% of production could be tested to find out if the parts produced are within specifications.

### Control and specification limits

Control chart limits indicate a different of conditions than the specification limits. Control limits are based on the distribution of sample averages，whereas specification limits are related to population distributions of parts. It is desirable to have the specification lim. It’s as large as possible compared to the process control limit.

The control limits represent the 3 5 points，based on a sample 〇f n observations. To determine the standard deviation of the product population, the central limit theorem can be used:

where

s = standard deviation the distribution of sample averages

σ= population deviation

n = sample size

Multiplying 173 the distance from the centerline of the X chart to one of the control limits by Vn will determine the total product population deviation. A simpler approximation is the use of the formula a = R/d2 from control chart factors in Table 3.1 to generate the total product standard deviation directly from the control chart data, d2 can be used as a good estimator for a when using small numbers of samples and their ranges.