# Defects Per Million Opportunities

**Independent, Identically Distributed and Charting:**

1. The central limit theorem (CLT) plays an important role in statistical quality control largely because it helps to predict the performance of control charts.

2. control charts are used to avoid intervention when no assignable causes are present and to encourage intervention when they are present.

3. The CLT helps to calculate the probabilities that charts will succeed in these goals with surprising generality and accuracy. The CLT aids in probability calculations regardless of the charting method (with exceptions including R charting) or the system in question, *e.g.*, from restaurant or hospital emergency room to traditional manufacturing lines.

**To understand how to benefit from the central limit theorem and to comprehend the limits of its usefulness, it is helpful to define two concepts.**

A. First, the term “**independent**” refers to the condition in which a second random variable’s probability density function is the same regardless of the values taken by a set of other random variables. For example, consider the two random variables: *X*_{1} is the number of boats that will be sold next month and *X*_{2} is their sales prices as determined by unknown boat sellers. Both are random variables because they are unknown to the planner in question.

B. The planner in question assumes that they are intendant if and only if the planner believes that potential buyers make purchasing decisions with no regard to price within the likely ranges. Formally, if *f *(*x*_{1} ,*x*_{2} ) is the “joint” probability density function, then independence implies that it can be written *f *( *x*_{1} , *x*_{2})= *f *( *x*_{1} ) *f *(*x*_{2} ) .

C. Second, “**identically distributed**” means that all of the relevant variables are assumed to come from exactly the same distribution. Clearly, the number of boats and the price of boats cannot be identically distributed since one is discrete (the number) and one is continuous (the price).

D. However, the numbers of boats sold in successive months could be identically distributed if

**(1) **buyers were not influenced by seasonal issues and

**(2) **there was a large enough pool of potential buyers. Then, higher or lower number of sales one month likely would not influence prospects much in the next month. In the context of control charts, making the combined assumption that system outputs being charted are independent and identically distributed (IID) is relevant.

I. Departures of outputs from these assumptions are also relevant. Therefore, it is important to interpret the meaning of IID in this context. System outputs could include the count of demerits on individual hospital surveys or the gaps on individual parts measured before welding.

II. To review: under usual circumstances, common causes force the system outputs to vary with a typical pattern (randomly with the identical, same density function). Rarely, however, assignable causes enter and change the system, thereby changing the usual pattern of values (effectively shifting the probability density function). Therefore, even under typical circumstances the units inspected will not be associated with constant measurement values of system outputs.

III. The common cause factors affecting them will force the observations to vary up and down. If measurements are made on only a small fraction of units produced at different times by the system, then it can be reasonably assumed that the common causes will effectively reset themselves. Then, the outputs will be IID to a good approximation. However, even with only common causes operating, units made immediately after one another might not be associated with independently

IV. distributed system outputs. Time is often needed for the common causes to reset enough that independence is reasonable.