Six Sigma

The normal, or Gaussian, distribution is a family of continuous probability distributions. The normal distribution function was first introduced in 1733, and since has become the most widely used family of distribution functions in statistical analysis. These distribution functions are defined by two parameters: a location (most commonly the "mean",$\mu$), and a scale (most commonly the "variance", $\sigma^{2}$). The use of normal distributions in statistics can be seen as advantageous due to its ability to maximize information entropy around a given mean and variance. Information entropy is the measure of uncertainty associated with a random variable.

For the purposes of this class and Chemical Engineering applications, you can model a given set of data using the probability density function which is shown below.

$\varphi=\frac{1}{\sigma \sqrt{2 \pi}} \exp \left(-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right)$                      (13.3.4)

$\varphi$ = probability density

$\mu$ = average

and the other variables are as defined in the above two sections.

The above PDF equation gives graphs, depending on mean and variance, similar to those below.

Above are 4 examples of different distributions given different values for mean and standard deviation. An important case is the standard normal distribution shown as the red line. The standard normal distribution is the normal distribution with a mean of 0 and a variance of 1. It is also important to note some special properties of probability density functions:

• symmetry about the mean,$\mu$
• the mode and mean are both equal to the mean
• the inflection points always occur one standard deviation away from the mean, at $\mu-\sigma \text { and } \mu+\sigma$.

Suppose we have a process where we make a product of a certain concentration and we have good control over the process. After analyzing a set of data from a time period we see that we have a standard deviation of only 0.01 and our product concentration is required to be within 0.05. In order to say our product is essentially defect-free, 4.5 standard deviations away from the average must be less than our required product tolerance (± 0.05). In this case 4.5 standard deviations is equal to 0.045 and our product tolerance is 0.05. This is more easily seen graphically, as seen in the figure below.

As you can see 4.5 standard deviations on either side of the averages falls just slightly inside of the required tolerances for our product. This means that 99.997% of our product will fall inside of our product tolerances.