Variance

Variance is the difference in values of data points in a sample or population. Many often refer to variance as the spread or dispersion of the data in a sample or population.

Technically, varinace is the square of the standard deviation of a sample or population. It is calculated as follows:

variance = average(sum((mean-X)^2)))

Where:

  • X = the individual data points in the sample/population
  • mean = the sample or population mean

 

1.5 Sigma Shift

Sigma shift is a method used to estimate long-term process capability from short term process capability. This method assumes that a process will drift 1.5 standard deviations (sigma) on a long term basis as compared to a short-term basis. Thus, a process that is within six standard deviations with a short-term sample, would be 4.5 sigma on a long-term basis.

Hypothesis Testing

Hypothesis testing is the method by which a theory is tested with statistical tests using observed data. Typically the investigator will have an idea or theory as to how a process works. They will create a null hypothesis which is the theory stated in the null (e.g., if my hypothesis is that applying a new process to technical support will improve customer satisfaction by 3 points, my null hypothesis is that application of this new process will not change customer satisfaction scores.)

Input and output data for the process are collected and analyzed using a statistical test. The statistical test assesses the likelihood that any differences, changes, or patterns were due to chance. If the p value for the test is less than the pre-determined target (often p < .05 or .01) the investigator will reject the null hypothesis and accept an alternative hypothesis instead. If the p value is greater than the pre-determined value, the investigator does not reject the null hypothesis–in practical terms this means that the sampled data do not support the alternative hypothesis.

Example: My alternative hypothesis is that having technical support agents collect specific data on each support inquiry will correlate with a difference in the satisfaction scores of our customers. The null hypothesis, then, is that implementing this new procedure will not relate with any difference in customer satisfaction scores.

I implement the new process in a test group and also observe a control group that continues to operate without the new process. I then collect customer satisfaction scores for each group prior to and during the test period. I run an ANOVA model that results in a p-value of .01.

In this case I reject the null hypothesis because the chances that the difference between the test and control groups was by chance is less than one percent.

Statistical Process Control (SPC)

Statistical Process Control is a methodology for improving business processes. One of the most widely known tools used with the methodology are statistical control charts such as X-bar and Individual-Moving Range. These charts use statistical methods to assess whether a business process operates within a stable range of variance. The charts are also often used to identify special or common cause variation in a process.

A statistical control chart typically sets upper and lower control limits (often at the third standard deviations above and below the mean for the sample). When data points exceed these control limits, the process is considered to be out of statistical control (and experiencing special cause variation.) Often, analysts will also employ other tests to determine whether a process is out of control by looking for various non-random patterns in the process data.

Special Cause Variation

Special cause variation refers to variation in an output or process that result from factors outside of the process. Special cause variation is often detected when output measures exceed the upper or lower control limits in a statistical process control chart. Special cause can also be detected when non-random patterns develop within the control limits.

An example of special cause of variation: Variation in the supply inputs for a process may change over time resulting in an process that goes out of statistical control.

Type II Error

When testing a hypothesis, this sort of error occurs when the null hypothesis is NOT rejected but should be rejected. The probability that this type of error occurred is represented by the Greek letter beta.

Examples:

  • A quality engineer passes a product when in fact it is defective
  • A defendant is acquitted but in truth, they are guilty
  • Employees of a company assume that the company does NOT have a particular problem but in fact the problem exists

 

Type I Error

When testing a hypothesis, this sort of error occurs when the null hypothesis is rejected but should not be rejected. The probability that this type of error occurred is represented by the Greek letter alpha.

Examples:

  • A quality engineer rejects a product as defective when in fact it is NOT defective
  • A defendant is convicted but in truth, they are innocent
  • Employees of a company assume that the company has a problem but in fact the problem does not exist