Because Six Sigma approaches things with a statistical mindset, it considers all problems as a function. Using mathematical symbols, this looks like:

The y=f(x) statement can be used in two ways. First, it is a general map for stating a problem. Y (the problem) occurs because some X (input or cause) is occurring. In reality, Y is usually occurring because of some group of causes or inputs, which means there are going to be more than one X inputs.

The idea can also be applied to specific processes and outcomes within the problem. As you get more and more granular, the y=f(x) concept becomes increasingly mathematical; in many cases, you can graph the relationship between the output (y) and the input (x).

To understand the concept of thinking of problems as a function, let’s look at a problem that might occur for a large internet service provider. The manager of a service team has discovered that service calls are taking much longer than expected; in fact, his five team members take 1.75 times longer on average than other service reps in the company to handle all types of calls.

To find out what might be causing the situation, the manager researches the problem by talking to the reps, talking to the customers, and going out on random calls with all five representatives. He makes the following observations:

- One representative is a native to the area the team services, which means he or she knows many of the customers personally. This results in friendly chatter that lengthens the time on the job.
- One representative is providing customers with very in-depth explanations and education about internet issues, sometimes over and beyond what the customers would ever need to know regarding their internet service.
- One representative is new to the job and takes longer to complete each task because he or she is unsure of the work, has to double-check the work, or calls another rep to ask questions about the work.
- The remaining two reps perform work in times that are on par with company averages.

The manager distills this data down to two overall causes for the problem:

- Too much talking (reps one and two)
- Inadequate training

The problem can now be stated as a function:

The extra time is a function of too much talking and inappropriate training.

The manager also now has two root causes to address. The example is simple, but it illustrates the basic concept in defining a y=f(x) relationship for a problem and its causes. It’s not always so easy to conduct the research and analysis to find the relationship, but the relationship is always present.

Some other examples of y=f(x) relationships include:

- Low customer satisfaction with hamburger taste is a function of an uncalibrated grill.
- Low employee morale is a function of a poor time-off approval system.
- Customer wait times are a function of technology distractions for employees.