Definition of Y = f(X):

« Back to Glossary IndexThe team looks over the problem statement for the new process improvement project.

One of the team members says, “What exactly are we trying to solve? I know that we are trying to get out more products over a shift, but I have no idea as to what we are trying to do to make that possible.”

“I’m still clueless regarding how we plan to figure this out. What are we even looking at changing? Are we changing where work is done, the tooling involved, or possibly the personnel? What factors influence the throughput rate, and how do we go about proving that our solution was the right solution?”

The project manager is pleased that the team is asking the right questions. It’s not enough to just see if the throughput rate improved, but it is important to understand how a change in the system inputs affect the throughput rate, or results of the process. In the world of Six Sigma, this concept is known as Y=f(x).

## An overview: What is Y=f(x)?

In the strictest view, Y=f(x) is a representation of a mathematical formula. It is one to use when examining different possible outcomes based on the inputs and factors used. The “Y” stands for the outcome, the “f” embodies the function used in the calculation, and the “X” represents the input or inputs used for the formula.

This formula, when associated with Six Sigma, is called the breakthrough equation. So, we use a formula to find out which inputs will send us to the optimal, or best, output.

The formula is used throughout a problem-solving process. For example, when using the DMAIC steps, we progress through the following five stages:

**Define:**Gather the desired outcome (Y)**Measure:**Gather the possible inputs (x) as well as measurements for inputs (x) and outputs (Y)**Analyze:**Use the formula (f) to test the relationship between inputs (x) and outcome (Y)**Improve:**Select and implements the best inputs (x) for achieving the desired outcome (Y)**Control:**Monitor the Y over time to see if any changes in inputs (x) are occurring

## The benefits of attending to Y=f(x)

### Assists in expressing the desired outcome as a measurable term

It can be different setting up the initial problem statement, and the various “who, what, when, where, why, and how many” questions. A key element of the problem statement is determining the metric we’re trying to improve.

When using Y=f(x) in the Define stage, it pushes us to clearly understand “Y,” which is the desired results we are working to achieve. By using the formula setup, it forces the team to view the outcome as a measurable term.

### Guides the results for process improvement efforts

It can be easy to lose track of how we measure process improvement goals during a process improvement project. Using Y=f(x) as a concept forces the project manager to consider metrics from the beginning of the project all the way to the end of the project. These projects need to show quantitative improvement, so understanding the desired outcome and how inputs affect reaching our project goals is critical.

### Makes it possible to monitor improvements over time

By understanding the inputs and outputs of a continuous improvement project, it becomes possible to understand how to verify the results once implemented, and the best way to monitor that the results are maintained over time.

## Best practices for using Y=f(x)

- By setting up the formula, it assists us in selecting the right tool to verify X-Y relationships.
- Be sure that as many inputs that are significant factors connected to the results are included. Consider statistical processing software, like Minitab, for examining whether the combination of specific inputs causes a greater (or lesser) impact on the results.
- Remember to use Y=f(x) at all stages of problem-solving. Forget to use it at the beginning, and you may be working on the wrong problem, or with the incorrect formula. Forget to use it at the end, and you may find out that either the anticipated savings were not met, or that they were not maintained.

## Frequently Asked Questions (FAQ) about Y=f(x)

### I have seen a version of Y=f(x) that included a “+E” component at the end of the equation. Is this a different concept?

No, this is still the Y=f(x) methodology, but it adds an additional mathematical quality into play. The “+E” stands for an element of random error related to the input “x” when transformed by the function “f.”

### Does Y=f(x) methodology only work with DMAIC?

While the example above matches up how Y=f(x) is used during the various stages of the DMAIC problem solving process, Y=f(x) is a concept that also matches up with the stages of other problem-solving methodologies.

### Do I need to monitor every x and Y from the project once I enter the control phase of DMAIC?

While there is nothing stopping you from doing so, it’s recommended to use the lessons learned from the event and only monitor the most important inputs (x) per the conclusions of the project.

It’s common for projects to fail at the control phase, so the easier you can make monitoring the metrics, the more likely the team is to successfully perform the monitoring steps.

## Y=f(x) guides the team to understand how inputs affect the results, leading to better solutions

Y=f(x) is a concept within Six Sigma and other problem-solving methodologies that connects two concepts: Results should be measurable, and there should be an understanding between how inputs affect the results. Problem-solving should involve the attempt to find the best set of inputs that help drive a process to the optimal result.

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