What Does Y=f(x) Mean? How to Use This Powerful Six Sigma Formula


Some people look at math and shut down. Managers are not usually those people. They know that there are formulas that must be used to solve certain business and process problems.

But there are different types of managers, just as there are different types of people. Some are more skilled as leaders than they are as mathematicians. Math is required, of course, but some managers still might be perplexed by certain equations, such as the commonly used Six Sigma formula Y=f(x).

Luckily, it doesn’t take a rocket scientist to understand and use Y=f(x) because it’s a cornerstone of the Six Sigma methodology and can be very useful when applying the acronym DMAIC (Define, Measure, Analyze, Improve, Control) to your project. We’ll show you how to use this formula in concert with your project management tools and processes when managing initiatives and outcomes.

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What Does Y=f(x) Stand For?

Using Y=f(x) can help determine the cause and effect in a project, and it can be used to measure performance and find areas for improvement. In Six Sigma this formula is considered the “breakthrough equation.”

To break it down into its components, let’s define each part of the formula:

  • Y: the outcome or outcomes, result or results, that you want
  • X: the inputs, factors or whatever is necessary to get the outcome (there can be more than one possible x)
  • F: the function or process that will take the inputs and make them into the desired outcome

Simply put, the Y=f(x) equation calculates the dependent output of a process given different inputs.

Here’s an example of some possible values for the variables in the “breakthrough equation.” In a marketing context, the outcome (Y) could equal the number of product sales, where the function (f) could be an email opened from a newsletter campaign, and the input (x) could be the number of coupons within the email. As you increase the number of coupons (x), then the function of email opens (f) yields more product sales (Y).

How Y=f(x) and DMAIC Work Together

If outcomes are the results of drivers within a process, and DMAIC is a vehicle to identify the process and input variables that influence the process output measurements, then the two combined can offer a powerful partnership. So, how does each word that makes up the acronym DMAIC work towards Y=f(x)?

  • Define is how to understand the Y, or outcome, and how to measure it.
  • Measure helps to prioritize the potential x’s and measure the x’s and Y.
  • Analyze is how to test the relationship between x and Y and verify and/or quantify what the important x’s are.
  • Improve is implementing fixes to help Y and addresses x’s.
  • Control is monitoring x and Y over time.

Let’s look at this process in sharper detail.


To begin you must determine if the project has a specific and defined business or process problem. Sometimes this is clear, but other times it is not. If not, get a clear understanding of the Y, such as the process problem in measurable terms within the context of project goals.

Start with the project charter, which defines the problem the project has been created to resolve. You need to determine the baseline of the project metrics to make measurements in relation to the project’s progress. Getting data from the customer helps.

The business case will help place the Y of the project into the larger context of the organizational strategy, financial targets, customer satisfaction and other relevant goals. Once these goals are identified, the defined part of the process is complete.

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Business Case Template

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Now you’re going to map your project, which will help to identify the potential causes of the x’s in the formula. You’ll see which x’s are going to have the greatest influence on Y. The list of x’s will likely have to be whittled down to a manageable number through prioritization.

This reduction of potential x’s will not be determined by data, as the project has yet to start. Therefore, brainstorming is needed to figure out what is most important. Then the next step will be to identify any patterns to create a baseline capacity for the project process.


Verify and quantify the x-Y relationship. There are graphical tools and statistical tools that can be applied to complete this phase of the process. This will bring you to a greater understanding of which x’s contribute the most to the problem of process Y.


How can you improve upon what you’ve got at this point in the process? Again, by brainstorming and thinking up creative solutions. Then you need to cut back on all the possible solutions you came up with to focus on just those that are most necessary.

To do this, you’ll want to rate each individual solution with a criterion, such as how much the solution adds to improving Y and how much it addresses x. However, there are other criteria you can use, such as ease, cost, etc. You can also help reduce the number by figuring out the potential for these solutions to fail and therefore how to make sure those failures don’t occur.

By the end of this phase in the process, you should notice improvements to Y.


The work you’ve done and the improvements you added are not set in stone. You need to make sure that they are sustainable. Therefore, you want to create a process management chart to visualize the new process flow.

Now you can see the critical checkpoints in the process and set up actions in case of the process strays from the plan.

When you’re making a process management chart, the x’s are called leading indicators or checkpoints in the process, and the Y is the lagging indicator or final checkpoint at the end of a process cycle. Basically, you’re dealing with a project dashboard set up to measure the specific metrics you want to monitor to keep the project on track. This will show the performance of Y over time.

At the close of the project, you can evaluate all this data. This information not only will help guide your project successfully but will also provide a precedent that will help when gauging the merit of future project proposals and their best possible implementation. If you’re interested in learning more about adopting Six Sigma, read our blog about Six Sigma certification.

ProjectManager Is the Perfect Tool for Your Six Sigma Management

Knowing the Y=f(x) formula is a great first step, but you’ll need the right tool in order to put it to use. ProjectManager is an award-winning project management software that is loaded with helpful tools to keep your project on track. For example, our Gantt chart lets you lay out all of your project tasks over a timeline, allowing you to view your entire project from start to finish.

ProjectManager Gantt chart

Use the Kanban Board Throughout Each Project Phase

Or, if you prefer to work in a more agile environment, you can switch to the kanban board view. Every task is a card, and the different stages of production are represented as columns. As you move the work through different stages of production, you move the card from one column to the next to represent its status.

A screenshot of the Kanban board project view

Oversee Data With the Real-Time Dashboard

As you and your project team put work into the project, you can track the progress and status of your tasks on the real-time dashboard. It’s a helpful tool that gives you a bird’s eye view of your progress in the form of colorful charts and graphs that display your KPIs.

ProjectManager’s dashboard view, which shows six key metrics on a project

Using a formula like Y=f(x) is just one more tool in a project manager’s toolbox. ProjectManager is another tool and a toolbox all in one. That’s because our online project management software has nearly all the features you’ll ever need to manage your project, including a real-time dashboard to better monitor progress. Try it today with this free 30-day trial.