The Transformation Function: Y = f(X)

algebraSix Sigma is focused on improving the performance of processes.  In simple terms, processes have inputs, activities, and outputs.  The process activities transform the inputs that come from suppliers into a product or service that is delivered to a customer.

Consider baking a cake.  The inputs are ingredients such as eggs, flour, milk, butter and frosting.  The steps or activities in the process are measuring the ingredients, mixing the ingredients, baking the cake, and frosting the cake.  The output is a finished cake ready for delivery to the customer.

In more technical terms, we represent the output variable of the process with the letter Y.  A process may have one or more output variables.  Each output variable has one or more input variables that are represented by the letter X.  The process transforms the X input variables into a Y output variable.   The equation is written as Y = f(X), and we understand this to mean that the output of a process is a function of the inputs.  If we want to get the right output, we need the right inputs.

These variables (inputs and outputs) may be either discrete or continuous in nature.  In the cake example, a discrete output variable would be the color of the finished product.  For a yellow cake, the requirement might be for the cake to be baked to a golden brown color, and the discrete variable is yes it is golden brown or not it is not.  The continuous input variables would be the baking time in minutes and the baking temperature in degrees Fahrenheit.  Please refer to article 17 in this series for a more detailed explanation of discrete and continuous data.

The better that we understand the relationship between inputs and outputs, the better job we can do of meeting customer requirements.  In order to improve the output, we don’t work on the output. We work on understanding the relationship between the inputs and the output, and how to better manipulate the input variables to get the desired output.

In the cake example, we could experiment with different amounts of baking time and different levels of temperature in order to find a combination of time and temperature that results in the desired golden brown output.  Designed experiments are an organized and structured way of defining the impact that input variables have on the output.

Another output variable for a cake might be the moisture level.  The moisture level could be described in discrete terms as dry, moist, or soggy.  The input variables that influence the moisture level are continuous variables:  the amount of milk in ounces and the amount of butter in ounces.  Again, we could experiment with different amounts of milk and butter to get the desired moisture level in the cake.  Not dry, not soggy, but just the right texture that best satisfies the customer.

The route to excellent output results is to completely understand the transformation process, figure out the best way to operate the process to get the desired results, and then enforce the discipline to operate the process in that manner every time.

Your comments or questions about this article are welcome, as are suggestions for future articles.  Feel free to contact me by email at

About the author:  Mr. Roger C. Ellis is an industrial engineer by training and profession.  He is a Six Sigma Master Black Belt with over 45 years of business experience in a wide range of fields.  Mr. Ellis develops and instructs Six Sigma professional certification courses for Key Performance LLC.   For a more detailed biography, please refer to