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X is the symbol used to denote the independent input variables to a process. These inputs could be material, manpower and/or machine and its settings etc. A combination of these Xs determine the output of the process and is usually represented by f(X) and read as 'function of x'. 




Y is the symbol used to denote the dependent output variable from a process. This output is usually a metric which is monitored for consistent performance (e.g. throughput, cycle time, quality, productivity etc.). The output is related to the inputs by the equation Y = f(X) and read as 'Y is a function of X'. 


An application-oriented question on the topic along with responses can be seen below. The best answer was provided by Prashanth Datta on 25th January 2019. 


Applause for all the respondents- Prashanth Datta, Hari Shyam Gupta, Santosh Chaurasiya, Vastupal Vashisth, Mathi, Shashank Parihar, Purbi Adak. 



Q. 129  Y=f(X) forms a very important part of a Six Sigma or DMAIC project. What are the various tools that can be used to list all possible Xs for an output Y.


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As the topic suggests, Y=f(X) forms a very important part of DMAIC Project. Before delving into the actual question on the tools required to identify Xs, a quick recap on "What Y=f(X) represents?" can help set the context to answer the question better.


Y=f(X) represents a more Data Driven Approach to summarize our business problem statement. The Y here is the Output and X is the Input which drives the Y.


There can be more than One X which drives the Output Y. In other words, we can represent the same as -- Y=f(X1,X2,X3....,Xn)


Apart from being identified as Output, Y is also referred to as Dependent Variable, Effect, Symptom, Monitor or Response.

Likewise, apart from being identified as Input, X is also referred to as Independent Variable, Cause, Problem, Control and Factor.


Simply putting together,

  • During the Define Phase, based on Voice of Customer [VOC] OR Voice of Business [VOB] OR Cost of Poor Quality [COPQ], we will identify the Critical To Quality [CTQ] metric, Y, which needs improvement.
  • During the Measure Phase we finalize the Y which we need to improve for the Business problem and also work on Standards and Measurement System for Y and have a Baseline Performance OR Current/As-IS performance documented against which an improvement will be tracked.
  • Improvement here refers to either
    • Shift in Mean (upwards of downwards based on the KPI tracked e.g. Reduction in Average handling time implies shifting the Mean down from Current Mean & Improve Occupancy % refers to shifting the Mean up from Current Mean)
    • Reduce the Variation in process i.e. Within the control limits

From the above prelude, we now have understood that the Output Y or Effect can have multiple Inputs X or Causes to drive.


The 3rd Phase of DMAIC which is Analyze will help us identify all Xs impacting Y and further helps narrow down Critical Xs which when controlled will help bring the desired improvement in the Output Y.


A simple example to illustrate, 

  • Y [Over Weight] = f((X1 [Calories of food consumed], X2 [Sleep  duration], X3 [Duration of Inactivity], X4 [Genetic Variations]))

The tools that help list all possible Xs can be categorized under 3 sections

1.Qualitative Analysis -- 

  • Brainstorming and Structured Brainstorming
  • Affinity Diagram >> Fishbone Diagram

2. Process Map Based Analysis -- 

  • Value Add and Non-Value Add Analysis
  • Detailed Process Mapping
  • Failure Mode and Effects Analysis [FMEA] by using Risk Prioritization Number concept.

3. Graphical Analysis

  • Historical input data trends can be analyzed by using Box Plots, Histogram, Scatter Diagram

For Identifying Critical Xs, 
1. We use Pareto along with other Graphical tools like Box Plots, Histogram and Scatter Diagram under Graphical Analysis
2. Value Add and Non-Value Data, Detailed Process Mapping and FMEA under Process Map Based Analysis
3. A more Statistical based approach using Hypothesis Testing

The purpose of identifying Critical Controllable Xs having an impact on Y is to help fix the problem at source itself

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Y (Output) is a Function of X (Input) and hence always focus on Xs while DMAIC project. Listing all possible Xs is very much important and it should not be basis one individual's thought/assumption.


Under DMAIC methodoly, there are certain tools, which we use during the Analysis phase to list all possible Xs


1. Brainstorming

2. 5 WHY's Analysis

3. Affinity Diagram (It is used, when the ideas/suggestions are too many, to diffuse the confusion after a brainstorming session by organizing the multiple ideas generated through the sessions. This categorizes a large amount of Ideas/Suggestions into logical groupings.

4. Cause & Effect Diagram



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DMAIC approach six sigma methodology provides a systematic approach to solve problems. This approach consist of proven tools and methods which applied correctly helps us to determine a desired outcome. during this approach the concept of Y= F(x) plays a very important role and it has different role phase by phase.

The term Y= F(x) , means Y is function of x illustrates that output Y is result of drivers x in process. let us see phase by phase importance and role of this concept Y = F(x)

Define Phase: in define phase we focus on Y, the outcome or our project Y and how to measure it . Various tolls are used in define phase to get a clear understanding of the project Y. main outcome in this phase is Project Charter which defines exactly what the business problem is . To get Y in this phase we use following  tools as our input which clearly relates the project Y. Some of the tools are:

1. VOC


3. CTQ's Tree

 4. AS-IS Process mapping

5. Kano Model

6. Stake Holder Analysis


Measure Phase:   in this phase we also focus on Y and main outcome is Current or Baseline performance. We measure X's and Y and prioritize potential X's. There are various tools which are used to prioritize potential X's and establish a relationship between X's and Y's. Some of them are given below:

  1. Fishbone Analysis
  2. Correlation
  3. data collection plan
  4. Gage R 7 R study
  5. FMEA
  6. Histogram
  7. MSA
  8. Probability Distribution
  9. Pareto Analysis
  10. Process Sigma
  11. Control Charts
  12. Regression Analysis

ANALYSE Phase:  This tool is simple and is used to test relationship between X's and Y's and to validate root cause to identify Critical X's. By using some tools we get to know the critical X's which are mostly impacting Y's. an some of the tools used in this phase are below;

  1. hypothesis Testing
  2. Regression
  3. Process capability Analysis
  4. Scatter Diagram
  5. Reliability modelling
  6. Design of experiments

IMPROVE Phase:  in this phase we focus on addressing important X's to improve our Y. Some of tools are given below which are used in this phase:

  1. Design of experiment
  2. PERT technique
  3. Fault Tree Analysis
  4. Design of experiment
  5. MSA
  6. Risk Analysis
  7. Taguchi Design
  8. Value stream mapping                                                                                                                                                                                                                                                                                Control Phase in this phase we monitor Y and important X's over a period of time. some of the tools are given below which are used in this phase to monitor : 

           1. SPC 

           2. MSA

           3. SOP's

           4. Run Charts

           5. Control Plan









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Y = f(x) formula to describe the relationships between the CTQs and the process variables that impact CTQs.

The following tools can be employed during the various phases of six sigma process



activities / Tools



Understand the process

Process Map- as is



Run Charts

Process inputs X variables  and outputs Y Variables


Identify the sources of variation

Process Map


Fish bone diagram


SPC on X’s and Y’s

MSA on X’s

Why Why Analysis

Identified sources of variation that impact the output(y)


Validate causes

Hypotheses testing

Multi -vari analysis

Correlation and regression analysis

Chi-square test

1-t test

2-t test

Paired T test

Potential X’s critical to the process output (Y)


Determine the variable relationship y=f(x)


Regression analysis



Scatter plots

Relationships between  input x and output y


Establish process operating specifications

New process capability

Cost / benefit analysis

Optimum process tolerances

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Prashanth Dutta's answer is the best answer as it is the most relevant response to the question asking for all tools that can be used to list the 'possible causes' or inputs.


Please keep in mind to read the question carefully and digress as little as possible. Other background details are also not required as your phrase your answer.

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