This document provides instruction on using the Mann-Whitney test to compare the medians of two independent samples. It discusses when to use the Mann-Whitney test, how to run it in Minitab, and provides an example comparing the medians of two columns of sample data labeled MetricC1 and MetricC2. The results of running the Mann-Whitney test on this example are interpreted to determine if the medians are statistically different between the two samples. The document encourages applying the test to factors identified in a previous lesson and discussing how the results could impact an organization.

1. Section & Lesson #:
Pre-Requisite Lessons:
Complex Tools + Clear Teaching = Powerful Results
Hypothesis Testing: Central Tendency –
Non-Normal (Compare 1:1)
Six Sigma-Analyze – Lesson 21
An extension on hypothesis testing, this lesson reviews the Mann-Whitney
test as a central tendency measurement for non-normal dist.
Six Sigma-Analyze #20 – Hypothesis Testing: Central Tendency –
Non-Normal (Compare 1:Standard)
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted by any means
(electronic, mechanical, photographic, photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.

2. Why do we need hypothesis testing?
o Remember, our project goal is to resolve a problem by first building a transfer function.
• We don’t want to just alleviate symptoms, we want to resolve the root cause.
 Remember Hannah? We don’t want to alleviate the arthritis pain in her leg, but heal the strep throat.
• If we don’t know what the root cause is, then we need to build a transfer function.
 By building a transfer function, we can know what changes (improvements) should fix the root cause.
o Remember, the Transfer Function is defined as Y = f(X).
• This is described as “output response Y
is a function of one or more input X’s”.
• It’s part of the IPO flow model where we
described the IPO flow model as one or
more inputs feeding into a process that
transforms it to create a new output.
o How does a transfer function fit with hypothesis testing?
• Hypothesis testing tells us which X’s (inputs) are independently influencing the Y (output).
 When we reject a null hypothesis, we’re building evidence proving which X’s are “guilty” of driving the Y.
 We’ll compile all the evidence in the Improve phase of DMAIC and begin to fix those root causes.
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
Y = f(X)
Input (X) > Process > Output (Y)

3. Review Hypothesis Testing: 4 Step Process
o Remember, the 4 high-level steps for hypothesis testing begin/end with being practical:
o As the heart of hypothesis testing, steps 2 & 3 can be drilled to the following 6 steps:
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
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Practical Statistical
Problem Problem
Solution Solution
 

Practical Problem
State the problem as a
practical Yes/No question.
 Statistical Problem
Convert the problem to an
analytical question identifying
the statistical tool/method.

Practical Solution
Interpret the analytical
answer in a practical way.
Statistical Solution
Interpret the results of the
hypothesis test with an
analytical answer.

1. Define the objective.
2. State the Null Hypothesis (H0) and Alternative Hypothesis (Ha).
3. Define the confidence (1-α) and power (beta or 1-β).
4. Collect the sample data.
5. Calculate the P-value.
6. Interpret the results: accept or reject the null hypothesis (H0).

4. Is the data type for both values discrete?
What are you measuring?
Is the data normal?
SpreadCentral Tendency
Compare 1:Standard
1 Proportion Test
Compare 1:1
2 Proportion Test
Compare 2+ Factors
Chi2 Test
Compare 1:Standard
1 Sample T Test
Compare 1:1
2 Sample T Test
or Paired T Test
Compare 2+ Factors
One-way ANOVA Test
Compare 1:Standard
1 Sample Wilcoxon
or 1 Sample Sign
Compare 1:1
Mann-Whitney Test
Compare 2+ Factors
Mood’s Median Test
or Kruskal-Wallis Test
Compare 1:Standard
1 Variance Test
Compare 1:1
2 Variance Test
Compare 2+ Factors
Test for Equal Variances
Yes No
Yes No
Proportions
Compare 1:1
Pearson Correlation
or Fitted Line Plot
Compare 2+ Factors
Multiple Regression or
General Linear Model
Is the data type for both values continuous?
Relationships
No Yes
Review Finding the Right Statistical Test
o What statistical test do I use for my hypothesis testing?
• The type of statistical test depends on the data to be tested, as described in the chart below:
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
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Lesson
Current

5. Confidence Intervals (CI) Redefined
o Remember, confidence intervals (CI) are an estimated upper/lower range of the mean.
• The CI narrows when you add more samples:
• Remember, statistics are intended to help you make inferences about a population.
 A mean of 57 with a 95% confidence interval of 55 and 59 implies that although the sample mean is only
57, you can be 95% confident that the population mean will fall somewhere between 55 and 59.
• The first step in hypothesis testing is to define the objective by asking “is there a difference…”
 Statistical tests generally compare the different CIs between factors to see if a difference exists.
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
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With fewer samples, the population
mean falls within a wide interval.
Add more samples and the population
mean falls in a more narrow interval.
With more samples, the more confident
(narrow) the mean interval becomes.
Factor A
Factor B
2.5 2.75 3.0 3.25 3.5
Large overlap indicates the population
mean may be the same for both
groups; therefore there is no difference
between them (high P-value).
N = 20
Factor A
Factor B
2.5 2.75 3.0 3.25 3.5
Small overlap indicates a smaller chance
the population mean may be the same for
both groups; therefore there is may be no
difference between them (small P-value).
N = 50
Factor A
Factor B
2.5 2.75 3.0 3.25 3.5
No overlap indicates the population
mean is different for both groups;
therefore there is a difference between
them (low or no P-value).
N = 100

6. Mann-Whitney Test: Introduction
o When should I use it?
• To compare median values from two random samples from two populations
having similar shape and variation.
• It’s also called the 2 Sample Rank test or 2 Sample Wilcoxon Rank Sum test.
o How do I find it in Minitab?
• Go to Stat > Nonparametrics > Mann-Whitney…
o What are the inputs for the test?
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
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Now
The columns containing the continuous data to test.
The desired confidence interval for the median.
Defines how the test will compare the median values.

7. Mann-Whitney Test: MetricC Example
o Example: MetricC sample values
• Background:
 Use the arbitrary values in the “MetricC” column of the Minitab Sample Data file.
 Copy/paste the values of MetricC and split them 50/50 into two new columns.
– In this example, the first 50 were labeled “MetricC1” and the remaining 50 as “MetricC2”.
• Practical Problem:
 Is the median of MetricC1 equal to the median of MetricC2?
• Statistical Problem:
 State the null (H0) and alternative (Ha) hypotheses:
– H0: η1 = η2 and Ha: η1 ≠ η2
 Define the confidence (1-α) and power (1-β):
– For confidence, we’ll accept the default of 95% (which means α = 5%) and power of 90% (which means β = 10%).
 Type the statistical problem into Minitab:
– In Minitab, go to Stat > Nonparametrics > Mann-Whitney…
– In the “First Sample” box, select MetricC1 from the list of columns and MetricC2 as the “Second Sample”.
– Ensure the “Confidence Level” is 95.0 and alternative is “not equal”.
• Statistical Solution:
 Refer to the session window results.
– P-value is the “Test is significant at” value.
– Since P-value is > 0.05 (α), then we fail to reject H0.
• Practical Solution:
 The sample is insufficient to prove that the median
of MetricC1 is different than the median of MetricC2.
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
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Now

8. Practical Application
o Refer to the critical metric (output Y) and at least 5 factors (input X’s) you identified in
a previous lesson for applying to this hypothesis testing.
• For any factor that is a continuous value, try applying the Mann-Whitney test.
 To do this, you’ll need to compare at least two different sets of that same factor (e.g., across multiple
periods of time, or by different locations, or by different groups, etc.).
 A non-parametric test like this is ideal for non-normal distributions, but you can still run it even if your
continuous value has a normal distribution.
 Other factors in your organization can be used for this exercise.
• Before running the Mann-Whitney Test, do the medians of the factors appear to be different?
• After running the Mann-Whitney Test, are the medians of the factors statistically different?
• If the answers to the above 2 questions are different, then how does that affect how you’d
typically measure and communicate that factor in the organization?
 For example, do the differences between the compared factors affect financial decisions (e.g., how people
are compensated), or process changes (e.g., how the process may be modified), or other critical actions?
 If so, then how should the results from this Mann-Whitney Test be used to influence your organization?
– Should they change how the factors are compared (e.g., across different times, locations, groups, etc.)?
– Should they change how the factor is measured?
– Should they change how they react when they compare the metric this way?
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
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