Mood’s Median Test and Its Use in Six Sigma

In Six Sigma, selecting the right statistical tool is crucial. Some processes involve non-normal data or outliers. In such cases, mean-based tests like t-tests or ANOVA can mislead. That’s where Mood’s Median Test becomes useful.

Mood’s Median Test helps compare the medians of two or more groups. It’s a non-parametric test, meaning it does not assume any specific data distribution. This makes it ideal for Lean Six Sigma projects dealing with real-world, messy data.

Table of Contents

What Is Mood’s Median Test?

Mood’s Median Test checks whether multiple independent groups share the same median. Instead of looking at means, it focuses on how values fall above or below the overall median. The test uses a Chi-square statistic to determine significance.

It’s most useful when:

Your data is ordinal or skewed
You have outliers
You are comparing more than two groups

Why Use the Median Instead of the Mean?

The mean is sensitive to outliers. A few extreme values can skew it. The median, however, is more robust. It reflects the true central tendency of skewed or non-normal data.

Measure	Description	Sensitive to Outliers?
Mean	Average of all values	Yes
Median	Middle value in sorted list	No

When process data includes spikes or irregular distributions, the median provides better insights. That’s where Mood’s Median Test shines.

Mood’s Median Test vs. Other Tests

Understanding where Mood’s Median Test fits among other tests is key.

Test	Compares	Assumes Normality	Sensitive to Outliers?	Uses Median?
t-test	Means (2 groups)	Yes	Yes	No
ANOVA	Means (2+ groups)	Yes	Yes	No
Mann-Whitney U	Medians (2 groups)	No	Less	Yes
Kruskal-Wallis	Medians (2+ groups)	No	Less	Yes
Mood’s Median	Medians (2+ groups)	No	No	Yes

Mood’s Median is particularly useful when data contains extreme values or when sample sizes differ greatly.

When to Use Mood’s Median Test in Six Sigma

In Six Sigma, decisions are driven by data. Choosing the right statistical method ensures valid conclusions. Use Mood’s Median Test when:

You compare two or more independent samples.
Your data is ordinal or skewed.
You suspect outliers that could distort the mean.
You want to test for median differences, not means.

Here are some Six Sigma scenarios where this test is useful:

Scenario	Description
Process shift analysis	Compare median defect levels before and after a change
Supplier comparison	Evaluate median delivery times of different vendors
Equipment performance	Compare output medians from different machines

How Mood’s Median Test Works

Mood’s Median Test is simple in logic. Here’s the breakdown:

Combine all values from all groups.
Find the overall median.
For each group, classify values as either:
- Above median
- Below median
Create a contingency table.
Perform a Chi-square test.

Interpreting Results

Like other hypothesis tests, Mood’s Median Test involves:

Null Hypothesis (H₀): All groups have the same median.
Alternative Hypothesis (H₁): At least one group has a different median.

P-Value

If p < 0.05, reject the null hypothesis.
If p ≥ 0.05, do not reject the null hypothesis.

The p-value indicates if differences are statistically significant.

Step-by-Step Example with Calculation

Let’s say we want to compare cycle times from three operators.

Data

Operator A	Operator B	Operator C
10	12	15
11	13	16
10	12	17

Step 1: Pool All Data

Combined data = [10, 11, 10, 12, 13, 12, 15, 16, 17]
Sorted = [10, 10, 11, 12, 12, 13, 15, 16, 17]

Step 2: Find the Overall Median

There are 9 values. The middle (5th value) is 12, so:

Overall median = 12

Step 3: Classify Each Value

We count how many values from each group are:

Below the median (<12)
Above the median (>12)

Values equal to the median (12) are excluded from the analysis.

Group	Values	<12	>12
A	10, 11, 10	3	0
B	12, 13, 12	0	1
C	15, 16, 17	0	3

Step 4: Create Chi-Square Table

Build a 2×3 contingency table:

Group	Below Median	Above Median	Row Total
Operator A	3	0	3
Operator B	0	1	1
Operator C	0	3	3
Column Total	3	4	7

Step 5: Calculate Expected Counts

We calculate expected values for each cell using:

\[E_{ij} = {(Row \space Total_{i} \times Column \space Total_{j}) \over Grand \space Total}\]

	Below Median	Above Median
Operator A	(3×3)/7 = 1.29	(3×4)/7 = 1.71
Operator B	(1×3)/7 = 0.43	(1×4)/7 = 0.57
Operator C	(3×3)/7 = 1.29	(3×4)/7 = 1.71

Step 6: Compute Chi-Square Statistic

Use the formula:

\[χ^{2} = {\sum {(O \space – \space E)^{2} \over E}}\]

Where O = observed count, E = expected count.

Group	Cell	O	E	(O-E)²/E
A	Below	3	1.29	2.24
A	Above	0	1.71	1.71
B	Below	0	0.43	0.43
B	Above	1	0.57	0.37
C	Below	0	1.29	1.29
C	Above	3	1.71	0.97
Total	–	–	–	6.99

\[χ^{2} ≈ {6.99}\]

Step 7: Determine Significance

Now let’s calculate the degrees of freedom to determine the significance:

Degrees of Freedom (df) = (Rows – 1) × (Columns – 1)
df = (3 – 1) × (2 – 1) = 2

Using a Chi-square table:

Critical value at α = 0.05 and df = 2 ≈ 5.99
Our χ² = 6.99 > 5.99

Conclusion: Reject the null hypothesis.

At least one operator has a different median cycle time.

Six Sigma Application

Problem

A Six Sigma team wants to improve throughput. They suspect Operator C is slower. But their data includes outliers from machine warmups.

Why Use Mood’s Median Test?

Data is not normally distributed
Contains extreme values
Median is more reliable than mean

Action Plan

Conduct Mood’s Median Test (as shown).
Determine significance.
Investigate Operator C’s work methods.
Apply improvement tools (5S, Standard Work, SMED).

When to Choose Mood’s Median Over Other Tests

Let’s compare popular non-parametric tests:

Test	Data Type	Groups	Measures	Robust to Outliers?
Mann-Whitney U	Ordinal	2	Ranks	Somewhat
Kruskal-Wallis	Ordinal	2+	Ranks	Somewhat
Mood’s Median	Ordinal	2+	Medians	Yes

Mood’s Median Test excels when outliers are present and median is the statistic of interest.

Advantages of Mood’s Median Test

Why use this test over others?

Handles outliers well
No assumption of equal variance
Works with ordinal or skewed data
Easy to interpret results

It’s ideal in environments where data quality varies or normality is questionable.

Limitations of Mood’s Median Test

Despite its benefits, the test has limitations:

Limitation	Explanation
Less powerful	Compared to ANOVA or Kruskal-Wallis
Ignores magnitude	Only considers counts above/below median
Excludes ties	Values equal to the median are not used
Requires independence	Groups must be independent

Understanding these limitations ensures appropriate use.

Tips for Using Mood’s Median Test in Six Sigma

Use during Analyze phase of DMAIC.
Test assumptions first—ensure independent samples.
Visualize data using box plots before testing.
Combine with other methods like Pareto charts or control charts.
Document your rationale for choosing the test in your project report.

Tools That Support the Test

You can use several software options:

Tool	Supports Mood’s Median Test?
Minitab	Yes (via nonparametric analysis)
JMP	Yes (through contingency setup)
Python	Yes (manual calculation or SciPy)
R	Yes (`mood.test()` or custom code)
Excel	Yes (manual chi-square test)

Real-World Examples

Manufacturing

A factory measures thickness from three coating machines. The data has some extreme values due to nozzle clogging. Mood’s Median Test helps compare medians without distortion.

Healthcare

A hospital compares patient wait times across three departments. Due to a few extreme delays, the median is more reliable. Mood’s Median Test highlights which department deviates.

Logistics

A warehouse compares shipment times by region. Some long delays skew the data. Mood’s Median Test reveals regional differences without letting outliers dominate.

Mood’s Median vs. Kruskal-Wallis

Both tests are non-parametric and compare multiple groups. But they differ:

Feature	Mood’s Median	Kruskal-Wallis
Focus	Median	Distribution rank
Outlier Impact	Very low	Low
Data Consideration	Above/below median	All ranks used
Power	Lower	Higher
Use Case	Strong outliers or median focus	Ranked comparisons

Choose Mood’s Median when the median is your key interest or when outliers are severe.

Python Code Example

Here’s a Python script to replicate our example:

pythonCopyEditimport pandas as pd
import scipy.stats as stats

# Operator data
a = [10, 11, 10]
b = [12, 13, 12]
c = [15, 16, 17]

# Combine and find median
all_data = a + b + c
median = pd.Series(all_data).median()

# Count above/below median
def count_above_below(group, median):
    above = sum(x > median for x in group)
    below = sum(x < median for x in group)
    return [below, above]

data = [count_above_below(a, median),
        count_above_below(b, median),
        count_above_below(c, median)]

# Chi-square test
chi2, p, _, _ = stats.chi2_contingency(data)
print("Chi-square:", chi2)
print("P-value:", p)

Output:

makefileCopyEditChi-square: 6.99
P-value: 0.030

Since p < 0.05, we reject the null hypothesis.

Conclusion

Mood’s Median Test is a valuable tool for Six Sigma practitioners. It compares medians across groups without being thrown off by extreme values or non-normal data. It’s best used when:

Outliers are present
The data is ordinal
Median is a better measure than the mean

By walking through the test manually and with Python, you now understand how to use it confidently in real projects. It provides a clear, robust alternative when traditional tools fall short.