Learning Objectives

• Explain the purpose of descriptive statistics in Six Sigma.
• Distinguish between measures of central tendency and measures of variation.
• Calculate and interpret the mean, median, and mode.
• Understand the range, variance, and standard deviation.
• Explain the importance of variation in process improvement.
• Understand percentiles and quartiles at a Green Belt level.
• Select appropriate graphical tools for different types of data.
• Interpret histograms and identify distribution patterns.
• Use box plots to understand center, spread, and potential outliers.
• Understand Pareto charts and their application to defect and problem prioritization.
• Interpret run charts for understanding performance over time.
• Use descriptive statistics and graphical analysis together to understand process behavior.
• Recognize the limitations of descriptive statistics.

1. Introduction

Once appropriate data has been collected, the next step is to understand what the data is telling us.

Before performing advanced statistical analysis, a Green Belt should first examine the data using descriptive statistics and graphical tools.

Descriptive statistics summarize the characteristics of a dataset.

Graphical tools provide a visual representation of the data and can reveal patterns that may not be obvious from numerical summaries alone.

For example, two processes may have the same average cycle time but very different levels of variation.

Therefore, looking only at the average can lead to an incomplete understanding of process performance.

A fundamental Six Sigma principle is:

Understand both the center and the variation of the process.

2. Concept / Theory

2.1 Descriptive Statistics

Descriptive statistics are numerical methods used to summarize and describe observed data.

They help answer questions such as:

• What is the typical value?
• How much variation exists?
• What is the smallest observation?
• What is the largest observation?
• Where are most observations concentrated?
• Are there unusually high or low observations?

Common descriptive statistics include:

• Mean.
• Median.
• Mode.
• Range.
• Variance.
• Standard deviation.
• Quartiles.
• Percentiles.

2.2 Measures of Central Tendency

Measures of central tendency describe the center or typical location of a dataset.

The three commonly used measures are:

• Mean.
• Median.
• Mode.

2.3 Mean

The mean, commonly called the average, is calculated by adding all observations and dividing by the number of observations.

For observations:

x₁, x₂, x₃, …, xₙ

the sample mean is:

x̄ = Σx / n

where:

• x̄ = sample mean.
• Σx = sum of all observations.
• n = number of observations.

Example

Suppose five customer-service response times are:

4, 5, 6, 7, 8 minutes

Mean:

(4 + 5 + 6 + 7 + 8) / 5 = 6 minutes

The mean response time is therefore 6 minutes.

2.4 Median

The median is the middle value when the observations are arranged in order.

For an odd number of observations, there is one middle value.

Example:

3, 5, 6, 8, 10

Median = 6

For an even number of observations, the median is the average of the two middle values.

Example:

3, 5, 6, 8

Median:

(5 + 6) / 2 = 5.5

2.5 Mode

The mode is the value that occurs most frequently.

Example:

2, 3, 3, 4, 5, 3, 6

Mode = 3

A dataset may have:

• One mode.
• More than one mode.
• No repeated value, in which case there may be no mode.

2.6 Mean, Median, and Mode

These measures provide different information.

Measure	Meaning
Mean	Arithmetic average
Median	Middle value when ordered
Mode	Most frequently occurring value

The mean can be strongly affected by extreme observations.

The median is generally less affected by extreme observations.

Therefore, the Green Belt should consider the distribution of the data before deciding which measure provides the most useful description of the center.

3. Key Topics

3.1 Measures of Variation

Six Sigma is fundamentally concerned with variation.

Two processes can have similar averages but different levels of variation.

Common measures of variation include:

• Range.
• Variance.
• Standard deviation.
• Interquartile range.

3.2 Range

The range is the difference between the maximum and minimum observations.

Range = Maximum − Minimum

Example:

Data:

10, 12, 13, 15, 18

Range:

18 − 10 = 8

The range provides a simple indication of spread.

However, it depends only on the two extreme observations and therefore does not describe the entire distribution.

3.3 Variance

Variance measures the average squared deviation from the mean, with the exact calculation depending on whether the data represents a population or a sample.

For sample data:

s² = Σ(x − x̄)² / (n − 1)

where:

• s² = sample variance.
• x = individual observation.
• x̄ = sample mean.
• n = number of observations.

Because deviations are squared, variance is expressed in squared units.

For example, if the original measurement is in minutes, variance is expressed in minutes².

3.4 Standard Deviation

The standard deviation is the square root of the variance.

For sample data:

s = √s²

Standard deviation is particularly useful because it is expressed in the same units as the original data.

A smaller standard deviation generally indicates that observations are more closely grouped around the mean.

A larger standard deviation indicates greater spread.

3.5 Example of Standard Deviation

Consider the dataset:

4, 5, 6, 7, 8

The mean is:

6

The deviations from the mean are:

−2, −1, 0, +1, +2

The squared deviations are:

4, 1, 0, 1, 4

The sum of squared deviations is:

10

For sample variance:

10 / (5 − 1) = 2.5

Sample standard deviation:

√2.5 ≈ 1.58

Thus, the sample standard deviation is approximately:

1.58 minutes

3.6 Why Variation Matters

Suppose two processes both have an average cycle time of:

10 minutes

Process A has relatively little variation.

Process B has substantial variation.

Although the averages are identical, the customer experience may be very different.

Process B may produce:

• Very long delays.
• Unpredictable performance.
• More service failures.
• Greater difficulty meeting customer requirements.

Six Sigma therefore seeks not only to improve the average but also to understand and reduce inappropriate process variation.

3.7 Quartiles

Quartiles divide ordered data into sections.

The commonly used quartiles are:

• Q1 — First quartile.
• Q2 — Second quartile, equivalent to the median.
• Q3 — Third quartile.

Approximately:

• 25% of observations are at or below Q1.
• 50% are at or below Q2.
• 75% are at or below Q3.

The exact calculation method can vary depending on the statistical convention and software used.

3.8 Interquartile Range

The Interquartile Range (IQR) is:

IQR = Q3 − Q1

It represents the spread of the middle portion of the data.

Because it focuses on the middle observations, the IQR is less affected by extreme values than the full range.

3.9 Percentiles

A percentile describes the relative position of an observation within an ordered dataset.

For example, the 90th percentile represents a value at or below which approximately 90% of observations fall, subject to the calculation convention used.

Percentiles are particularly useful in service and process-performance analysis.

For example:

• Median response time.
• 90th percentile response time.
• 95th percentile delivery time.

Averages alone may hide occasional extreme delays.

4. Tools / Methodology

4.1 Graphical Analysis

Graphs allow the Green Belt to visualize:

• Center.
• Spread.
• Shape.
• Patterns.
• Trends.
• Outliers.
• Differences between groups.

Important graphical tools include:

• Histogram.
• Box Plot.
• Pareto Chart.
• Run Chart.
• Bar Chart.
• Scatter Plot.

The appropriate graph depends on the data and the question being investigated.

4.2 Histogram

A histogram displays the distribution of numerical data using intervals called bins.

A histogram can help reveal:

• Center.
• Spread.
• Shape.
• Skewness.
• Multiple peaks.
• Potential unusual observations.

For example, a cycle-time histogram may show that most transactions are completed within a narrow range while a smaller number take considerably longer.

4.3 Interpreting a Histogram

A Green Belt should examine:

Center

Where are observations concentrated?

Spread

How widely are observations distributed?

Shape

Is the distribution approximately symmetric, skewed, or otherwise unusual?

Peaks

Is there one dominant peak or multiple peaks?

Unusual observations

Are there observations that appear far from the main group?

A histogram should be interpreted together with process knowledge.

4.4 Box Plot

A box plot summarizes the distribution using key positional statistics.

It typically displays:

• Minimum or lower whisker.
• First quartile.
• Median.
• Third quartile.
• Maximum or upper whisker.
• Potential outliers according to the plotting convention.

The box represents the middle 50% of the data.

The median is displayed within the box.

Box plots are particularly useful for comparing distributions across groups.

For example, a Green Belt could compare cycle times across:

• Shift A.
• Shift B.
• Shift C.

4.5 Pareto Chart

A Pareto Chart displays categories in descending order of frequency or impact, often with a cumulative percentage line.

It is useful for prioritizing categories that contribute substantially to a problem.

For example, customer complaints may be classified as:

• Delivery delay.
• Billing error.
• Product damage.
• Incorrect documentation.
• Other.

A Pareto Chart can help the team identify which categories contribute the largest portion of the observed complaints.

The chart supports prioritization, but it does not by itself establish root cause.

4.6 Run Chart

A Run Chart displays observations in time sequence.

It helps the Green Belt identify:

• Trends.
• Shifts.
• Changes in performance.
• Patterns over time.
• Possible effects of process changes.

For example, daily defect rates can be plotted over several weeks.

Time order is important because randomly ordered data can hide process changes.

4.7 Bar Chart

A Bar Chart is commonly used to compare categorical data.

For example:

• Number of complaints by region.
• Defects by product type.
• Orders by department.

The categories are distinct groups rather than continuous measurement intervals.

4.8 Choosing the Graph

Data / Objective	Useful Graph
Distribution of numerical measurements	Histogram
Compare distributions between groups	Box Plot
Prioritize categorical causes or defect categories	Pareto Chart
Examine performance over time	Run Chart
Compare categorical counts	Bar Chart
Examine relationship between two numerical variables	Scatter Plot

The Green Belt should select the graph based on the question being asked.

4.9 Descriptive Statistics and Graphs Together

Numerical summaries and graphs complement each other.

For example, a dataset may have:

Mean = 20 minutes

but the histogram may reveal that the data contains two distinct groups.

Similarly:

Mean = 10 minutes

may appear acceptable, but a box plot could reveal several unusually long observations.

Therefore:

Do not rely on a single numerical statistic when the distribution of the data provides additional information.

5. Worked Example / Case Study

Case Study — Customer-Service Response Time

A customer-service department records the response times for 10 customer requests:

4, 5, 5, 6, 6, 7, 8, 8, 9, 22 minutes

5.1 Mean

Total:

80 minutes

Number of observations:

10

Mean:

80 / 10 = 8 minutes

The average response time is:

8 minutes

5.2 Median

There are 10 observations.

The two middle values are:

6 and 7

Median:

(6 + 7) / 2 = 6.5 minutes

Therefore:

Median = 6.5 minutes

5.3 Interpretation

The mean is:

8 minutes

The median is:

6.5 minutes

The unusually high value of 22 minutes increases the mean.

This demonstrates why a Green Belt should examine both numerical summaries and the underlying distribution.

5.4 Graphical Analysis

A histogram would show that most response times are concentrated between approximately 4 and 9 minutes, with one substantially higher observation.

A box plot would help visualize the center, spread, and potential outlier.

A run chart could be used if the observations were associated with specific dates or time order.

5.5 Six Sigma Interpretation

The Green Belt should not immediately conclude that the 22-minute observation is an error.

Possible explanations should be investigated.

The observation may represent:

• A genuine unusual event.
• A different type of customer request.
• A special process condition.
• A data-recording problem.
• A process failure.

The data should therefore be investigated using process knowledge before removing or modifying any observation.

6. Practical Application

Exercise 1 — Delivery Time

A logistics team records delivery times for a group of orders.

The Green Belt wants to understand:

• Typical delivery time.
• Variation.
• Distribution shape.
• Potential unusual deliveries.
• Performance over time.

Select appropriate descriptive statistics and graphical tools.

Possible tools include:

• Mean.
• Median.
• Standard deviation.
• Histogram.
• Box plot.
• Run chart.

The Green Belt should use the tools together rather than relying on only one measure.

Exercise 2 — Defect Categories

A manufacturing process produces several defect categories.

The team wants to identify which categories contribute the largest number of defects.

A Pareto Chart would be an appropriate graphical tool for prioritizing the categories.

The Green Belt should remember that the Pareto Chart identifies important categories but does not automatically establish why those defects occur.

Exercise 3 — Comparing Shifts

Three production shifts have different cycle-time observations.

The Green Belt wants to compare the distributions.

A box plot can provide a useful visual comparison of:

• Median.
• Spread.
• Quartiles.
• Potential outliers.

Additional statistical analysis may be required to determine whether observed differences are statistically significant.

6.4 Interpreting Outliers

An outlier is an observation that is unusually distant from the main pattern of the data according to an appropriate analytical or graphical criterion.

An outlier should not automatically be deleted.

The Green Belt should ask:

1. Is the value recorded correctly?
2. Was the measurement procedure followed?
3. Did a special event occur?
4. Does the observation represent a different process condition?
5. Is the observation part of the population being studied?

The decision to retain, investigate, or exclude an observation should be based on evidence and the purpose of the analysis.

7. Lesson Summary

Descriptive statistics provide numerical summaries of process data.

Measures of central tendency include:

• Mean.
• Median.
• Mode.

Measures of variation include:

• Range.
• Variance.
• Standard deviation.
• Interquartile range.

Quartiles and percentiles provide additional information about the position of observations within a distribution.

Graphical tools provide visual information about the data.

Important tools include:

• Histogram.
• Box Plot.
• Pareto Chart.
• Run Chart.
• Bar Chart.
• Scatter Plot.

The Green Belt should examine both center and variation.

Averages alone may not adequately describe process performance, particularly when data is skewed or contains unusual observations.

Outliers should be investigated rather than automatically removed.

Descriptive analysis provides an important foundation for the more advanced statistical analysis used later in Six Sigma.

8. Lesson Learnt / Conclusion

Descriptive statistics answer the basic question:

What does the data look like?

Before attempting advanced statistical analysis, the Green Belt should understand:

• Where the data is centered.
• How much it varies.
• What its distribution looks like.
• Whether there are unusual observations.
• Whether performance changes over time.
• Whether different groups appear to behave differently.

The combination of numerical statistics and graphical analysis provides a stronger understanding than either approach alone.

Green Belt Takeaway:

Do not look only at the average. Understand the center, variation, distribution, and patterns in the data.