Many of the analysis required in the Green Belt and Black Belt Body of knowledge assumes a normal distribution. There’s a good reason for this; normal distributions have a nice, symmetrical shape that makes working with them much easier (like in an ANOVA, for example!) But how do you know if your process is following a normal distribution? Use these tests to find out.

Visual Normality Tests / Graphical Analysis

• Visual examination. Make a histogram or another bar graph. Only reject normality in the presence of “gross non-normality”–extreme departures from symmetry.
• If you see a bell curve, a distribution is approaching normal.
• Tall, thin curve = smaller standard deviation.
• Fatter, lower curve = larger standard deviation.
• You can test using a Normal Probability Plot. The probability plot transforms the data into a normal distribution and plots it as a scatter diagram.
  • Normal data will follow the trend line.
  • Non-normal data will have more points farther from the trend line.
• The peak of the normal curve is an indication of the average, which is the center of process variation. An average of a group of numbers is an indication of the central tendency.

Quantifiable Normality Tests

• Use the Chi Square Goodness of Fit test.
• Follow the Anderson-Darling Normality test or Critical Value Method.
  • The output includes the Anderson-Darling statistic, A-squared, and both a p-value and critical values for A-squared.
  • “Null hypothesis” is that the data is normal. The “alternative hypothesis” is that the data is non-normal. Reject the Null hypothesis (i.e., accept the alternative) when p<=alpha or A-squared>critical value.
  • if p > alpha then the data is normal.
  • if A-squared < Critical Value, then the data is normal
• Mean is the inverse of the Poisson distribution.
• The smaller the standard deviation, the tighter the grouping of data around the mean.

Worked Example

Say you have 30 cycle-time measurements and want to know if they’re normal enough to run an ANOVA. Plot them on a normal probability plot: if the points hug the straight trend line with only minor scatter at the very ends, treat the data as normal and proceed. If a chunk of points curves noticeably away from the line, especially in the middle of the range rather than just the tails, that’s a sign of real non-normality, not just sampling noise. Follow up with the Anderson-Darling test for a formal yes/no answer: if the p-value comes back above your alpha (typically 0.05), you don’t have enough evidence to call the data non-normal, so it’s reasonable to treat it as normal for the analysis you’re about to run.

A Batting Average Example

Batting averages across a full league tend to cluster into a bell shape: a handful of exceptional hitters at one extreme, a handful of struggling ones at the other, and most players somewhere in the thick middle. That’s the visual, intuitive version of what a histogram or probability plot is checking for: are the extreme values genuinely rare compared to the common middle, the way they should be if the process generating them is well-behaved?

What If It’s Not Normal?

Confirming non-normality isn’t the end of the analysis, it’s a fork in the road. Options include a Box-Cox transformation to make the data approximately normal, switching to a non-parametric test that doesn’t assume normality in the first place, or checking whether the non-normality itself is meaningful (a bimodal distribution, for instance, often means you’re actually looking at two different processes mixed together, which is itself a useful finding).

Exam Tip

Watch the direction of the Anderson-Darling decision rule carefully, it trips people up: a LOW p-value (below alpha) means you reject the assumption of normality, not confirm it. “Low p, not normal” is the version worth memorizing cold.

Frequently Asked Questions

How many data points do I need to test for normality? There’s no hard minimum, but very small samples (under 20 or so) make any normality test unreliable, since there isn’t enough data to distinguish real non-normality from ordinary sampling noise. With very small samples, visual inspection combined with practical/engineering judgment often matters more than the formal test’s p-value.

Does “normal enough” mean perfectly normal? No real-world data is perfectly normal. The practical question is whether the departure from normality is large enough to distort whatever analysis you’re about to run (an ANOVA, a capability study), not whether the data matches a textbook bell curve exactly.

Can a process be “in control” but not normal? Yes, these are separate questions. A control chart checks for stability over time (no unusual signals), while normality is about the shape of the distribution at a given point. A stable process can still have a skewed or otherwise non-normal distribution, which is exactly why Six Sigma treats them as distinct checks, not one combined test.

What’s the most common real-world cause of non-normal data? A natural lower or upper bound is one of the most frequent culprits, cycle times, defect counts, and wait times all tend to pile up near zero and can’t go negative, which produces a right-skewed shape rather than a symmetric bell curve. Mixing two different processes together (two machines, two shifts, two suppliers) into one dataset is another common cause, often producing a bimodal (two-humped) shape that no single transformation will fully fix, since the real issue is that you’re looking at two populations at once, not one.

What’s the most common real-world cause of non-normal data? A natural lower or upper bound is one of the most frequent culprits, cycle times, defect counts, and wait times all tend to pile up near zero and can’t go negative, which produces a right-skewed shape rather than a symmetric bell curve. Mixing two different processes together (two machines, two shifts, two suppliers) into one dataset is another common cause, often producing a bimodal (two-humped) shape that no single transformation will fully fix, since the real issue is that you’re looking at two populations at once, not one.

What’s the most common real-world cause of non-normal data? A natural lower or upper bound is one of the most frequent culprits, cycle times, defect counts, and wait times all tend to pile up near zero and can’t go negative, which produces a right-skewed shape rather than a symmetric bell curve. Mixing two different processes together (two machines, two shifts, two suppliers) into one dataset is another common cause, often producing a bimodal (two-humped) shape that no single transformation will fully fix, since the real issue is that you’re looking at two populations at once, not one.