Six Sigma Chapter 12: Normal Distribution

BUSSCM 1780 Kimpel 2025

Standard Normal Distribution

μ = 0, std dev 𝝈 = 1, symmetrical, area = 1 

probability

Normal-shaped variation allows us to predict the ___________ (likelihood) of future events

Vertical axis

measures the probability density and peaks at μ = 0

Horizontal axis

is scaled in standard deviations (𝝈) 

Convex, concave

𝝈 = 1 is the distance from μ where the curve changes from _______ to ________

dot plot, histogram

If you have hundreds of data points….you can create a _______, __________ and visually inspect for symmetry and bell shape 

normal probability plot

If you do NOT have hundreds of data points….you can create a __________________.

Traditional Yield (Y) 

• The proportion of good vs. acceptable items (conforming to specification) you get out of a process compared to the number of raw items

• Typically employed on the LAST final inspection step of a process

• This is a misleading measure because it hides the impact of inspection and rework!

First Time Yield (FTY)

• First time yield accounts for rework in a process

• Inspection is the doorway to the hidden factory

• This measure shows the LIKELIHOOD of an item passing through one process step successfully the first time!

Rolled Throughput Yield (RTY) 

• Most processes have more than one step

• How do you calculate the overall yield for a string of process steps? 

• Multiple the first time yields for each step together

• This measure is the COMBINED overall yield of and end-to-end process and tells you the likelihood of an item passing through all process steps successfully the first time   

Defects 

• DPU = number of defects observed / number of units inspected

• DPO = number of defects / (number of opportunities for error per unit)*(number of units) 

• DPMO = DPO * 1,000,000

Defects per Unit (DPU) 

• A basic assessment of a characteristic or process capability is to measure the total number of defects that occur over a given number of units 

• DPU provides a measure of the AVERAGE number of defects on a single unit

Defects per Opportunity (DPO) 

• A DPU of 0.478 for an automobile is viewed differently than the same rate on a bicycle because of different levels of complexity

• To “level the playing field” create a defects per opportunity rate

• DPO allows you to fairly compare the defect rates of things with very different levels of complexity

Defects per Million Opportunities (DPMO) 

• When the number of opportunities on a unit grows LARGE and the number of observed defects grows SMALL, DPO becomes very SMALL

• Additionally, you may want to estimate the number defects after running the process or observing the characteristic after a long time

• A simple way to solve both problem is to measure the defects over a large number of opportunities 

Sigma (Z) score 

• Z = |SL - x-bar| / 𝝈

• A Six Sigma level of quality is defined as 3.4 DPMO 

• A sigma score tell you HOW MANY standard deviations can fit between the mena and specification limit of any process or specification

• Statisticians call this Sigma (Z) score or NORMAL SCORE

Low

• A ______ sigma (Z) score means a significant portion of the tail extends beyond the SL 

High

• A _________sigma (Z) score when the variation distribution is far from SL 

Changes to Sigma (Z) Scores

• A sigma (Z) score can change in one of three ways: 

1. The central tendency can shift

2. The width of the distribution, as defined by 𝝈, can change

3. The location of the specification limit (SL) can change 

Short-term Sigma (Z) score

the best performance you can expect from your currently configured process

• Short-term entitlement variation is an idealistic measure of capability 

Long-term Sigma (Z) score

• Six Sigma practitioners measure short-term variability of a process or characteristics and calculate the sigma (Z) score

• Then they translate the short-term sigma (Z) score to long-term defect rate performance using a 1.5 short-term standard deviation shift

Capability ratios & indices

a set of measures that directly compare the voice of the customer (VOC) with the voice of the process (VOP)

• Six Sigma practitioners have defined the “effective limits” of any process as being THREE standard deviations (𝝈) from the mean which implies 99.7%

Short-Term Capability (Cp)

• Compares the width of a two-sided specification to the effective process limits

• Notes: 

  • A capable process must have a Cp of at least 1.0

  • Six Sigma quality requires a Cp = 2.0

  • Does not look at how well the process is centered in the specification range

Short-Term Capability Index (Cpk)

• Accounts for off-center processes 

• Notes: 

  • When the upper and lower indices are the same then the process is centered

  • A capable process must have a Cpk of at least 1.0 

  • Six Sigma quality requires a Cpk = 2.0