Measure Phase

Measurement is perhaps one of the most important aspects of Six Sigma since the overall idea is to reduce variation and eliminate defects.

First, let's review what a statistic is since there are definitely instances where statistical measures are used in Six Sigma. A statistic is a fact or piece of data from a study of a large quantity of data, often numerical.  Often statistics are based on a sample (preferably random) from a population (the entire group).  There are two basic types of statistics: qualitative and quantitative.

• Quantitative statistics are information about quantities and is therefore numbers.
• Qualitative statistics are descriptive and relates to phenomenon which can be observed but not measured

Quantitative Statistics Introduction

Throughout statistics we denote:

• N : the number of observations in the population, that is the population size
• n : the number of observations in a sample, that is, the sample size
• x_i: the i- th observation

Common types of data

There are four common types of data you will see frequently in measurement:

• Nominal / Qualitative / Categorical (e.g. names, colors, types, varieties, etc.)
• Ordinal (ordered:  e.g. rankings, places, order)
• Discrete (Integer values: e.g. number of students in class)
• Quantitative (value has meaning, amounts, usually continuous)

Measures of Central Tendency

Measures of central tendency (center, location) measures the middle point of a distribution or data; these include mean and median.

Mean. The arithmetic average of a group of values. The mean is a measure of the center in the sense that it “balances” the deviations from the mean.   It can be thought of as the "center of gravity" of a distribution.

Mean = average [sum(x) / n]

x’s are the data values

n = total number of x’s

Population mean: the mean of all the values in a population.

Sample mean: the mean of sample values collected. If the sample is random and the sample size is large then the same mean is a good estimate of the population mean.

Note that the population mean and sample mean are arithmetic means or average, and computed in the same manner.

Median. The middle value of data when ordered from smallest to largest. The median is a measure of the center in the sense that it “balances” the number of observations on both sides of the median.

(3,6,8,9,12) median = 8

(3,5,6,8,9,12) median = 7 (avg. of middle two numbers)

Mode.  The most frequent number.

(3,5,6,6,7,9,12) = 6

Measure of Dispersion

Measure of dispersion (variability, spread): measures the extent to which the observations are scattered; these include standard deviation and range.

Spread.  In addition to knowing where the center is for a given distribution, we often want to know how "spread out" the distribution is -- this gives us a measure of the variability of values taken from this distribution. The below graphic shows the general shape of three symmetric unimodal distributions with identical measures of center, but very different amounts of "spread".

(Emory Oxford College Mathematics and Computer Science, n.d.)

Range. Largest observation minus smallest observation (max minus min)

Population variance. The (population) variance is the average of the squared deviations from the population mean.

Sample variance. The (sample) variance is the “average” of the squared deviations from the (sample) mean.

Standard deviation. The standard deviation is a measure of variability in a process. Defined as the root mean square (RMS) deviation from average it indicates how much a process can be expected to vary from the average. The standard deviation is normally fixed for a process that is under statistical control and can only be affected by a process change that affects the variability in a process.

Coefficient of variation.  The coefficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. It shows the extent of variability in relation to the mean of the population.

Outliers. Extreme observations not conforming to the rest of the observation. As a rule of thumb observations that are three standard deviations above or below the mean are considered as outliers.

Parameter

Parameter (in statistics): a numerical descriptive measure computed from the population measurements (census data where everyone is surveyed). Basically it is a value that tells you something about the population.  A parameter does not change since everyone (or everything) was surveyed to find the parameter. 

Process Performance Metrics

• Defects per unit (DPU)
• Parts per million (PPM)
• Defects per million opportunities (DPMO)
• First pass yield (FPY)
• Rolled throughput yield (RTY)
• Cost of poor quality (COPQ)

DPU = Defects / Units

parts per million (PPM or ppm): PPM = DPU x 1,000,000      

The quoted defect rate of a 6σ process is 3.4 parts per million (PPM or ppm), or 3.4 defects per million opportunities

Defects per million opportunities (DPMO)

DPMO = (Defects/Total Opportunities) x 1,000,000 OR DPMO = DPO x 10 6

Benefits

• Defines the current baseline performance
• Provides a common measurement to use as a baseline in assessing the ability of a process to produce defect-free products
• Provides data to help prioritize future quality projects

Yield is defined as the percentage of products that successfully complete the production process

First pass yield (FPY) 

First pass yield (FPY) is the percentage of units that successfully complete a process with no rework. This statistic only considers the initial input data and the final output. FPY is typically used to express the process yield for an entire operation. In the case of processes with many steps, it is calculated by multiplying the yield of each step in a process.

Example: 

You start with 100 units.  There are four sub-processes it they must get through

Process one 95% make it to the next step with no rework so throughput is 95

Process two 95% make it to the next step with no rework so throughput is 90 (result is 90.25 but one cannot have .25 of a unit)

Process three 90% make it to the next step with no rework so throughput is 81

Process four 99% make it to the next step with no rework so throughput is 80 (result is 80.19 but one cannot have .19 of a unit)

Total first pass yield, therefore, is 80 of 100 or 80%

This can also be calculated as 95% x 95% x 90% x 99% = 80.412% so 80%

FPY can also be used to calculate across a number of sub-processes, simply by multiplying the yield of each step

First pass yield is only used for an individual sub-process. Multiplying the set of processes would give you Rolling throughput yield (RTY). 

Rolled throughput yield

Rolled throughput yield (RTY or Yrt) is the probability that a single unit can pass through a series of process steps free of defects. RTY is calculated as the overall quality level after several steps in a process have been completed by multiplying the throughput yield of each step within the process.

Calculating Rolled Throughput Yield:

Apply the formula below. Note that rework is just that: rework. There is no distinction made regarding the complexity or number of times a product or service is reworked. RTY signals when a process or sub-process must be improved.

yield = [(N units entering process) – (# of reworks + # in scrap)] / N units entering process

Example:

A unit must go through three processes:

Of 4,000 units started, 200 defective units were scrapped and 600 additional units had one defect each that were successfully reworked.

yield of step 1= (4,000 – (200 scrap +  600 rework))/4,000

= (4,000-800)/4,000

= 3,200/4,000

= 0.80

At step two, we have 3,800 units that came in (since we have both the good units and the reworked ones). 10 are scrapped and 40 are reworked so

yield step 2 = (3,800 - (10 + 40) / 3,800) = .9868

At step three, we have 3,790 units that came in (since we have both the good units and the reworked ones).  24 are scrapped and 19 are reworked

yield step 3 = (3,790 - (24 + 19) / 3,790) = .9886

RTY = .8 x .9868 x .9886 = .78 or 78% yield

The probability of this 3 step process producing a defect free product is 78%

Example 2: