Statistical Solutions, LLC

General Instructions

You will need:

1. Total possible number of events (i.e. the whole population)
2. Time over which these events were measured
3. Number of events sampled out of the whole population
4. Number of defects - count two defects on one event twice
5. Number of possible defects per event (usually one)

Sample - A subset of the entire population, preferably randomly selected and fully representative of the whole set of all possible events.

Population - The entire set from which the sample has been taken. Theoretically the population is all entities that have, are, or will be produced, however we often take the population to be all entities from which the current sample has been drawn.

The period - Time period of when the sample was collected is used here to calculate the annualized defect rate and cost, both of which are useful in evaluating the customer and business experience of any defective process.

Defects - Any critical characteristic of the sampled entity that fails to meet customer expectations.

Opportunities - Critical characteristics of the sampled entity that could either meet or fail customer expectation. In many manufacturing situations there may be multiple opportunities per entity, whereas transactional situations often either meet the customer's requirements or they don't. When evaluating potential opportunities (and defects) it is important to work at the customer-significant functional level. Defect opportunities must be critical  to the customer, be independent of each other, and only increase numerically with increased complexity. For example, in a book, defects could be misspelled words, grammatically incorrect sentences, missing text or pages and so on, however readers may regard the 'sentence' as the functional entity of interest, and a defect is a spelling or grammatical error in each sentence.

Cost - Evaluate the sum total of all costs from the current defect rate, and divide out to obtain a unit cost per defect. This allows for easy calculation of the improvement, and enforces the Six Sigma concept of customer defect reduction as the end goal!

Shift - A complex subject in Six Sigma, and traditionally includes a value of 1.5 sigma short to long-term shift in the calculations. The shift is the degradation experienced between the short-term best process capability and the long-term process performance when all possible process time-related causes of variation have been added in. From work undertaken at Motorola an empirical figure of 1.5 sigma is now taken as 'standard' for the worst case shift, although some today now advocate that this figure could be lower.

• In theory:  Process Sigma (short-term) = 1.5 + Process Sigma (long-term)

Capability sigma (short-term, within) - This is often taken as the best capability figure, and by convention the Process Sigma value of every process is stated as the best short-term value. The expected "within" performance values quantitatively represent the potential process performance if the process did not have the shifts and drifts between subgroups. The expected values are calculated using the within-subgroup variation.

Performance sigma (long-term, overall) - This is often taken as the worst capability figure, and by convention the short-term Process Sigma is related to the DPMO at the long-term state, typically shifted by 1.5 sigma from the short-term state. The expected "overall" performance values quantitatively represent the actual process performance. The expected values are calculated using the overall sample variance.

If your data is short-term, then the DPMO figure can be related back to (short-term) process sigma without any shift, however the customer will experience the DPMO defect rate for the long-term process state.

If your data is long-term, then the DPMO figure must have the 1.5 sigma shift added in to return to the short-term Process Sigma value. Transactional process data is typically long-term.

How this calculator works

The sample and population sizes are used to extrapolate the total defects in the population. The population and defect opportunity are used to calculate the total number of opportunities. DPMO is then calculated using

DPMO = (total defects in the sample * 1,000,000) / total opportunities in the sample

The DPMO is then converted to the number of standard deviations for the equivalent right-hand tail fraction of the normal distribution. This number of standard deviations is the base process sigma value.

The base process sigma is then adjusted to obtain the long-term process sigma value; if the data is long-term then no adjustment is needed, however for short-term data the assumed 'short' long-term shift is added in. This is 1.5 sigma by default, but can be adjusted to any value of choice. Intermediate states between short and long-term apply this shift in proportion.

The (official) short-term process sigma value is then calculated by adding back a standard 1.5 sigma shift.

- Where the process shift is not known the default 1.5 shift should be used -

Since 6 sigma = 3.4 DPMO as standard, the 1.5 correction factor will always return any long-term DPMO back to an 'official' short-term Sigma Value

Sigma value - The following table shows the conversion from sigma value to Cpk

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