# Variation, the Normal Distribution, DPMO and Sigma Level - Six Sigma

Variation and the Normal Distribution

Variation reduction is the key mechanism for Six Sigma to deliver business benefit. By focusing on product, service or process variation (depending on circumstances) projects create consistency of performance and improved conformance to customer
requirements.

Six Sigma focuses on the concept of defects per million opportunities (DPMO). It uses the standard normal distribution as its measurement system. From the standard normal distribution, the mean is µ and the standard deviation is denoted by σ. From figure, 68.2% of the population lies within ±1.0σ of the mean, 95.45% of the population lies within ±2.0σ of the mean and 99.73% of the population lies within ±3.0σ of the mean

Standard normal distribution

When addressing variation it is important to remember the effects of special and common cause variation. The normal distribution and DPMO cannot apply if special causes are dominant within the process.

Defects per Million Opportunities

Six Sigma uses the DPMO level of a process to generate a Sigma level for the process. The idea of a Sigma level is that it compares the variation in process performance to the acceptable levels set by the customer, The higher the Sigma level the better; a Six Sigma performance indicates DPMO.

A one - sided normal distribution

So for example, from figure , when σ = 3 there are 1350 DPMO ( (1 - 0.998) * 1000000).

According to the standard normal distribution a process a six sigma performance would actually produce a DPMO of 0.002, but Sigma levels are calculated using an inbuilt 1.5 σ shit for the process average. This is effectively an allowance for the natural propensity of processes to dirt and, although debate still rages as to the validity of the exact assumption this is the commonly used approach.

The basic idea is to create a process quality metric which allows comparison of any type of process; Goh (2010) described this as one of the six triumphs of Six Sigma. The DPMO are calculated first and then translated into a Sigma value via a conversion table (see table below).

Table Process sigma table

The precision of the numbers in this table is something of an illusion as they are based on a perfect normal distribution which, being infinite, never occurs in practice.

DPMO Example:

If we invoice our customers for products they have bought we have a document with several fields which must be filled in correctly. On a particular company's invoices these eared: Customer name, customer address, order identification number, value of goods and payment due date. There are thus 5 distinct pieces of information on the invoice and thus 5 opportunities for error (e.g. missing, incorrect, illegible) per invoice.

Last month the company sent out 1,000 invoices so that the total number of defect opportunities that month was 5,000. To establish the existing Sigma level for the process all were intercepted and inspected prior to dispatch. In total 105 errors were found.

DPMO = (105 / 5,000) x 1,000,000 = 21,000

Using the table above we can see the nearest match is 22,750.3 corresponding sigma level of 3.5. This would be the baseline for this process.