might be any number in theory.
Figure 1-2 shows the likely percentage of the population within plus one and minus one standard deviation from the mean, plus two and minus two standard deviations from the mean, and so on. Assuming your sample is representative, you can see how your information provides a good picture of the heights of all the people in your organisation. You find that approximately two-thirds of them are between 5 foot 6 inches and 5 foot 8 inches tall, about 95 per cent are in the range 5 foot 5 inches to 5 foot 9 inches, and about 99.73 per cent are between 5 foot 4 inches and 5 foot 10 inches.
© John Morgan and Martin Brenig-Jones
Figure 1-2: Standard deviation.
In reality, the calculation is a little more involved and uses a rather forbidding formula – as shown in Figure 1-3.
© John Morgan and Martin Brenig-Jones
Figure 1-3: Standard deviation formula.
Using n – 1 makes an allowance for the fact that we’re looking at a sample and not the whole population. In practice, though, when the sample size is over 30, there’s little difference between using n or n – 1. When we refer to a ‘population’ this could relate to people or things that have already been processed, for example a population of completed and despatched insurance policies or hairdryers.
The process sigma values are calculated by looking at our performance against the customer requirements – see the next section.
In the real world you probably don’t measure the height of your colleagues. Imagine instead that in your organisation you issue products that have been requested by your customers. You take a representative sample of fulfilled orders and measure the cycle time for each order – the time taken from receiving the order to issuing the product (in some organisations this is referred to as lead time). Figure 1-4 shows the cycle times for your company’s orders.
© John Morgan and Martin Brenig-Jones
Figure 1-4: Histogram showing the time taken to process orders.
You can see the range of your company’s performance. The cycle time varies from as short as one day to as long as seven days.
But the customer expects delivery in five days or less. In Lean Six Sigma speak, a customer requirement is called a CTQ – Critical To Quality. CTQs are referred to in Chapter 2 and described in more detail in Chapter 4, but essentially they express the customers’ requirements in a way that is measurable. CTQs are a vital element in Lean Six Sigma and provide the basis of your process measurement set. In our example, the CTQ is five days or less, but the average performance in Figure 1-4 is four days. Remember that this is the average; your customers experience the whole range of your performance.
In the example provided in Figure 1-4, all the orders that take more than five days are defects for the customer in Six Sigma language. Orders that take five days or less meet the CTQ. We show this situation in Figure 1-5. We could express the performance as the percentage or proportion of orders processed within five days or we can work out the process sigma value. The process sigma value is calculated by looking at your performance against the customer requirement, the CTQ, and taking into account the number of defects involved where you fail to meet it (that is, all those cases that took more than five days).
© John Morgan and Martin Brenig-Jones
Figure 1-5: Highlighting defects.
We explain the process sigma calculation in the next section.
Process sigma values provide a way of comparing performances of different processes, which can help you to prioritise projects. The process sigma value represents the population of cases that meet the CTQs right first time. Sigma values are often expressed as defects per million opportunities (DPMO), rather than per hundred or per thousand, to emphasise the need for world-class performance.
Not all organisations using Six Sigma calculate process sigma values. Some organisations just use the number of defects or the percentage of orders meeting CTQs to show their performance. Either way, if benchmarking is to be meaningful, the calculations must be made in a consistent manner.
Figure 1-6 includes ‘yield’ figures – the right first time percentage. You can see that Six Sigma performance equates to only 3.4 DPMO.
© John Morgan and Martin Brenig-Jones
Figure 1-6: Abridged process sigma conversion table.
So, for example, you could have a situation whereby the speed of delivery CTQ was met, but the accuracy and completeness CTQs were missed. The outcome would be one defective (see the bullet list below) as a result of these two defects. In calculating sigma values for your processes, you need to understand the following key terms:
✔ Unit: The item produced or processed.
✔ Defect: Any event that does not meet the specification of a CTQ.
✔ Defect opportunity: Any event that provides a chance of not meeting a customer CTQ. The number of defect opportunities will equal the number of CTQs.
✔ Defective: A unit with one or more defects.
In manufacturing processes you may find that the number of defect opportunities is determined differently, taking full account of all the different defects that can occur within a part. The key is to calculate the process sigma values in a consistent way.
You can work out your process sigma performance against the CTQs as shown in Figure 1-7. We have a sample of 500 processed units. The customer has three CTQs,
