Before we get into a dis- cussion of how to set busi- ness metrics, we should return to a concept basic to Six Sigma—sigma. In Chapter 1 we defined it as
“A term used in statistics to represent standard deviation, an indicator of the degree of variation in a set of measurements or a process.” So, now we should elaborate a little on standard deviation and variation.
Variation
As defined in Chapter 1, variation is “any quantifiable difference between individual measurements.” Any process improvement should reduce variation, so that we can more consistently meet customer expectations. But in order to reduce it, we must be able to measure it. So, how do we measure variation?
There are several ways, each with advantages and disad- vantages. Let’s take an example to show how these methods work.
Your company produces widgets. There are two lines that assemble the components, A and B. You want to reduce the variation in assembly times, so that the workers who package the components can work most efficiently—not waiting for fin- ished widgets, not falling behind, and not being forced to work so quickly that they make mistakes.
The first step is to track assembly times. You gather the fol- lowing data:
Process A: 3.7, 6.5, 3.2, 3.2, 5.7, 7.4, 5.7, 7.7, 4.2, 2.9 Process B: 4.7, 5.3, 4.7, 5.4, 4.7, 4.4, 4.7, 5.8, 4.2, 5.7 Now, what do those figures mean? We can compare the two processes in several ways, using common statistical concepts.
(In reality, you would be collecting much more data than our 10 sample values, but we’ll keep this example simple.)
Get Data
Traditional management often operates by the “seat of the pants”—by tradition, by impres- sion, by reaction to events, by gut instincts.The essence of Six Sigma management is to use objective data to make decisions.
If we use the mean, we find that line A averages 5.02 minutes and line B averages 4.96 minutes.
That’s very close. But we don’t know which process varies more.
We can calculate the medianvalue (the mid- point in our range of data).
For A it’s 4.95 and for B it’s 4.7. That’s close.
We can also calculate
the mode(the value that occurs most often): for A, it would be either 32 (two times) or 5.7 (two times) and for B it’s 4.7 (four times). The mode doesn’t help us much here.
Based on these three measurements, what do we know about the variations in our two widget assembly lines? How do they compare? Which statistical concept best represents the variation in each line?
We don’t know much about our variation at this point.
Fortunately, there are two more concepts that we can use:
range and standard deviation.
Range is easy to calculate: it’s simply the spread, the differ- ence between the highest value and the lowest value. The range for A is 4.8 (7.7 – 2.9) and the range for B is 1.6 (5.8 – 4.2).
Now, that’s a considerable discrepancy between A and B! The variation in process A is much greater than in process B.
But range is a rough measure, because it uses only maxi- mum and minimum values. It seems to work OK in this case.
But what if we had the following values?
Process C: 3.2, 6.5, 3.4, 6.4, 6.5, 3.3, 3.7, 6.4, 6.5, 3.5 The range for this set of values is 3.3, which suggests that there’s less variation in process C than in process A, with a range of 5.1. But common sense tells us that the values in process C vary more, even if less widely.
Mean Average (more specifically called the arith- metic mean), the sum of a
series of values divided by the num- ber of values.
Median Midpoint in a series of values.
Mode Value that occurs most often in a series of values.
Range Difference between the high- est value and the lowest value in a series, the spread between the maxi- mum and the minimum.
We need another concept, something more accurate than range for calculating and representing process variation. That concept is standard deviation.
Standard Deviation
Standard deviation measures variation of values from the mean, using the following formula:
where Σ= sum of, X = observed values, X bar (X with a line over the top) = arithmetic mean, and n = number of obser- vations.
That formula may seem complicated, but it’s actually simple to understand if we break it down into steps:
1. Find the average of the process values.
2. Subtract the average from each value.
3. Square the difference for each value (which eliminates any negative numbers from the equation).
4. Add all of these squared deviation values.
5. Divide the sum of squared deviations by the total number of values.
6. Take the square root of the result of that division.
So, when we do the calculations for process A and process B (fortunately, there are software applications that can crunch these numbers for us!), we get the following results:
A: standard deviation = 1.81 B: standard deviation = 0.55
These figures quantify what we observed, that the variation in process A is greater than the variation in process B. Our simple example uses only 10 val- ues for each process, so Standard deviation
Average difference between any value in a series of values and the mean of all the values in that series.This statistic is a measure of the variation in a distribution of values.
σ = Σ (x - x)2 n
calculating standard deviation doesn’t help us much more than simple observation. But when our measurements give us many more values, the concepts are more useful—and we appreciate even more the software that does all of these calculations for us.
If we plot enough values on a control chart (we’ll discuss this type of chart in Chapter 7), we’ll likely find that the distribu- tion of values forms some variant of a bell-shaped curve. This curve can assume various shapes. However, in a normalcurve, statisticians have determined that about 68.2% of the values will be within 1 standard deviation of the mean, about 95.5% will be within 2 standard deviations, and 99.7% will be within 3 stan- dard deviations.
Your goal is to reduce the variation in your widget assembly processes. So you first need to determine how much variation is acceptable to your customer. Then, you use those values to set your lower specification limit(LSL) and your upper specification limit(USL). These are the upper and lower boundaries within which the system must operate.
In our example, we might determine that the customer (the widget packaging group) would be happy if the assembly lines took between 4 and 6
minutes (within 1 minute either way of the ideal time of 5 min- utes). That would set an LSL of 4.0 and an USL of 6.0 around a mean of 5.02 minutes for line A and a mean of 4.96 for line B.
Since the standard deviation for A is 1.81, we’re quite far from our goal, because the standard deviation is greater than the interval between the LSL and the mean (1.02) and the inter- val between the USL and the mean (.98) specification limits.
On the other hand, since the standard deviation for B is 0.55, we’re already meeting our goal, because the standard deviation is greater than the interval between the LSL and the mean (.96) and the interval between the USL and the mean (1.04) specification limits. (Of course, we already knew that line
Specification limit One of two values (lower and upper) that indicate the
boundaries of acceptable or tolerated values for a process.
B was meeting the customer’s expectations, because none of the 10 times recorded in our measurement is above 6 minutes or below 4 minutes.)
Process Capability
So, how does all of this discussion of variation and standard deviation and curves relate to Six Sigma? The goal of Six Sigma is to reduce the standard deviation of your process variation to the point that six standard deviations (six sigma) can fit within your specification limits.
You may recall the concept of process capability from Chapter 1, defined as “a statistical measure of inherent variation for a given event in a stable process.” The capability index(Cp) of a process is usually expressed as process width(the differ- ence between USL and LSL) divided by six times the standard deviation (six sigma) of the process:
Cp = USL – LSL/6σ
The higher your Cp, the less variation in your process. In our case of the widget assembly lines, with an acceptable allowance of +/–1 minute from the ideal mean difference, reaching a six
LSL 4.0
USL 6.0
LSL 4.0
USL 6.0 Mean
5.02
Mean 4.96
A B
Figure 3-1. Bell curves showing results from two sets of measure- ments
Process width Spread of values +/-3 sigma from the mean—process width, also known asnormal variation.
sigma level of quality would mean reducing the standard devia- tion of both process A and process B to about .166. The Cp of process A is 1.105 and the Cp of process B is 3.636.
There’s a second process capability index, Cpk. In essence, this splits the process capability of Cp into two values.
Cpk = the lesser of these two calculations:
USL – mean/3σ or mean – LSL/3σ
In addition to the lower and upper specification limits, there’s another pair of limits that should be plotted for any process—the lower control limit(LCL) and the upper control limit (LCL). These values mark the minimum and maximum inherent limits of the process, based on data collected from the process. If the control limits are within the specification limits or align with them, then the process is considered to be capable of meeting the specifications. If either or both of the control limits are outside the specification limits, then the process is
considered incapableof meeting the specifications.
“Sadistics”
A friend who was studying statistics had a young
daughter who referred to the subject as “sadistics.” From the mouths of babes....
At this point, you may not be able to perform all of these calculations and others used in Six Sigma. That’s why there’s statistical software. You may not understand all the ins and outs of these concepts. That’s why training is essential to any Six Sigma initiative.
What’s important here is that you understand the basic con- cepts of Six Sigma measurements and better appreciate the importance of establishing metrics to track variation so you can improve processes. With that quick overview of the essentials, we can leave our imaginary example of widgets and return to the very real situation of your business.
Control limit One of two values (lower and upper) that indicate the inherent limits of a process.