Читать книгу Lean Six Sigma For Dummies - Martin Brenig-Jones - Страница 22

Introducing a simple example

Suppose you want to understand the cycle time (lead time) of a process in your organization in days. You could collect a representative sample of data (more on sampling in Chapter 7), and from that sample, calculate the average (or mean) number of days. Calculating the average difference between each cycle time in your data set and the overall average cycle time will give you the standard deviation. The standard deviation is always expressed as the same unit as the “thing” you are measuring; in this case we’re talking about days.

Figure 1-2 shows the time taken to process the orders. The cycle time varies from as short as one day to as long as seven days. Each of the data points represents a customer’s experience of the process. As well as seeing how much performance varies, you can also see the “shape” of the data. This data looks normally distributed. In a normal distribution, the shape is symmetrical around the mean, and the data has a 50-percent chance of falling either side of it. Sometimes this shape is referred to as a bell curve or a Gaussian distribution. Many things that are measured fall into this shape — for example, the heights of people, the size of snowflakes, and IQ scores.

FIGURE 1-2: Histogram showing the time taken to process orders.

When the data is normally distributed, we can understand the likely percentage of the population within plus one or minus one standard deviation from the average (or mean), plus two or minus two standard deviations from the average, and so on. Assuming your sample is representative, you can see how your data provides a good picture of the process cycle time. You find that approximately two-thirds of them are between 3 days and 5 days, about 95 percent are in the range of 2 days to 6 days, and about 99.73 percent are between 1 day and 7 days. This is illustrated in Figure 1-3.

The formula used to calculate the standard deviation is shown in Figure 1-4. It looks a little scary at first glance, but as with all formulas, when you start to break it down it becomes more accessible.

x in the formula represents your individual data points

xi represents each x from x1 up to xn

n represents the number of data points in your data set

represents the average (mean) of your data points

represents the “sum of”

FIGURE 1-3: Standard deviation.

As with most things in life, tackling the formula in stages makes it easier, so let’s have a go.

Start by adding up all of the data points you have. For example, if the data points are 5, 6, 7, 8 and 9 (we’ll demonstrate with some simple ones), the sum is 35. The number of data points we have is 5, so the value of n in the formula is 5. We can then work out the mean (average or ) of those data points, and the mean () is 7 (because ). Are you with us so far? Great!

Next we need to work out the part. This means subtracting the value (7) from each of our data points. , and so on. All is going well, but we now have some negative values in the mix, and we need to get rid of them. This is done by squaring the numbers we’ve just worked out. So , and so on. Next, add together all of the answers we just got. If you’re still with us and have done that bit, you should get 10. So now we have real numbers to work with, and the numbers that we put into the formula are (Yes, we did all that work to get to 10 divided by 5.) 10 divided by 5 is 2. We then, finally, take the square root of 2 to get the standard deviation value of 1.41. Hooray!

Note that there are two versions of the standard deviation formula included in Figure 1-4. The first is used when we have a sample of data, and the second is used when we have the entire population. The data set we used to work through the example above was very small! If we were using the formula for a sample, instead of using n we’d use . So we’d end up with , which is 2.5. The square root of 2.5 is 1.58.

FIGURE 1-4: Standard deviation formula.

In practice, when the sample size is more than 30, there’s little difference between using n or . When we refer to a “population,” this could relate to people or things that have already been processed, like, for example, a population of completed and dispatched policy documents.

Now for some good news: You can use a scientific calculator, Excel, or any number of online calculators to work out the standard deviation of your data, without having to worry about this formula.

The Process Sigma values are calculated by looking at our process performance against the customer requirements, which we cover in the next section.

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