The Sigma Rating System

Despite what this highly informative entry on Urban Dictionary might have you believe, the sigma rating system is not a way of rating the relative attractiveness of another person, but a way of rating the quality of a manufacturing process. The sigma rating system is based on the idea that the number of defects that could potentially occur in a production process is statistically predictable. That means that by observing the number of times, and how severely someone botches the production of an item during a relatively short sample period; you can predict how many, out of a million units, he’s going to totally bugger up.

The sigma scale ranges from one to six. 1 sigma is at the bottom of the scale with about 69%, or 687,672 units per million, of units produced being defective. This is probably about equivalent to a Tesco shopping bag which, roughly seven times out of ten, will fail to hold the wait of your shopping, causing it to liberate its contents all over the pavement. In contrast, six sigma signifies a DPMO (defects per million opportunities) of 3.4, which I have kindly illustrated below.

Every white pixel in this image represents non-defective output, and every black pixel represents defective output. You might need to click on the image and enlarge it to see the black pixels.

Assuming you understand what I’ve said so far, you might be wondering how they come up with this figure; it’s not as if a manufacturer can just do something a million times and then count the how often they balls it up. This is where the maths comes in. I wouldn’t normally bore you with this level of detail, but the graphs I found whilst researching this stuff were so mesmerising I just had to find out what they meant, and after a good few hours trying to get my head around it, I think I’ve earned the right to try and explain it to you.

Let’s say you want to go into manufacturing cups of tea from your kitchen. God knows why you’d want to do that but for the sake of argument let’s say I’ve come to visit with my chai chugging girlfriend. She needs a constant supply of tea, and you need to supply it. She’s fussy about her tea but more than that, she doesn’t want to have to wait around all day to get it since she must drink them at a rate of about one every ten minutes (yeah, this is what I have to deal with). So you need to be producing tea at a set rate to a fairly specific standard. For this example we’ll use the amount of milk in the tea as the thing being measured. We’ll say that Tea Monster likes roughly 50ml of milk in her tea. Since it wouldn’t make much difference to be a few ml over or under, we’ll assume a buffer of about 3ml over or under 50ml; these are our upper and lower specification limits (USL, LSL). If, after closely monitoring the amount of milk being put into each cup of tea for a while, you find that the amount of milk being added is at a fairly consistent and predictable standard, we could make a good estimate of the mean amount of milk used, and the standard deviation from that amount. Put simply: add up the total amount of milk used so far and divide it by the number up cups of tea made (this is the mean); then work out how much, on average, you go over or under that amount (this is the standard deviation). Let’s pretend that these work out as 49.4ml for the mean and 0.5ml for the standard deviation. At this point, you can imagine a scale stretching between your upper and lower specification limits, with a range of -6 to 6. The middle of this scale is what you’re aiming for, a distance of six sigma from either specification limit. Check out this graph for where we’re up to.

OK, so it’s a little squiffy but what you should be able to see on the graph is this:

• The black scale along the bottom represents the amount of milk in the tea ranging from 47ml to 53ml, and is divided up into twelve levels; six above the target 50ml and six below.
• The green line in the middle represents our target mean, the 50ml we want to be putting into each cup of tea.
• The red lines either side are our upper and lower specification limits of 47ml and 53ml of milk.
• The blue curve represents the measurements taken so far with the mean being at the peak of this line.

In this graph you can see that the largest amount of measurements (the peak of the blue line) is in the dead centre of the scale at 0, and drops off at about 3.5 points from the centre. This means that on average the cups of tea are getting 50ml of milk each with roughly a 3.5 point margin of error on each side. On this particular scale 1 point is about half an ml of milk. The maths all gets very complex here and there are certain formulas to work out the specific values, but basically this graph is translated into what is called a process capability index, with measurements of Cp and Cpk. A six sigma rating requires a Cp of 2 and a Cpk of 1.5 so with a bit of quick mathalizing, I can work out work out the sigma rating of our tea production process using the mean of 49.4ml and the standard deviation of 0.5ml. Check out my working below.

Assuming these chicken scratchings are correct, we can see that we have a Cp of 2 and a Cpk of 1.6. We could push our standard deviation to .75ml and it would still be a six sigma process as this would only change the Cpk to 1.5, which is within the formal definition of six sigma.

So there you have it, a rather thorough run-down of the sigma rating system. You might be wondering why stop at six sigma? Why not keep pushing and go to seven or eight sigma (whatever that would be)? Why not aim to eliminate all defects completely? Well those are very good questions that I will have to cover in a future blog post, but for now, I’m going to go and do something very dumb to make up for all the smart stuff I’ve just offloaded. Maybe I’ll stick a fork in the toaster.