statistical tolerance analysis basics: Root Sum Square (RSS)

In my last post on worst-case tolerance analysis I concluded with the fact that the worst-case method, although extremely safe, is also extremely expensive.

Allow me to elaborate, then offer a resolution in the form of statistical tolerance analysis.

cost

A worst-case tolerance analysis is great to make sure that your parts will always fit, but if you’re producing millions of parts, ensuring each and every one works is expensive and, under most circumstances, impractical.

Consider these two scenarios.

1. You make a million parts, and it costs you $1.00 per part to make sure that every single one works.
2. You make a million parts, but decide to go with cheaper, less accurate parts. Now your cost is $0.99 per part, but 1,000 parts won’t fit.

In the first, scenario, your cost is:

$1.00/part * 1,000,000 parts = $1,000,000

In the second scenario, your cost is:

$0.99/part * 1,000,000 parts = $990,000,

but you have to throw away the 1,000 rejects which cost $0.99/part. So your total cost is:

$990,000+1,000*$0.99=$990,990. Which means you save $9,010.

Those actual numbers are make-believe, but the lesson holds true: by producing less precise (read: crappier) parts and throwing some of them away, you save money.

Sold yet?  Good.  Now let’s take a look at the theory.

statistical tolerance analysis: theory

The first thing you’ll want to think of is the bell curve.  You may recall the bell curve being used to explain that some of your classmates were smart, some were dumb, but most were about average.

The same principle holds true in tolerance analysis. The bell curve (only  now it’s called the “normal distribution“) states that, when you take a lot of measurements, be it of test scores or block thicknesses, some measurements will be low, some high, and most in the middle.

Of course, “just about” and “most” doesn’t help you get things done.  Math does, and that’s where the normal distribution (and excel… attachment below) come in.

sidebar: Initially I planned on diving deep into the math of RSS, but Hileman does such a good job on the details, I’ll stick with the broad strokes here. I highly suggest printing out his post and sitting down in a quiet room, it’s the only way to digest the heavy stuff.

the normal distribution and “defects per million”

Using the normal distribution, you can determine how many defects (defined as parts that come in outside of allowable tolerances) will occur.  The standard unit of measure is “defects per million”, so  we’ll stick with that.

There are two numbers you need to create a normal distribution, and they are represented by μ (pronounced “mew”) and σ (pronounced (“sigma”)

• μ is the mean, a measure of the “center” of a distribution.
• σ is the standard deviation, a measure of how spread out a distribution is. For example, the number sets {0,10} and {5,5} both have an average of 5, but the {0,10} set is spread out and thus has a higher standard deviation.

Using one of our blocks (remember those?) as an example…

Let’s say you measure  five blocks like the one above (in practice it’s best to measure 30 at the very least, but we’ll keep it at 5 for the example) and get the following results:

• x1 = 1.001″
• x2 = 0.995″
• x3 = 1.000″
• x4 = 1.001″
• x5 = 1.003″

The average (μ) is 1.000 ( and the standard deviation () is .003. Plug those a normal distribution, and your tolerances break down like this. (see attached excel file for formulas)

If you require the blocks to be 1.000±.003 (±1σ), the blocks will pass inspection 68.27% of the time… 317,311 defects per million.

If you require the blocks to be 1.000±.006 (±2σ), the blocks will pass inspection 95.45% of the time… 45,500 defects per million

If you require the blocks to be 1.000±.009 (±3σ), the blocks will pass inspection 99.73% of the time…2,700 defects per million

and so on.

Using the data above you can say with confidence (assuming you measured enough blocks!) that if you were to use a million blocks, all but 2700 of them would come in between 0.991 and 1.009.

Just remember to treat those numbers with the respect that they deserve. Trying to push a manufacturer to hold tolerances they aren’t comfortable with us a draining and often futile excercise.

The tolerances dictate the design, not the other way around.

Related posts:

1. statistical tolerance analysis basics:
   worst-case tolerance analysis
2. First Cut prototype
3. a quick tip for preventing PCB-induced stress

Tagged as: example, manufacturing, root sum square, RSS, rss tolerance stackup, rss tolerancing, spc, statistical tolerance analysis, tolerance analysis fundamentals