Statistics: how does standard deviation measure quality - please answer everyone?

Question

There are many types of measures of variation/dispersion from the mean, but standard deviation can be confusing.

If we are measuring error/variability, then 1σ is better than 2σ. That is to say that most data are lying within only 1σ, is better than if the data were scattered within 2σ.

_________ . _________ . _________ . _________ . _________

BUT, if we are measuring quality standard - i.e. acceptable range of deviation from the mean - then higher acceptable deviations, the better.

the larger range of acceptable deviation, the less error:

μ ± 1σ → error 1 in 3

μ ± 3σ → error 1 in 370

μ ± 6σ → error 1 in 506797346 http://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule#Higher_deviations

Is this a correct description of what standard deviation measures in quality measurement?

All comments and answers are welcome.

Thank you so much for everyone who will answer.

Answer 1

There are at least two senses in which you can use the standard deviation in quality measurement:

1. Manufacturing precision: How closely are we able to meet a manufacturing specificiation (e.g., bearing diameter).
2. Fraction falling within a given tolerance interval: Continuing the example, if bearings must be within $\pm 1$ mm, how many standard deviations does this represent GIVEN our observed manufacturing variability (i.e., (1) from above).

The more fundamental use of the standard deviation is in (1), where you are characterizing how well-controlled your manufacturing process is. In this case, the larger the standard deviation is, the lower the quality of your manufacturing process. This is regardless of the actual standards you need to meet - if process A has higher standard deviation than process B, than for any tolerance interval $\pm a$ mm, process A will generate more bad products than process B.

Now, the second use of standard deviation, (2) from above, is probably what is causing you confusion. In this case, you are correct that more standard deviations indicate a higher quality process, but these standard devaitions are NOT the same as in (1). To illustrate, imagine that you have a process that produces bearings with mean diameter 0.2 mm with standard deviation 0.01 mm (and normally distributed). Now, your tolerance interval is $\pm .05$ mm. How many standard deviations does $\pm .05$ represent, given that you observed a process standard deviation of 0.01 mm? In this case, you would say that your process is $5\sigma$, in that the predetermined tolerance limits represent 5 standard deviations away from your mean process output (which I assumed to be unbiased), as calculated from the underlying process data.

So, you see, you want the standard devation of your manufacturing process to be low, which will increase the number of standard deviations that can fit in your pre-specified tolerance interval. The two uses of $\sigma$ do not mean the same thing.

Answer 2

Eupraxis was smart to say that a normal distribution was assumed. In the real world, we aren't so lucky. For example, go to a financial time series. Take the daily percent change of a security. You can calculate the mean and standard deviation, but you will find some data points beyond 6 sigma (which, according to current dogma, is supposed to be impossible). This is a real world example of a "tail-heavy" distribution.

In fact there are some distributions, like the tangent, that don't even have a standard deviation! Exercise for the student: on a spreadsheet, generate random numbers x from x=tan(pi*rand()). Take the mean and standard deviation for various sample sizes, and then find the extreme values. Some are going to be WAY out there. This forms the basis for the "back of the envelope" test for a normal distribution. Take the extreme of 1000 random numbers, subtract the mean, divide by the standard deviation, and look for a score in the range of 3 to 4. That means the distribution could be normal. Of course a stronger test would be to take the extreme of larger sample, but 1000 is about all we want to wait for on a handheld calculator.

Also, in science measurements there is such a thing as error. Some of the error may be systematic, and some of it random. But there is also such a thing as BLUNDER, where and obvious mistake has been made by the scientist. That can be responsible for observations beyond 6 sigma.