Whaddya mean 68%? I want 95%!

I opened up an individual serving container of yogurt the other day. You know, the kind of yogurt container that gives yuppies a bad name? Not only was it the fancy "Greek" yogurt, but the kind that uses 5.4 ounces of packaging to give you 5.3 ounces of yogurt? Honestly, I only eat it because my wife buys it at the grocery store.

But I didn't mean for this blog to be about me. The contents of the yogurt container reminded me of a blog post that I have been meaning to write. You see, that yogurt container had only 2.6 ounces of yogurt. Yes, it was half full.

Picture of me, ordering a yuppiegurt

Now, I'm a yogurt-container-half-full kinda guy, but if my wife had opened that container, I am sure she would have immediately been on the phone to the customer service department at Chobani. She would wait on hold for an hour just to give them a piece of her mind. Oh... and she would get a bunch of coupons. What did I do? I smiled, shook my head, and said "sampling and manufacturing tolerances!" Yes, I got the short tail of the Gaussian curve when it came to manufacturing tolerances.

--- Now I'm going to change the subject from the stuff in the package to the package itself ---

This all reminded me of a discussion that I had with my good buddy, Steve Smiley. We were talking about one of our favorite topics - ISO 12647-2. Yes, we can be a couple of wild and crazy guys when you get us wound up. A little beer and a little hot Mexican food, and we're quoting stuff like this: 

The variability of the process colour solids in production is restricted by the following condition. For

at least 68 % of the prints, the colour differences between a production copy and the OK print shall not exceed the pertinent variation tolerances specified in Table 7.

(Table 7, you will recall, has a lot of 4 DE and 5 DE stuff in it.)

Steve has taught me to always check the halftone dot structure before partaking

The last time Steve and I chatted, he told me that the brand owners that he consulted with weren't happy with 68% of the product being within a certain tolerance. They want 90% or even 95% within some tolerance. They know that my wife will be inspecting the labels on the Chobani packages. If just one of them is just a slight bit pale in color, they know that they will be sending yet another packet of free coupons to Milwaukee.

Seriously... while my wife is on a first name basis with the customer support people for almost all the consumer product companies, the real reason that the color of a package is so important is that if a package is a bit off color, then people will pick over it in favor of the brightly colored box. Eventually that off-color package is going back to the factory with its handy fold out pouch between its clam shell blister pack. Just like I found out when I tried stand-up comedy at the Baptist's convention, off-color doesn't sell.

I had a simple answer to Steve's question. I'm sure that I didn't articulate it well, since we were on our third bottle of sriracha sauce by then. It was an application of a blog post I did called "Assessing Color Difference Data". The whole point of this blog is that for distributions of color difference data, there is a simple conversion between the 68th percentile and any of the other percentiles. I copy the table from this remarkably insightful blog post:

P-tile	Multiplier	r-squared
10	0.467	0.939
20	0.631	0.974
30	0.762	0.988
40	0.883	0.997
50	1.000	1.000
60	1.121	0.997
68	1.224	0.993
70	1.251	0.991
80	1.410	0.979
90	1.643	0.947
95	1.840	0.903
99	2.226	0.752
Max	2.816	0.378

It's not obvious from the table, but the conversion is pretty simple. If a color tolerance is stated like this: "68% of production shall be within 5 DE of the target color", then you can convert this to a 95% statement with the following calculation:  (5 DE / 1.224) X 1.840 = 7.52 DE. The first part, dividing by 1.224, converts from a 68th percentile to a 50th percentile. The second part, multiplying by 1.840, converts the 50th percentile DE to a 95th percentile. To put it simply... you multiply by 1.5.

So by my careful and erudite analysis, the following two specifications are equivalent:

"68% of production shall be within 5 DE of the target color"

"95% of production shall be within 7.5 DE of the target color"

If a press run meets the first criteria, then it will pretty well meet the second, and vice-versa. Unless of course, the press has decided to not follow the laws of statistics.

I have intentionally left a topic un-discussed: How do you tell what percentage of the press run was in tolerance? I'm gonna save that thrilling question for my next blog post.