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### Roger's Equations

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# Engineering Product Insight: Focal Lengths of Convex Lenses

Posted June 08, 2015 9:00 AM by Bayes

A Database of Insight

As many of the older CR4 members know, I've worked at IHS Engineering360 for over a decade (Joined CR4 in February of 2005!). One of the central pieces of IHS Engineering360 is our Engineering Products database. This is a database of millions of engineering products such as motors, lenses, bearings, etc.

When you're around a large database for a long time, you start to notice patterns. One thing we noticed a long time ago was that the specs of products like motors, lenses, bearings, etc. had a strange tendency to form Log-Normal distributions. For instance, see the graph below generated from data from our database of products. This particular graph shows the log-normal distribution of focal lengths in convex lenses.

Log-Normal Distribution of the Focal Lengths of Convex Lenses

Understanding the Graph Above

It's a bit hard to read so I'll tell you what the axes are. The x-axis consists of a series of focal length ranges starting from the very small (<1.5mm) and ending at the very large (>1000 mm). The y-axis represents the percentage of parts in the database within that focal length range. So in other words, of all the convex lenses in our database, less than 1% have focal lengths smaller than 1.5 mm, and less than 1% have focal lengths larger than 1000 mm. The graph's peak corresponds to focal lengths falling within the bin of 50 to 100 mm. Products with focal lengths in that range represent about 23% of the convex lenses in our database.

Since this is a log-normal distribution, the bins get larger as you move left to right. You might expect larger bins to hold more products, but this isn't the case. There tends to be a sweet spot for all product specs. I think intuitively all engineers know this. If someone were to mention a focal length, or a torque, or laser power, if you're familiar with the product you would likely know whether that spec were fairly typical or atypical. The graph above is simply a representation of that, except rather than it being a nebulous "feel" thing, we can actually quantify it thanks to our extraordinary sized database of parts.

Why Care?

Nowadays data visualization is everywhere. The internet is filled with infographics that attempt to help one better understand data. This isn't necessarily a bad thing as sometimes data can become obtuse when presented as a set of numbers. However, not all infographics are equal. If you could weigh the information hidden in the graphic above, it would weigh a ton. Why? Because it represents the entire market for convex lenses in terms of focal lengths.

The free market, when functioning correctly, is efficient. When we took suppliers' products and entered their specs into a database so that they could more easily be found by Engineers, we didn't realize we'd be capturing a snap shot of the market as well. As our database grew we quickly realized we could tell what specs were "typical" just by binning the specs of the part (and getting a log-normal distribution). Does it always work? No. Specs such as temperature range, or voltage tend to not form distributions due to standardized values and ranges. But for so many other products, insight is just a graph away.

-R

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Guru

Join Date: Apr 2010
Location: About 4000 miles from the center of the earth (+/-100 mi)
Posts: 9048
#1

### Re: Engineering Product Insight: Focal Lengths of Convex Lenses

06/09/2015 9:12 PM

Any idea why?

The Engineer

Join Date: Feb 2005
Location: Albany, New York
Posts: 5170
#2

### Re: Engineering Product Insight: Focal Lengths of Convex Lenses

06/10/2015 8:41 AM

That's a good question. Why do the torques of motors, focal lengths of lenses, power of Lasers, radius of bearings, and tens of thousands (at least) of other specifications follow Log-Normal Distributions?

In this case, it's important to note that the focal length is dependent upon the size of the optic to a certain degree. Big things are expensive to make (material costs), small things are expensive to make (precision costs), so in a sense you may be simply looking at chart of the cost to construct products with the peak being the cheapest (and thus most commoditized) products.

It could also simply be that human beings like to build human scale devices, and the further you get from the human scale, the less likely you are to find devices. In fact, one could argue you could measure technological progress by scales. For instance, the progress of pulsed lasers from pico to femto to attosecond pulses. Or power plant output from kW to MW to GW. Or computer processor speed, or whatever.

In fact, I would argue, that if it were possible to look at this log-normal distribution over a long period of time (hundreds of years), we'd probably see that it started very narrow and has gradually gotten wider over time as lens technology has improved.