Saturday, November 21, 2015

Benford's Law and Package Sizes

I'm putting this up quickly because Andrew Gelman has a nice blog post about Benford's Law.

My intent has been to write this up for formally, but that's been my intention for a while. I generated this data in 2008.  So it's obviously not tops on my bucket list.

The question here is whether the distribution of package sizes (for packages you'd find in a grocery store) follow Benford's Law.

What is Benford's Law?

Benford's law, also called the First-Digit Law, is a phenomenological law about the frequency distribution of leading digits in many (but not all) real-life sets of numerical data. That law states that in many naturally occurring collections of numbers the small digits occur disproportionately often as leading significant digits.[1] For example, in sets which obey the law the number 1 would appear as the most significant digit about 30% of the time, while larger digits would occur in that position less frequently: 9 would appear less than 5% of the time. If all digits were distributed uniformly, they would each occur about 11.1% of the time.[2] Benford's law also concerns the expected distribution for digits beyond the first, which approach a uniform distribution.
It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants,
and a nice graph of the expected distribution:

Does this fit products you'd find in a grocery store?

Products come in different sizes, and often product sizes are standardized for either historical or logistical reasons.  For example, you'd typically find bottles of beer in a 6 pack, 12 pack, or 24 pack. No 7 packs!

This makes it unlikely that individual categories follow Benford's Law, and in fact the individual categories do not show much of a fit. (the Benford's Law expectation is the thick blue line).

But when we combine the results across categories, we do get a pretty good fit. The Unweighted Average counts each product category equally (so the fact that there are a lot more beers than razors doesn't affect the results). The Grand Total just looks across products, so beer counts a lot more than razors. The total number products tabulated is 134,484 (387 for razors, 10,417 for beer).

The fit is clearly not perfect: there are two many 5s and too few 2s for example. The 5's are easy to see: where the volume equivalent is pounds, an 8 ounce or 9 ounce package will have a first digit of 5. A 1, 10, or 11 ounce package will have a  first digit of 6.

How this was done:

To look at this issue, we used the product data from the IRI Marketing Data Set [1]. This contains data from 31 large consumer product goods categories. IRI keeps standardized size data, called a volume equivalent for the category. This might be 1 pound, 288 ounces, or some other unit appropriate to the category.  For example, the volume equivalent for beer is 288 ounces, so a 6 pack of 12 ounce bottles (72 ounces) would have a volume equivalent of .25, and a first digit of 2.

This may be different from the various measurements that might be found on the package, which might be 18 ounces, 1 pound 2 ounces, 4 packages, 8 servings, 510.3 grams, or all of the above.

We excluded deodorant and toothbrushes because the volume equivalent is 1 package, so 100% show up as 1. We excluded cigarettes for similar reasons (the volume equivalent is 1 carton, so 1 pack is .1 of a carton; 93% of the items are either cartons or packs.

We are not weighting by sales volume, so unusual sizes that may have been sold only briefly in a few stores are weighted just as heavily as the standard items in the category.

[1] Bronnenberg, B. J., Kruger, M. W., & Mela, C. F. (2008). Database Paper—The IRI Marketing Data Set. Marketing Science, 27(4), 745–748. http://doi.org/10.1287/mksc.1080.0450