Does Benford's law reveal that smaller natural numbers are more common than bigger ones in real life?

Question

I have executed two experiments to verify whether smaller natural numbers are more common than bigger ones, spired by some high-performance database software that stores smaller natural numbers with a unique(memory-saving) approach.

Experiment one extracted all natural numbers from a text corpus like Jeopardy(Over 200,000 questions from the famed tv show) and computed each number's frequency.

The other experiment extracted all natural numbers from the Fibonacci sequence as follows.

integers_freq = {}
integers_cnt = 0

# get all substrings from a numeric string
# e.g. n = 144 substrings = ['1', '4', '4', '14', '44', '144']
def get_substrings(s):
    # omit...

# fibonacci numbers
fibonacci = [1, 1]
for i in range(2, 20):
    fibonacci.append(fibonacci[i-1] + fibonacci[i-2])

for n in fibonacci:
    s = str(n)
    substrings = get_substrings(s)
    for substring in substrings:
        integers_cnt += 1
        if substring in integers_freq:
            integers_freq[substring] += 1
        else:
            integers_freq[substring] = 1

# sort the integers by integer value
integers_freq = sorted(integers_freq.items(), key=lambda x: int(x[0]))

for integer, freq in integers_freq:
    print("count = {} Pr({}) = {:.4f}".format(freq, integer, freq/integers_cnt))

And I plot the natural numbers(extracted from top 20 Fibonacci sequence) distribution with a bar chart

Both experiments expressed that smaller natural numbers are more common than bigger ones. Can we use Benford's law(Generalization to digits beyond the first) to explain this phenomenon?

Answer 1

Benford's law is not necessary to explain this. Any distribution over the natural numbers - including the empirical distribution of the numbers found in Jeopardy - will privilege smaller natural numbers in some sense.

One way to make this precise: for any threshold $\epsilon > 0$, we can find a natural number $N$ such that all numbers with a probability of more than $\epsilon$ of being encountered are in the range $\{1,2,\dots,N\}$. (If we think of "probability-more-than-$\epsilon$" as "common" and "less than or equal to $N$" as "small", then only small numbers are common.)

There are, of course, other ways to try to say "smaller natural numbers are more common" in a formal way. Many of them will be true of any distribution over the natural numbers. Some will be false for the Jeopardy distribution; for example, I bet there's many instances where a large number is likelier than a smaller one, just because round numbers are more likely to be encountered.

Answer 2

Your two examples have very different causes. The Fibonacci numbers obey Benford's law as the mantissas of their logarithms are equally distributed in $[0,1)$. Your extraction of substrings of the Fibonacci numbers will be biased by the fact that the leading digits are more often $1$ than $9$. If you consider much larger Fibonacci numbers the effect will be diluted because more of your substrings will not include the leading digits.

The Jeopardy example is a classic for Benford's law but probably even more extreme. Many of the numbers they quote have dimensions and we like to choose units that make the numbers small. They also choose examples of things that there are only one to four of to ask about. This is a strong bias in favor of small numbers.

Answer 3

Benford's law isn't some universal principle that applies to "real life" regardless of context. There are lots of situations where it works and lots where it doesn't, and the Wikipedia article you link actually does an okay job at outlining this.

Simply put, different data sets will be distributed differently based on the underlying mechanisms related to where that data came from. You have to look at it on a case-by-case basis.

For example, the questions/answers from a general knowledge game show over a long period of time will presumably cross a few orders of magnitude, and therefore the criteria of Benford's law are more likely to be met. Conversely, we should expect Joe Biden's precinct results in Chicago to be closer to a normal distribution than dominated by small numbers, just because most of the precincts are similar sizes.

As for the general question about whether smaller numbers are more common than larger numbers, I'm not sure how that can be answered in a non-arbitrary, non-subjective way. You could take all the numbers from Jeopardy or, say, a daily newspaper, but how do you determine that that's representative of "real life"? People don't experience the world as a random variable picking numbers from a newspaper. Perhaps in some situations we can be modelled that way, but that would have to be demonstrated first.