The Law of Medium Numbers

There’s a law of large numbers, a law of small numbers, and a law of medium numbers in between.

The law of large numbers is a mathematical theorem. It describes what happens as you average more and more random variables.

The law of small numbers is a semi-serious statement about how people underestimate the variability of the average of a small number of random variables.

The law of medium numbers is a term coined by Gerald Weinberg in his book An Introduction to General Systems Thinking. He states the law as follows.

For medium number systems, we can expect that large fluctuations, irregularities, and discrepancy with any theory will occur more or less regularly.

The law of medium numbers applies to systems too large to study exactly and too small to study statistically. For example, it may be easier to understand the behavior of an individual or a nation than the dynamics of a small community. Atoms are simple, and so are stars, but medium-sized things like birds are complicated. Medium-sized systems are where you see chaos.

Weinberg warns that medium-sized systems challenge science because scientific disciplines define their boundaries by the set of problems they can handle. He says, for example, that

Mechanics, then, is the study of those systems for which the approximations of mechanics work successfully.

He warns that we should not be mislead by a discipline’s “success with systems of its own choosing.”

Weinberg’s book was written in 1975. Since that time there has been much more interest in the emergent properties of medium-sized systems that are not explained by more basic sciences. We may not understand these systems well, but we may appreciate the limits of our understanding better than we did a few decades ago.

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4 thoughts on “The Law of Medium Numbers

  1. You’re missing a word in there:

    “For medium number systems, we can expect that large fluctuations, irregularities, and discrepancy with any theory will more or less regularly.”

    I’m guessing it’s “will *occur* more”?

  2. Thanks for sharing and for the great summary, John. I went ahead and tried to hash out how this might apply to software, The Law of Medium Software Projects

  3. Hmm… inspiring post. Thanks for sharing. But…
    What is the clue? Be prepared? Or there is something else?

    Noticing that “medium things are hard” may be brain-teasing, interesting, but it “just happens” on your way in science (physics in my case). In science however you just move on. Solve some equations, make some generalization – if it is possible, or – like in this case – put sticker “too hard to really invest my time” and leave.

    I would say that “medium things” are responsible for occurrence of “butterfly effects” – and THAT is important thing to see – on many levels.

    If I would consider this three things: “medium size”, “butterfly effect” (caused by medium size), and IT (especially problems in IT) it’s worth to see that in small system you just don’t have enough domino pieces, and in really big there is enough other bricks on your way to stop bigger problems.

    I’m looking for some clue, but the only one which comes to my mind is just “see it!”. And maybe that’s enough. Because when you see that you’re going to fall into “medium numbers” field you know that either:
    – you should split your problem into smaller parts
    – or you should get bigger – gather more issues together

    And that’s probably clue I was looking for
    Thanks for your post.


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