#2 Learning Spanish and The Power Law Curve

Normal distributions and Power law curves rule our lives. They are everywhere. SAT scores, human height, newborn baby weight, IQ scores, etc., follow the normal distribution. In contrast, the distribution of wealth, venture capital returns, earthquake magnitude, website popularity, and retail sales all follow a power law curve.

Power law curves are non-linear relationships between two variables. The 80/20 rule or Pareto principle is an example of a power law that states that the top 20% literally command 80% of the value. For example, a very small percentage of individuals would capture the majority of the wealth in society, and similarly, the top few companies in the stock market would account for the majority of the valuation of the stock market.

Recently, I realized that learning a language is also impacted by the power law curve. I have been learning Spanish for a few years now and in the beginning I made great progress by memorizing a few hundred words but lately my progress has been very slow. In fact, during one of the class tests in November, our Spanish teacher gave us a prompt and we had to respond with a few talking points in Spanish. I floundered the quiz as I could not even comprehend what the prompt was.

As it turns out, learning a language is indeed impacted by the power law. In all known languages, a small percentage of words make up the majority of usage. This is known as Zipf’s law. The most used word in a language may make up as much as 7% of all words used, while the second-most-used word is used half as much, and so on. As few as 135 words can together form half of a language (as used by native speakers).

I asked ChatGPT to make me a power law curve of the frequency of common vs rare words and below is what I got.

The Spanish language is made up of roughly ~90,000 words but a few thousand most common words can help anyone do a reasonable job at communication. However to be a complete master, one has to learn tens of thousands more words, which also explains the law of diminishing returns. It means that one would get to reasonable progress very soon by putting in little effort but a lot more investment is needed to get to complete mastery and additional new investment will likely not produce an immediate noticeable output. This is the stage where learners often get frustrated and give up. However, understanding the power law can motivate one to adjust their expectations. Progress may feel slower, but every new skill one acquires—like conjugating verbs in the subjunctive mood—builds the foundation for fluency.

The power law curve isn’t just a concept for economists or scientists; It's everywhere and is a reminder that progress in learning Spanish—or any skill—is uneven yet predictable. By embracing the early gains and pushing through the plateaus, one can keep climbing the curve to fluency.