The first-digit law or Benford's law https://en.wikipedia.org/wiki/Benford's_law is about a mysterious mathematical phenomena that does not have a convince proof yet. It says the distribution of the first-digit in many number sets following a logarithmic probability distribution instead of even distributions between 1 to 9. For example, consider only the first digit of all county populations regardless of the country population size. This law says the probability of "1" is log10((1+1)/1); for "2", log10((2+1)/n); ... 1 0.301029995664 2 0.176091259056 3 0.124938736608 4 0.0969100130081 5 0.0791812460476 6 0.0669467896306 7 0.0579919469777 8 0.0511525224474 9 0.0457574905607 In the stock market, there are also many examples. Out of curiosity, I checked some of the data regarding stocks using a few thousands of stocks. Observed many data sets follow this law. I checked the following, and they all follow this law. 1. the numbers of employees for 4302 companies; (note: not per company, but all companies in the data set, the same for all the following in the list). 2. market cap values for 4365 companies; 3. the latest revenue data for 4245 companies; 4. the net income 5. trading volume 6. 20-day or 50-day trading volume 7. EPS 8. outstanding shares total 9. daily change and yearly change of stock prices Also checked per stock data, for example, DOW index since 1960, the change value per day (ignore leading 0 and negative sign and dot), the distribution of the first digit from 1 to 9 is: total data 15211 digit num_data distribution 1 4595 0.302084 2 2502 0.164486 3 1786 0.117415 4 1465 0.096312 5 1291 0.084873 6 1081 0.071067 7 956 0.062849 8 831 0.054632 9 704 0.046282
For the 16,033 daily closing prices of the S&P 500 from yahoo finance since March 4, 1957, the distribution of first digits is similar except for a small rise with 8 and 9.
Yes. I got the almost the same for SPY closing prices as you did. The closing prices for one stock does not really following the first-digit law because they are not random enough across multiple magnitude. But consider the closing prices for a large number of stocks per day, it almost perfectly follows that pattern.