### What is the Law of Large Numbers?

The law of large numbers, in probability and statistics, states that as a sample size grows, its mean gets closer to the average of the whole population. In the 16th century, mathematician Gerolama Cardano recognized the Law of Large Numbers but never moved to prove it. In 1713, Swiss mathematician Jakob Bernoulli proved this theorem in his book Ars Conjectandi. It was later refined by other noted mathematicians, such as Pafnuty Chebyshev, founder of the St. Petersburg mathematical school. In a financial context, the law of large numbers indicates that a large entity that is growing rapidly cannot maintain that growth pace forever. The biggest of the blue chips, with market values in the hundreds of billions, are frequently cited as examples of this phenomenon.

### Understanding the Law of Large Numbers

In statistical analysis, the law of large numbers can be applied to a variety of subjects. It may not be feasible to poll every individual within a given population to collect the required amount of data, but every additional data point gathered has the potential to increase the likelihood that the outcome is a true measure of the mean.

In business, the law of large numbers relates to growth rates, stated as a percentage. The law of large numbers indicates that, as a business expands, the percentage rate of growth becomes increasingly difficult to maintain.

The Law of Large Numbers is not to be mistaken with the Law of Averages, which states that the distribution of outcomes in a sample (large or small) reflects the distribution of outcomes of the population.

### The Law of Large Numbers and Statistical Analysis

If a person wanted to determine the average value of a data set of 100 possible values, he is more likely to reach an accurate average by choosing 20 data points instead of relying on just two. For example, if the data set included all integers from 1 to 100, and sample-taker only drew two values, such as 95 and 40, he may determine the average to be approximately 67.5. If he continued to take random samplings up to 20 variables, the average should shift towards the true average as he considers more data points.

### Law of Large Numbers and Business Growth

In July 2015, the revenue generated by Wal-Mart Stores Inc. was recorded as $485.5 billion while Amazon.com Inc. brought in $95.8 billion during the same period. If Wal-Mart wanted to increase revenue by 50%, approximately $242.8 billion in revenue would be required. In contrast, Amazon would only need to increase revenue by $47.9 billion to reach a 50% increase. Based on the law of large numbers, the 50% increase would be deemed more difficult for Wal-Mart to accomplish than Amazon.

The same principles can be applied to other metrics, such as market capitalization or net profit. As a result, investing decisions can be guided based on the associated difficulties that companies with very high market capitalization can experience as they relate to stock appreciation.