Law of Large Numbers

When using statistics to make predictions or test hypotheses about a population, the more members of that population that can be included in the sample, the more accurate your predictions or tests are likely to be. (If you use the entire population in your study, there would be no need to predict or test, but this rarely is the case.) The reason bigger is better in statistical analysis is the law of large numbers.

Law of Large Numbers

Basically, the law of large numbers says that the more data you have, the more likely you will achieve the “true” values or statistics that represent the population. Let’s consider a simple experiment, that of flipping a fair coin. Each time you flip the coin, there is a 50% chance it will land head up and a 50% chance it will land tail up. So, over time, 50% of the coin flips should be heads and 50% tails.

Let’s start with 10 flips of a coin using a computer simulation. Flipping the coin 10 times resulted in the following number of heads each time:

7              7              2              4              4              1              4              5              3              6

This gives an average of 4.3 heads. That is close to the expected value of 5 heads but still has a 14% error.

Now let’s see what happens if the coin is flipped 100 times (and repeat this 10 times). The result may be:

55           43           50           61           55           55           50           59           41           53

The average is 52.2 heads. That is very close to the average of 50, but still has an error of 4.4%.

If the coin is flipped 1,000 times and the 10 times is repeated. The result may be:

508         512         465         514         498         519         511         466         472         481

The average is 494.6 heads. This is close to the 500 heads average we would expect. The error is only 1.08%. As we continue with larger and larger numbers of coin flips, the average number of heads will get closer and closer to 50% of the total number of flips. That is, we will approach 0% error. The graph below shows how the percentage of heads in each trial stays closer to 50% when the number of coin flips is greater.

Law of Large Numbers

The Law of Large Numbers in Action

Let’s take a look at a few real life examples of the law of large numbers in action.

How does a company that produces light bulbs determine how long a bulb should last? They light up a lot of light bulbs and wait for them to burn out. Or, more accurately, they determine how many light bulbs have burnt out after a certain amount of time. For a test like this, they may use 100,000 light bulbs.

Casinos take advantage of the law of large numbers to make a profit. Every game of chance in some way favors the casino. So with a large number of gamblers, the casino will take in more money than it pays out. A simple example of this is the game of roulette. In this game, there are 38 numbered slots. Of these, 18 are even numbers, 18 are odd numbers, and two are numbered 0 and 00. So for each “even” and “odd” bet placed, the chance of winning is 18 out of 38, which is 47.37%. (If there were only 36 slots, 18 even and 18 odd, then the chance of winning is 50%, and the casino will not make money on the game in the long run.)

Roulette at the casino

The American roulette wheel has 38 slots, 18 with even numbers, 18 with odd numbers, and two numbered 0 and 00. There are also 18 black slots, 18 red slots, and two green slots. The two green slots are numbered 0 and 00; the red and black slots are divided evenly between the even and odd numbers.

The 47.37% figure means that, overall, the casino is going to win 52.67% of the time on these types of bets, giving the casino a 5.3% advantage. So while any individual player has a good chance of winning (after all, 47.37% isn’t too far from 50%), with a large number of players, the casino will keep a great deal of money. For example, if 10,000 individual bets are placed on even or odd, the casino will win 5.3% x 10000 = 350 more times, on average, than it loses.

Personal Financing

The law of large numbers can also be considered in more personal terms. As most financial experts know, it’s not good to put all your eggs in one basket. Investing your money in a few stocks, for example, is very risky because the variation in growth and losses can be great. While you can make a killing if a stock does great, with just a few stocks, you are going to experience wide fluctuations and the potential for great loss is just as good as the potential for great gain.

Mutual funds are aimed at employing the law of large numbers to counteract those fluctuations and to provide more modest overall growth over the long run. So while an individual may invest in a small number of different companies, a mutual fund can invest in hundreds of companies. The law of large numbers says that, in the long run, mutual fund returns will be at a steady figure that is close to the average return of all funds in the category in which the fund invests or one that tracks a certain standard, such as the S&P 500.

Research Implications

For anyone doing research, the implications of the law of large numbers is that the more subjects (e.g., people) you can include in your sample, the better your results will represent what is really happening with the population. For example, if you are doing a study on income in a city, you will not only want to collect income figures from all types of neighborhoods but also from a large number of people or households. In a city of 200,000 people, for example, your data will be much more accurate if you collect the necessary information from 5,000 people as it would be if you collect it from 500 people. So think big when it comes to samples.

If you are interested in learning more about the law of small numbers, consider pursuing a degree in data science or data analysis.