The law of large numbers is a concept that is often misunderstood in statistics. In this lesson, you will learn the real meaning of the law of large numbers and how it is employed.
The Law of Large Numbers
Sharon is an insurance agent for a large company. Her company claims they've run the numbers and can save you 17% on your puppy insurance in 20 minutes or less. Because everyone needs to insure their puppy, right? Right, but I don't know if Sharon's company can really keep costs down because they have the best data. How do they know?
Sharon tells me that their rates are based on the statistic that 1 in every 20 puppies has an accident every day. But how do I know this is true?
In this lesson, you will learn about the law of large numbers and how it applies to real-life scenarios.
Defining the Law of Large Numbers
The law of large numbers is a theorem that states that the larger your sample size, the closer the sample mean will be to the mean of the population. It is also called Bernoulli's Law after Jacob Bernoulli, a Swiss mathematician. Sharon says that her company uses the law of large numbers to determine puppy insurance rates.
Let's show an example of the law in action. You probably know the probability of selecting a king out of a deck of cards is 4/52, or approximately 8%.
Let's say you and a friend were to randomly start pulling one card out of a deck, replacing it each time and tracking which card you pulled. Maybe you pull a king out two out of the first four draws. Then your percentage of pulling a king would be 50%. Your friend pulls no kings out in the first four draws.
However, after 20 draws, you have still only pulled the king twice. Now your percentage of pulling a king would be 10%. Meanwhile, your friend only pulls a king out one time out of twenty, giving a percentage of 5%. This illustrates the law of large numbers: the more cards you pull, the closer your percentage will match the probability of pulling a king. In this example, the population mean is the probability that the king will be drawn from the deck of cards (again, roughly 8%). The sample mean results from the experiment being repeated over and over, and the mean changes as the experiment is repeated.
Essentially, the larger your sample size, the more accurate your information. This is very different from the law of averages, which is a made-up and illogical law. Many people confuse the law of large numbers as meaning that the more often you conduct an experiment, the chance of a certain outcome happening to 'balance' or average out the odds increases. This is not true. If you pull a card from a deck of 52, put the card back, and then draw a second card, your chances of drawing a king are still 4 out of 52 - they're not any better or worse depending on what you drew before. For more information on this topic, please check out our chapters on probability.
Another way to put this is that the law of large numbers shows a relationship between the theoretical probability and the experimental probability of something happening. The law of large numbers states that the more often an experiment is repeated, the difference between the theoretical probability and the experimental probability is closer to zero.
The Law of Large Numbers in Life
I want you to think of the law of large numbers as a person walking on a tight rope. Think of perfect balance as the theoretical average. The person represents the experimental mean. The more an experiment is repeated, the better balance the person has. Mathematically, the experimental mean can be either larger or smaller than the theoretical mean, but that difference gets smaller and smaller as the experiment is repeated.
The law of large numbers is very helpful in determining the accuracy of certain predictions. In the case of Sharon and her puppy insurance, we can assume that a big company can collect data over millions of puppies and then use that data to make predictions about the population. According to the law of large numbers, the more data Sharon's company collects, the more accurate their sample mean is to the population mean.
Overall, it's important to remember the larger a sample, the more accurate the prediction.
Sharon's company claims that they know how to predict puppy accident rates. She can predict this by using the the law of large numbers, which is a theorem that states that the larger your sample size is, the closer the sample mean will be to the mean of the population. The law of large numbers shows a relationship between the theoretical probability and the experimental probability in an experiment. It states that the more often an experiment is repeated, the difference between the theoretical probability and the experimental probability grows closer to zero.
The law of large numbers is used in many areas of life. It can help determine accuracy in predictions that are being made, such as predicting the number of puppy accidents that occur every day. Oops, I think I'll go get that insurance now.
Once you've finished with this lesson, you will have the ability to:
- Describe the law of large numbers
- Explain how the law of large numbers can be used to make predictions