The Law of Large Numbers means that the error of a mean value will become insignificant as the number of observations increases.
The error of the mean is important because it indicates the average outcome. When enough data points are gathered, these averages can provide a useful description of the system.
For example, if you run a restaurant and you want to know how much revenue you can expect over the next year, you can use the Law of Large Numbers to estimate the amount of revenue you will generate.
Or, if you are a stock market analyst and you want to know the future price of a stock, you can use the Law of Large Numbers to estimate the average market price.
Or, if you are a sports manager and you want to know how likely your team is to win a game, you can use the Law of Large Numbers to estimate the likelihood of your team winning.
It doesn’t matter what you want to know. The Law of Large Numbers can be used to estimate anything. All you need are enough data points.
When you have a large number of data points, the Law of Large Numbers ensures that your data will be precise (within the margin of error) and, therefore, will not over-estimate or under-estimate the true value of the system.
There is one more concept that is essential to understand before moving on. This is the term covariance. You will learn more about covariance in the next lesson.
Covariance (or Variance)
Let’s take a step back for a moment and discuss the meaning of the word covariance (which, by the way, is also the plural form of covariance).
Covariance (cov in short) is a measure of the relationship between two variables. If you have two measurements (like temperature and light brightness) of the same location (like your house), you can calculate the covariance. This will tell you how much the measurements (e.g., light brightness) vary together (e.g., rise and fall, or increase and decrease).
If you have two measurements (like temperature and light brightness) of different locations (like your bedroom and living room), you can also calculate the covariance. This will tell you how much the measurements vary from each other (e.g., bedroom temperature and living room temperature).
In astronomy, the covariance is important because it indicates how well two variables (e.g., light and temperature) can be used to determine the third variable (e.g., the position of a planet).
The Error Of The Mean
What if I were to tell you that the Law of Large Numbers doesn’t apply to you?
What if I were to tell you that, as a rule, you should never make any kind of statistical estimate for anything, ever?
Or, what if I were to tell you that sometimes, your data might even cause you to under-estimate the true value of the system?
The error of the mean is more specific than the law of large numbers and, in many cases, it can be calculated with ease. If, for example, you have 100 observations and the mean value is 55, the error of the mean will be (55–100)/55 = (–4)/(4–1) = –4%.
The error of the mean is a very important concept to grasp. Many times, without realizing it, you will make the mistake of assuming that the data point with the mean value is the best representation of the entire population. This is far from true. In the example above, the data point with a mean of 55 does not represent the entire population. In fact, depending on what you want to estimate and how many data points you have, this number could be wildly inaccurate.
Law Of Small Numbers
While the Law of Large Numbers is useful and reliable in many cases, there are times when you will need to use the Law of Small Numbers. This is especially important if you have a small data set and you want to make sure that your numbers are precise.
The Law of Small Numbers can be used if you want to make sure that any estimate you make is as accurate as possible. For example, if you want to know what is the population of a certain country, you can use the Law of Small Numbers to get an idea of how many people are in that country without having to rely on inaccurate population figures.
In the above example, let’s say you are interested in the population of England. You have no idea how many people are in England, so you decide to make an informed guess and you decide to use the Law of Small Numbers. In this case, you would want to pick (at random) a value from the following range:
- 10 to 15
- 20 to 25
- 30 to 35
- 40 to 45
- 50 to 55
- 60 to 65
- 70 to 75
- 80 to 85
- 90 to 95
- 100 to 105
This will give you an idea of what is the most accurate estimate possible for the population of England. (You will need a higher number range if you are trying to be precise).
Mean Vs Marginal Analysis
When we talk about the Law of Large Numbers, we usually think about calculating the mean value of a group of data points. However, the Law of Large Numbers can also be applied to a single data point, called a marginal value.
For example, let’s say you have a population of 100 and you want to know what is the GDP per capita (Gross Domestic Product) for that country. You can use the Law of Large Numbers to get an idea of the mean value for the country. Or, you can use the Law of Large Numbers to get an idea of how much your own data point contributes to the mean value. (In this case, your own data point will have a mean value of 50 and a standard deviation of 10, so you will get a 50% contribution from your own data point to the mean value of the country).
Marginal analysis is important because it allows you to isolate and identify which data point (or values) contribute the most to the overall mean or sum value. In some cases, it can be difficult to identify which data point contributes the most to the mean value. For example, if we have two data points that are close to each other, it can be difficult to determine which one influences the mean value the most.
The Variance
If you are familiar with the concept of standard deviation, you know that it is the square root of the covariance. This is one way of thinking about variance. Another way is to think of variance as “the difference between the overall value and the mean value.” If you have two measurements (like temperature and light brightness) of the same location (like your bedroom), you can calculate the variance. This will give you a clear idea of how far (or close) your measurements are from the true value (the mean).
Let’s say you want to know the variance for two measurements (like temperature and light brightness) of different locations (like your kitchen and living room). You can use the Law of Large Numbers to get an idea of how far your measurements are from the mean. In this case, you would want to choose (at random) a value from the following range:
- 10 to 15
- 20 to 25
- 30 to 35
- 40 to 45
- 50 to 55
- 60 to 65
- 70 to 75
- 80 to 85
- 90 to 95
- 100 to 105
This will give you an idea of how accurate your estimate is (by isolating which data points contribute the most to the overall variance).
More Than Meets The Eye
There are a lot more important concepts to understand about the Law of Large Numbers than what we have discussed so far. However, before we move on, let’s discuss one more concept: convergence.
Convergence happens when the values of a given variable (e.g., the market price of a stock) become closer and, as a result, the variance (i.e., the difference between the maximum value and the minimum value) of that variable decreases.