# The Expected Value of M: What Happens as Sample Size Increases?

As sample size increases, the expected value of M (the mean of the distribution of M values) approaches the population mean. This is due to the law of large numbers, which states that as a sample size gets larger, the sample mean gets closer to the population mean.

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## Introduction

When we talk about the expected value of M, we are talking about the long-run average value of M. This value can be thought of as the average of all the possible values of M that could be realized. As the sample size increases, the expected value of M will get closer and closer to the population mean.

### What is the Expected Value of M?

In probability theory and statistics, the expected value of a random variable is the weighted average of all possible values that this random variable can take. In other words, if you have a six-sided die and you roll it, what is the average of all the numbers that come up? The answer is 3.5. This average value here is known as the expected value.

### What is the Law of Large Numbers?

The law of large numbers is a formal statement of the intuitive idea that the average of a sequence of random numbers will tend to be close to the expected value as the number of trials becomes large. More precisely, if X1,…,Xn is a sequence of independent and identically distributed random variables with expected value μ, then the sample mean converges in probability to μ as n approaches infinity:

limn→∞P(|X¯n−μ|>ε)=0 for all ε>0.

## Sample Size and the Expected Value of M

If you’re anything like me, you’ve probably wondered how the expected value of M changes as sample size increases. Does it go up? Does it go down? Does it stay the same? To answer these questions, we’ll need to take a quick look at the definition of expected value.

### How Does Sample Size Affect the Expected Value of M?

The Expected Value of M is a statistical tool that measures the center of a data set. It is also known as the mean or average. The Expected Value of M can be affected by two things: the population mean (μ) and the sample size (n).

The population mean is the average of all possible values in a population. For example, if you were to take the average height of all American adults, that would be the population mean. The population mean does not change when new data is added.

The sample size is the number of values you have in your sample. For example, if you were to take the heights of 100 American adults, that would be your sample size. The sample size can change when new data is added.

As the sample size increases, the expected value of M approaches the population mean. This happens because there are more values in the sample and so they better represent the entire population.

### What Happens as Sample Size Increases?

As sample size increases, the expected value of M approaches the population mean. This is because, as the number of samples gets larger, the distribution of M becomes more and more Normal. Recall that the Normal distribution is centered at the population mean. Therefore, as the number of samples gets larger, M will tend to be closer and closer to the population mean.

## Conclusion

As the sample size increased, the value of M tended to approach 3. This was to be expected, since M is the average of a random sample of numbers drawn from a population with a mean of 3. If the sample size is large enough, the value of M will be very close to 3.