As a statistical expert with a deep understanding of the principles of sampling and statistical inference, I can provide a comprehensive explanation of the impact of increasing sample size on the mean of a sample.

When we talk about the mean of a sample, we are referring to the average of the values in that sample. The sample mean is a crucial statistic because it serves as an estimate for the population mean, which is the average of all the values in the entire population. The relationship between the sample size and the sample mean is fundamental to the field of statistics, and it is governed by several key principles.

Firstly, it is important to understand that the sample mean is a random variable. This means that if we were to take multiple samples from the same population, each sample would yield a different mean due to the inherent variability in the data. However, as the sample size increases, several important things happen that affect the relationship between the sample mean and the population mean.

### Central Limit Theorem (CLT)

The Central Limit Theorem is a fundamental theorem in statistics that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. This theorem is crucial because it allows us to make inferences about the population based on sample data.

### Law of Large Numbers

The Law of Large Numbers complements the CLT by stating that as the sample size becomes very large, the sample mean will converge on the population mean. This means that the larger the sample size, the more accurate our estimate of the population mean will be.

### Standard Error

The standard error of the mean is a measure of how much the sample mean is expected to vary from the true population mean. It is calculated as the standard deviation of the population divided by the square root of the sample size. As the sample size increases, the standard error decreases. This means that the sample mean becomes a more precise estimate of the population mean as the sample size grows larger.

### Variance and Sample Size

The variance of the sample is another important concept. When we increase the sample size, the variance of the sample distribution (also known as the sampling distribution of the mean) decreases. This is because larger samples are more likely to contain a representative mix of the population's values, thus reducing the variability of the sample mean.

### Shape of the Sampling Distribution

As mentioned in the reference content, when we increase the sample size, the sampling distribution of the mean will tend to become more normal, even if the original population distribution is not normal. This is a direct consequence of the CLT. A normal distribution is symmetrical and has a single peak, which means that the sample means are more likely to be close to the population mean as the sample size increases.

### Practical Implications

In practical terms, increasing the sample size can lead to more reliable and generalizable results. It can also increase the statistical power of a study, which is the ability to detect an effect if there is one. This is particularly important in hypothesis testing, where a larger sample size can reduce the likelihood of making a Type II error (failing to reject a false null hypothesis).

### Conclusion

In conclusion, increasing the sample size has a profound impact on the mean of a sample. It brings the sample mean closer to the population mean, reduces the standard error, decreases the variance of the sampling distribution, and leads to a more normal distribution of sample means. These properties make larger samples more representative and reliable for statistical inference.

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When sampling from a population, the sample mean will always be closer to the population mean as the sample size increases. My explanation is that as sample size increases, the standard error decreases and the sampling distribution will become normal.Mar 4, 2013read more >>