How does sample mean change with sample size?

As sample size increases, the sample mean becomes a more accurate estimate of the population mean, meaning its variability decreases.

In more detail, the relationship between sample size and the behavior of the sample mean is governed by the Central Limit Theorem and the Law of Large Numbers. Here's a breakdown:

Law of Large Numbers: This theorem states that as the sample size increases, the sample mean converges towards the population mean. Essentially, with more data points, the sample mean provides a more reliable representation of the true population average.
Central Limit Theorem (CLT): The CLT states that regardless of the distribution of the population, the distribution of sample means will approach a normal distribution as the sample size increases. Furthermore, the standard deviation of this distribution of sample means (known as the standard error) is inversely proportional to the square root of the sample size.

Therefore:

Standard Error Decreases: The standard error (SE) quantifies the variability of the sample means around the population mean. The formula for standard error is:
```
SE = σ / √n
```
Where:
- σ (sigma) is the population standard deviation.
- n is the sample size.
As n (sample size) increases, SE decreases. This means the sample means from larger samples are clustered more closely around the population mean.
Increased Accuracy: A smaller standard error implies higher precision. The sample mean becomes a more reliable and accurate estimator of the population mean with larger sample sizes.

Example:

Imagine trying to estimate the average height of all adults in a city.

Small Sample (n=10): If you randomly select only 10 people, your sample mean height might be quite different from the true average height due to random chance (you might happen to pick mostly tall people or mostly short people).
Large Sample (n=1000): If you randomly select 1000 people, your sample mean height is much more likely to be close to the true average height of all adults in the city. The larger sample size smooths out the effects of random variation.

In summary: As the sample size grows larger, the standard error of the sample mean decreases, and the sample mean provides a more precise and reliable estimate of the population mean. This is a fundamental principle in statistical inference, allowing us to make accurate predictions and draw valid conclusions about populations based on sample data.