What happens to the variance as the sample size increases?

As a statistical expert with a deep understanding of probability distributions and statistical inference, I can provide a comprehensive explanation on the impact of increasing sample size on variance. When we talk about variance in the context of statistics, it's important to distinguish between the variance of the population and the variance of the sample. The variance of a population is a measure of how much the values in the population differ from the mean. On the other hand, the sample variance is a measure of how much the values in the sample differ from the sample mean. Now, let's delve into the effects of increasing the sample size: 1. Population Variance: The population variance itself is a fixed property of the population and does not change with the sample size. It is calculated using the entire population of data points. 2. Sample Variance: As the sample size increases, the sample variance might initially appear to increase as well because you are including more data points which could potentially have more variability. However, if the sample is truly representative of the population, the sample variance will tend to stabilize around the population variance as the sample size grows larger. This is due to the Law of Large Numbers, which states that as the sample size grows, the sample mean will converge to the population mean. 3. **Variance of the Sample Mean (Standard Error)**: The variance of the sample mean, often referred to as the standard error, is a measure of how much the sample mean is expected to vary from the true population mean. As the sample size increases, the standard error decreases. This is because the larger the sample size, the more information you have, and the sample mean becomes a more precise estimate of the population mean. The formula for the standard error of the mean is the standard deviation of the population (σ) divided by the square root of the sample size (n): \[ SE = \frac{\sigma}{\sqrt{n}} \] As you can see, the standard error is inversely proportional to the square root of the sample size. Therefore, as n increases, SE decreases. 4. Law of Large Numbers: The Law of Large Numbers is a principle that explains why the sample mean approaches the population mean as the sample size becomes very large. It implies that the larger the sample size, the more likely it is that the sample mean will be close to the population mean, thus reducing the variance of the sample mean. 5. Practical Implications: In practical terms, a decrease in the standard error means that you can be more confident in your estimates. For instance, in hypothesis testing or constructing confidence intervals, a smaller standard error allows for narrower confidence intervals, indicating greater precision in the estimation of the population parameter. 6. Caveats: It's important to note that while increasing the sample size generally leads to a decrease in the standard error, the quality of the sample is also crucial. If the sample is not representative of the population (due to bias in sampling), then even a large sample size will not yield accurate results. In conclusion, as the sample size increases, the variance of the sample mean (standard error) decreases, leading to more precise estimates of the population parameters. However, the population variance remains constant, and the sample variance will tend to stabilize around the population variance if the sample is representative.

I believe that the Law of Large Numbers explains why the variance (standard error) goes down when the sample size increases. ... As a sample size increases, sample variance (variation between observations) increases but the variance of the sample mean (standard error) decreases and hence precision increases.Aug 22, 2017