How do we compute sample size fluctuations? Suppose we have a population of 16000 and a sample size of 30, and the number of samples 1000, what will be the impact on population mean and standard deviation when the population is normally distributed? When the population is skewed? And when the population shows uniform distribution what will happen to the standard deviation and mean (increase/decrease)?

To compute sample size fluctuations, we can use the formulas for standard deviation and standard error.

1. Population Mean:
The population mean remains the same regardless of the sample size. So the impact on the population mean will be zero in all cases.

2. Population Standard Deviation:
When the population is normally distributed, the standard deviation of the sample mean (also known as the standard error) can be calculated using the formula:
Standard Error = Population Standard Deviation / √(Sample Size)

So, in your case, for a sample size of 30 and 1000 samples, the impact on the standard deviation of the sample mean will be as follows:

a) Normally Distributed Population:
The standard deviation of the sample mean will be Population Standard Deviation / √30 for each sample. Fluctuations in the sample mean will follow a normal distribution.

b) Skewed Population:
If the population is skewed, the Central Limit Theorem generally implies that as the sample size increases, the distribution of the sample mean will approach normality. However, the skewness of the population may still impact the precision of the sample mean estimation.

c) Uniformly Distributed Population:
For a uniformly distributed population, the standard deviation of the sample mean is given by the range of the population values divided by √(12 * Sample Size). As the sample size increases, the standard deviation of the sample mean decreases, leading to a reduction in the variability of the sample mean.

In summary, for a normally distributed population, the standard deviation of the sample mean decreases as the sample size increases. For skewed populations, the impact can be more complex, as the skewness of the population may affect the precision of the sample mean estimation. Lastly, for a uniformly distributed population, the standard deviation of the sample mean decreases with an increase in the sample size.

To compute sample size fluctuations, we need to understand the characteristics of the population and the properties of the sample. Let's break down each scenario and explain how to analyze the impact on population mean and standard deviation.

1. Population is normally distributed:
- In this case, the sample mean is an unbiased estimate of the population mean. So, the sample mean will be close to the population mean on average.
- As the number of samples increases, the sample means will tend to fluctuate less around the population mean. The standard deviation of the sample means, also known as the standard error of the mean, decreases as the square root of the sample size increases. It follows the formula: Standard Error of the Mean = Standard Deviation / √(Sample Size).
- Therefore, as the number of samples increases while keeping the sample size constant, the population mean remains the same, but the standard deviation of the sample means decreases.

2. Population is skewed:
- If the population distribution is skewed, the sample mean may still be an unbiased estimate of the population mean, but it may not represent the population accurately.
- As the number of samples increases, the sample means will still converge closer to the population mean. However, the standard error of the mean may not decrease as much as in the case of a normal distribution.
- Due to the skewness of the population, the sample means may vary more, which can result in larger fluctuations around the population mean. This is particularly true if the sample size is small.

3. Population shows a uniform distribution:
- In a uniform distribution, every value in the population has the same probability of occurring. It is a flat distribution without any skewness.
- When we take multiple samples from a uniformly distributed population, the sample means will fluctuate around the population mean, but they will tend to cluster symmetrically.
- The standard deviation of the sample means will depend on the sample size. As the sample size increases, the standard deviation of the sample means decreases, following the same relationship as in the case of a normal distribution.

In summary:
- For a normally distributed population, the standard deviation of the sample means decreases as the number of samples increases.
- For a skewed population, the sample means may still be an unbiased estimate of the population mean, but the standard error and fluctuations can be larger, especially for smaller sample sizes.
- For a uniformly distributed population, the standard deviation of the sample means decreases as the sample size increases, with symmetric fluctuations around the population mean.

Remember, these explanations are based on general principles, and the precise calculations for sample size fluctuations can vary depending on the specific statistical analysis and assumptions made.