What is the difference between the population distribution, the distribution of the same, and the sampling distribution and what are the formulas to get the normal distribution of each?
The population distribution, the distribution of the sample, and the sampling distribution are different concepts in statistics.
1. Population Distribution: The population distribution refers to the distribution of a specific variable in an entire population. In other words, it represents the pattern of values that a variable takes within a particular population. To understand the population distribution, you can collect data from each individual in the population and analyze it. However, in practice, it is often not feasible or practical to collect data from an entire population.
2. Distribution of the Sample: The distribution of the sample refers to the distribution of a specific variable in a sample drawn from a population. A sample is a subset of individuals from a population. By analyzing the sample, we can make inferences about the population. The distribution of the sample provides insights into the characteristics of the variable within the sample, and it can often resemble the population distribution. When analyzing a sample, we use statistical techniques to estimate population parameters based on sample statistics.
3. Sampling Distribution: The sampling distribution refers to the distribution of a sample statistic calculated from multiple samples taken from the same population. It provides information about the behavior of a particular statistic when repeatedly sampling from the population. The concept of the sampling distribution is important in statistical inference because it helps us understand the sampling variability of a statistic.
Now, let's discuss the formulas for obtaining the normal distribution of each:
1. Population Distribution: If you have access to the entire population data, you can construct a histogram or use statistical software to plot the distribution of the variable. However, there is no specific formula to obtain the normal distribution of the population since it depends on the data available.
2. Distribution of the Sample: To examine the distribution of a sample, you can also create a histogram using the sample data. Similarly, there isn't a specific formula for obtaining the normal distribution of the sample. However, based on the Central Limit Theorem (CLT), if the sample size is sufficiently large (typically n ≥ 30) and the sample is drawn randomly from the population with a finite mean and standard deviation, the distribution of the sample mean tends to be approximately normal.
3. Sampling Distribution: The sampling distribution depends on the specific statistic of interest. For example, the most common statistic used is the sample mean (x̄). If the population distribution is normal, the sampling distribution of the sample mean will also be normal, regardless of the sample size. The formula for the mean of the sampling distribution is simply the population mean (μ). The standard deviation (σ) of the sampling distribution, also known as the standard error (SE), is determined by dividing the population standard deviation by the square root of the sample size (n). So, the formula for the standard deviation of the sampling distribution is σ/√n.
In summary, while the population distribution and the distribution of the sample depend on the available data, the sampling distribution can be approximated to a normal distribution if certain conditions are met, particularly when the sample size is large or the population distribution is already normal.