In the context of statistical inference, why is it necessary to distinguish between parameters and statistics when seeking information about a population?

In the context of statistical inference, it's necessary to distinguish between parameters and statistics when seeking information about a population for several reasons:

1. Definition: A parameter is a numerical characteristic of a population, while a statistic is a numerical characteristic of a sample. For example, the mean of a population is a parameter, while the mean of a sample is a statistic.

2. Purpose: Parameters are used when we want to describe the entire population, while statistics are used when we only have a sample of the population.

3. Variability: Parameters are fixed as they represent the entire population, while statistics can vary from sample to sample.

4. Estimation: We often use statistics to estimate parameters. For example, we might use the mean of a sample (a statistic) to estimate the mean of the population (a parameter).

5. Uncertainty: Since statistics are based on samples, they come with a degree of uncertainty. This uncertainty is quantified using confidence intervals and margins of error. Parameters, being fixed values, do not have this uncertainty.

In summary, distinguishing between parameters and statistics is crucial in statistical inference because it helps us understand the difference between the information we have (statistics from our sample) and the information we want to know (parameters about our population).