Sampling and Estimation

Definition and Basic Theory

Sampling refers to the process of selecting a subset from a statistical population to estimate characteristics of the whole population.
An estimator is a statistic used to infer the value of an unknown parameter in the population based on sampled data.
The sampling distribution is the probability distribution of a given estimator based on all possible samples of a fixed size.
The standard error measures the statistical accuracy of an estimator. It’s the standard deviation of the sampling distribution.
Bias is the difference between the expected value of an estimator and the true value of the parameter being estimated. An estimator is unbiased if its expected value equals the true parameter value.

Key Properties

The law of large numbers states that as the sample size increases, the sample mean (or total) will get closer to the population mean (or total).
Central Limit Theorem (CLT) states that for a sufficiently large sample size, the distribution of the sample mean will approximate a normal distribution, regardless of the shape of the population distribution.
The confidence interval for a parameter is an interval computed from the sample data. The chosen level of confidence represents the frequency (i.e., the proportion) of possible confidence intervals that contain the true value of the unknown population parameter.
Margin of error is the range of values below and above the sample statistic in a confidence interval. It’s usually defined by the standard error and the chosen level of confidence.

Practical Applications

Sampling and estimation techniques are used in a variety of real-world applications including market research, quality control, and population studies where it’s impractical or impossible to collect data from every individual.
It’s used in hypothesis testing to draw conclusions about populations based on sample data.

Connection to Other Topics

Sampling and estimation are basic tools in statistical inference, the process of using data from a sample to make estimates or test hypotheses about a population.
It’s closely connected with other concepts such as probability, distributions, and hypothesis testing.

Understanding sampling and estimation is essential for conducting and interpreting a variety of statistical analyses. These topics underpin much of the methodology in fields that rely on data analysis, from social sciences to economics and business.