Inferential Statistics Terminology

This post is about Inferential Statistics (or statistical inference) and some of its related terminologies. This is a field of statistics that allows us to understand and make predictions about the world around us.

Parameter and Statistic

Any measurable characteristic of a population is called a parameter. For example, the mean of a population is a parameter. OR

Numerical values that describe the characteristics of a whole population are called parameters, commonly presented in Greek Letters.

Any measurable characteristic of a sample is called a statistic. For example, the mean of a sample is a statistic. OR

Numerical measures describing the characteristics of a sample are called statistics, represented by Roman Letters.

Population and Sample

Population: The entire group of individuals, objects, or data points that one is interested in studying. A population under study can be finite or infinite. However, often too large or impractical to study directly.

Sample: A smaller, representative subset of the population. It is used to gain insights about the population without having to study every member. A sample should accurately reflect the characteristics of the population.  

Inference

A Process of drawing conclusions about a population based on the information contained in a sample taken from that population

Estimator

An estimator is a rule (method, formula) that tells how to calculate the value of an estimate based on the measurements contained in a sample. The sample mean is one possible estimator of the population mean $\mu$.

An estimator will be a good estimator in the sense that the distribution of an estimator is concentrated near the value of the parameter.

Estimate

Estimate is a way to use samples. There are many ways to estimate a parameter. Estimates are near to reality (biased or crude). Decisions are very accurate if the estimate is near to reality.

$X_1, X_2, \cdots, X_n$ is a sample and $\overline{X}$ is an estimator. $x_1, x_2, \cdots, x_n$ are sample observation and $\overline{x}=\frac{\Sigma x_i}{n}$ is an estimate.

Estimation

Estimation is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable.

Statistical Inference (or Inferential Statistics)

Any process (art) of drawing inferences (conclusions) about the population based on limited information contained in a sample taken from the same population is called statistical inference (or inferential statistics). It is difficult to draw an inference about the population because the study of the entire universe (population) is not simple. To get some idea about the characteristics (parameters) of the population, we choose a part of a reasonable size, generally, referred to as a sample (by some appropriate method).

Statistical inference is a powerful set of tools used to conclude a population based on data collected from a sample of that population. It allows us to make informed decisions and predictions about the larger group even when we have not examined every single member.

Why Estimate?

• Speed: Often, an estimate is faster to get than an exact calculation.
• Simplicity: It can simplify complex problems.
• Decision-Making: Estimates help one to make choices when one does not have all the details.
• Checking: One can use estimates to check if a more precise answer is reasonable.

Why is Statistical Inference Important?

• Decision-making: It helps us make informed decisions in various fields, such as medicine, business, and social sciences.
• Research: It is crucial for conducting research and drawing meaningful conclusions from data.
• Understanding the World: It allows us to understand and make predictions about the world around us.

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