Estimation and Types of Estimation in Statistics

The Post is about Introduction to Estimation and Types of Estimation in the Subject of Statistics. Let us discuss Estimation and Types of Estimation in Statistics.

The procedure of making a judgment or decision about a population parameter is referred to as statistical estimation or simply estimation.  Statistical estimation procedures provide estimates of population parameters with a desired degree of confidence. The degree of confidence can be controlled in part, by the size of the sample (larger sample greater accuracy of the estimate) and by the type of estimate made. Population parameters are estimated from sample data because it is not possible (it is impracticable) to examine the entire population to make such an exact determination.

The Types of Estimation in Statistics for the estimation of the population parameter are further divided into two groups (i) Point Estimation and (ii) Interval Estimation

Point Estimation

The objective of point estimation is to obtain a single number from the sample which will represent the unknown value of the population parameter. Population parameters (population mean, variance, etc) are estimated from the corresponding sample statistics (sample mean, variance, etc).
A statistic used to estimate a parameter is called a point estimator or simply an estimator, the actual numerical value obtained by an estimator is called an estimate.

A population parameter is denoted by $\theta$ which is an unknown constant. The available information is in the form of a random sample $x_1,x_2,\cdots,x_n$ of size $n$ drawn from the population. We formulate a function of the sample observation $x_1,x_2,\cdots,x_n$. The estimator of $\theta$ is denoted by $\hat{\theta}$. The different random sample provides different values of the statistics $\hat{\theta}$. Thus $\hat{\theta}$ is a random variable with its sampling probability distribution.

Interval Estimation

A point estimator (such as sample mean) calculated from the sample data provides a single number as an estimate of the population parameter, which can not be expected to be exactly equal to the population parameter because the mean of a sample taken from a population may assume different values for different samples. Therefore, we estimate an interval/ range of values (set of values) within which the population parameter is expected to lie with a certain degree of confidence. This range of values used to estimate a population parameter is known as interval estimate or estimate by a confidence interval, and is defined by two numbers, between which a population parameter is expected to lie.

For example, $a<\bar{x}<b$ is an interval estimate of the population mean $\mu$, indicating that the population mean is greater than $a$ but less than $b$. The purpose of an interval estimate is to provide information about how close the point estimate is to the true parameter.

Types of Estimation

Note that the information developed about the shape of a sampling distribution of the sample mean i.e. Sampling Distribution of $\bar{x}$ allows us to locate an interval that has some specified probability of containing the population mean $\mu$.

Interval Estimate formula when $n>30$ and Population is normal $$\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}$$

Interval Estimate formula when $n<30$ and Population is not normal $$\bar{x} \pm t_{(n-1, \alpha)}\,\, \frac{s}{\sqrt{n}}$$

Which of the two types of estimation in Statistics, do you like the most, and why?

The Types of Estimation in Statistics are as follows:

  • Point estimation is nice because it provides an exact point estimate of the population value. It provides you with the single best guess of the value of the population parameter.
  • Interval estimation is nice because it allows you to make statements of confidence that an interval will include the true population value.

Read about the Advantages of Interval Estimation in Statistics

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