Mainly used to estimate the population mean when population mean is not known as they have the same expected value.

The concept you are referring to is the use of the sample mean ($\bar{x}$) as an estimator for the population mean ($\mu$) when the population mean is unknown.

1. Definition of Sample Mean: The sample mean is calculated by taking the sum of the individual observations in a sample and dividing it by the number of observations. Mathematically, it can be represented as: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$ where $n$ is the number of observations, and $x_i$ represents each individual observation.

2. Properties of the Sample Mean: One of the key properties of the sample mean is that it is an unbiased estimator of the population mean. This means that the expected value of the sample mean equals the population mean: $E(\bar{x}) = \mu$ This property holds true under random sampling, making the sample mean a reliable estimator of the population mean.

3. Application: In practice, when researchers gather a sample from a larger population (where population parameters are unknown), they can use the sample mean to estimate the population mean. This is particularly useful in various fields, including statistics, social sciences, and quality control.

In summary, when the population mean is unknown, the sample mean serves as a robust estimator with the same expected value as the population mean, facilitating statistical analysis and decision-making based on sample data.