A Population Parameter is a value that describes a characteristic of an entire population. Unlike a sample statistic, which is derived from a subset of the population, a population parameter encompasses data from every member of the population under consideration. Population parameters can include measures such as the population mean, variance, and proportion.
Types of Population Parameters
Population Mean (\(\mu\))
The population mean, denoted as \(\mu\), is the average of all individual values in the population. It is calculated as:
where \(N\) is the population size and \(X_i\) represents each individual value in the population.
Population Variance (\(\sigma^2\))
The population variance, denoted as \(\sigma^2\), measures the dispersion or variability of the population. It is defined as:
Population Proportion (\(P\))
The population proportion, denoted as \(P\), refers to the fraction of the population that possesses a certain attribute. It can be expressed as:
where \(X\) is the number of members with the characteristic of interest and \(N\) is the total number in the population.
Significance in Statistics
Population parameters are critical in the field of statistics as they provide a true measure of the population characteristics. However, estimating these parameters is often challenging due to the impracticality of accessing and measuring every member of the population. Hence, sample statistics, which are estimates of population parameters, are commonly used.
Sample Statistic vs. Population Parameter
- Sample Statistic: An estimate derived from a sample subset of the population (e.g., sample mean (\(\bar{x}\)), sample variance (\(s^2\))).
- Population Parameter: A true, fixed value describing the characteristic of the entire population (e.g., population mean (\(\mu\)), population variance (\(\sigma^2\))).
Examples and Applications
Example 1: Population Mean
If a company wants to know the average age of all its employees (924 employees), the actual computed average age represents the population mean.
Example 2: Population Proportion
In a survey about voting preferences conducted in an entire city, the proportion of people favoring a particular candidate represents the population proportion.
Historical Context
The concept of population parameters has been fundamental to the development of statistical theory and methodologies. Pioneers such as Karl Pearson and Ronald Fisher developed key statistical techniques, allowing estimation and inference about population parameters using sample data.
Related Terms
- Parameter Estimation: The process of using sample data to estimate population parameters.
- Census: A complete enumeration of the population, often used to gather population parameters directly.
- Inferential Statistics: A branch of statistics that focuses on making inferences about population parameters based on sample data.
FAQs
What is the difference between a sample and a population?
How do you find the population parameter?
Why are population parameters important?
References
- L. Wasserman, “All of Statistics: A Concise Course in Statistical Inference”, Springer, 2004.
- D. Freedman, R. Pisani, R. Purves, “Statistics”, W.W. Norton & Company, 2007.
Summary
Population parameters are essential for understanding the entire population’s characteristics and are vital in both theoretical and applied statistics. While directly obtaining these parameters can be challenging, statistical inference and sample data provide mechanisms for estimating and making informed decisions based on these estimates.
