Formula of Population Variance
The population variance is a fundamental statistical measure that quantifies the dispersion or variability of a dataset around its mean. Whether dealing with grouped or ungrouped data, understanding the population variance formula is essential for analyzing and interpreting the spread of data points within a population.
Ungrouped Data
Ungrouped data, also known as raw data, consists of individual data points that are not categorized or grouped into intervals. Each data point in ungrouped data represents a distinct value or observation.
Formula of Population Variance in Ungrouped Data:
[Tex]\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i – \mu)^2 [/Tex]
Where:
- σ2 is the population variance.
- N is the total number of data points in the population.
- xi represents each individual data point.
- μ is the population mean (average of all data points).
- ∑ denotes the summation of all terms from i = 1 to N.
Grouped Data
Grouped data refers to a dataset where individual data points are grouped or categorized into intervals or classes. Each interval represents a range of values, and the frequency of data points falling within each interval is recorded.
Formula of Population Variance in Grouped Data:
[Tex]\sigma^2 = \frac{1}{N} \sum_{i=1}^{n} f(m_i – \bar{x})^2 [/Tex]
Where:
- σ2 is the population variance.
- N is the total number of data points in the population.
- f is the frequency of occurrence of an observation.
- mi is the midpoint of the ith interval.
- [Tex]\bar{x}[/Tex] is the mean for grouped data.
Population Variance
Population variance is a fundamental concept in statistics that quantifies the average squared deviation from the mean of a set of data points in a population. It is a measure of how spread out a group of data points is.
There are two types of data available, namely, ungrouped and grouped data. Thus, there are two formulas to calculate the population variance. In this article, we will learn more about population variance, its formulas, and various associated examples.
Table of Content
- What is Population Variance?
- Formula of Population Variance
- Ungrouped Data
- Grouped Data
- Population Variance and Sample Variance
- Population Variance and Standard Deviation
- Solved Problems on Population Variance
- Practice Questions on Population Variance
- FAQs on Population Variance