Variance in Statistics: Definition, Formula, Calculation
Table of Content:
Variance - Statistics
The formula for the variance of a population is:
\begin{equation} \sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2 \end{equation}where:
- \( \sigma^2 \) is the population variance
- \( N\) is the total number of observations in the population
- \( x_i\) is the value of the $i\) th observation
- \( \mu\) is the population mean
Note that the only difference between the population variance formula and the sample variance formula is that the sample variance divides by \(n-1\) instead of \(N\) . This is because using \(n-1\) instead of \(N\) gives an unbiased estimate of the population variance when the sample size is small.
The relationship between standard deviation and variance:
Here's an example of how to write about the relationship between standard deviation and variance:
The variance and standard deviation are both measures of the spread of a dataset. The variance is the average of the squared differences of each data point from the mean, while the standard deviation is the square root of the variance. The formula for the population variance is:
\begin{equation} \sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2 \end{equation}where \(\sigma^2\) is the population variance, \(N\) is the total number of observations in the population, \(x_i\) is the value of the \(i\) th observation, and \(\mu\) is the population mean. The formula for the population standard deviation is:
\begin{align} \sigma &= \sqrt{\frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2} \\ \sigma &= \sqrt{\sigma^2} \\ Standard \: Deviation &= \sqrt{ variance } \end{align}In other words, to find the standard deviation, you take the square root of the variance. Conversely, to find the variance, you square the standard deviation. This relationship holds for both population and sample data.