sqrtwidehatsigma2

The symbol $\sqrt{\widehat{\sigma}^2}$ represents the estimated standard deviation of a sample. The hat symbol (^) above the sigma indicates that $\sigma^2$ is an estimate of the population variance, not the true population variance itself. The sigma squared ($\sigma^2$) typically denotes variance, which is a measure of the spread or dispersion of data points around the mean. Taking the square root of the estimated variance, $\widehat{\sigma}^2$, results in the estimated standard deviation, $\sqrt{\widehat{\sigma}^2}$. This value quantifies the typical deviation of individual data points from the sample mean. It is a crucial statistic in inferential statistics, used for constructing confidence intervals, performing hypothesis tests, and understanding the variability within a dataset. Unlike the population standard deviation, which is usually unknown and denoted by $\sigma$, the sample standard deviation $\sqrt{\widehat{\sigma}^2}$ is calculated directly from observed data. The calculation of $\widehat{\sigma}^2$ typically involves summing the squared differences between each data point and the sample mean, and then dividing by the degrees of freedom, which is the sample size minus one ($n-1$). This division by $n-1$ instead of $n$ provides an unbiased estimate of the population variance.