in my first course in Statistics when I took the measure of variation the first thing intoduced to me is :(The variance) which has this formula :
\begin{gather*} \sigma^2=\frac{1}{N}\sum_{i=1}^{n}(x_{i}-\mu)^2 \end{gather*}
and The variance tells us the average of squared distances from the mean, but the standard deviation is the square root of the variance given by this formula : \begin{gather*} \sigma=\sqrt{\frac{1}{N}\sum_{i=1}^{n}(x_{i}-\mu)^2} \end{gather*}
My question is : what did standard deviation tell us ? it takes only the square root of variance
I've searched online for the answer of my question but all the answers are like this :
Standard deviation tells us about the variability of values in a data set, It is a measure of dispersion
or like this
The standard deviation is the average amount of variability in your data set
but I don't understand How they came with these answers , all what i understand that is :standard deviation is just the square root of variance but i don't know what it really tell us