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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

Mans
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  • @AaronHendrickson

    indeed in centimeters but what i don't understand how it describes the average variability in data set and it just takes the square root of variance ?

     – Mans Feb 12 '23 at 22:19
  • It is an estimate of the probability that the average is near the true mean. For normal dist., it is about 67%. Note - you are using N and n. If $\mu$ is true mean is known, N=n. If not N=n-1. – herb steinberg Feb 12 '23 at 22:41
  • Is this helpful? – J.G. Feb 12 '23 at 23:13
  • We want the average distance from the mean but if we took that we would get a zero standard deviation because values above the mean would cancel with values below the mean. We square the difference and then take the square root to get the average distance away from the mean, disregarding whether the values are above or below. – John Douma Feb 12 '23 at 23:43
  • @JohnDouma But when we take square root we do it for the $N$ too, which means that we took square root of the number of data points, if so how it could gave The average? – Mans Feb 13 '23 at 10:46

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