Why do we need Standard deviation when we already have variance and vice versa?


When I learnt about the variance and Standard deviation, I could not stop myself from wondering why do we even need one when we already have another, as they are completely related to each other. Because knowing one is equivalent to knowing the other.

As we know, \(\sqrt[]{variance}\) = \(std\:dev\)

Variance is the average squared differences from the mean. But, do you know why do we take squared differences instead of taking absolute differences? Comment below and I will write about it (link).

Now back to the main question, the answer is there are many situations where the standard deviation is far more useful than the variance and vice versa.

One of the reason is that the standard deviation is expressed in the same units as the mean, whereas the variance is expressed in squared units, but for analysing a distribution, you can use whatever suits you.

e.g. Assume we have heights of students in a class, and the mean height is 165 cm. The unit of Mean is the same as the unit of data i.e. cm. Now, the variance is having the unit as square centimetre, which is hard to interpret in this case, whereas the unit of std dev is centimetre.

let in this case we find that std dev is 20 cm, so we can easily interpret the data compared to knowing that the variance is 400 square centimetre.

More Reasons:

You use Std dev to explain the percentage of data points lying in a range. (68.2% of the data points lie in the 1st standard deviation from the mean in a normal distribution).

Okay, so std dev is very useful, then when do we use variance, one of the main reason is that the variance has a property that if we add two independent variables, then their variance also adds up. Note: it only applies to independent random variables.

But it does not happen with the standard deviation.

\(variance(x+y) = variance(x) + variance(y)\)

(property of variance)

\((SD(x+y))^2 = (SD(x))^2 + (SD(y))^2\)

but,  \(SD(x + y)\) < \(SD(x)\) + \(SD(y)\)

because by the laws of algebra square roots cannot be added.

e.g.  \(\sqrt[]{16 + 25}\) < \(\sqrt[]{16}\) + \(\sqrt[]{25}\)


To sum up it all, we can say that we need both of them to make our lives easier so that we can use whatever is more appropriate in the given conditions.

Thank you for reading. Leave a comment if you have any question or suggestion.

Image src: https://mathbitsnotebook.com/Algebra2/Statistics/STstandardDeviation.html

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