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# Standard Deviation

Standard deviation is a number used to tell how measurements for a group are spread out from the average (mean or expected value). It measures how spread out a set of data is.

A low standard deviation means that most of the numbers are close to the average, while a high standard deviation means that the numbers are more spread out.

As we can see, in the left figure, all data points are very close to the mean and so less spread is there while in the right figure some data points are very far from the mean and so more spread is present there.

We calculate it by-

1. Find the squared difference of every data point from the mean `(Data value - mean)²` to check how far the data points are from the mean. We square the term so that it doesn't matter whether it is negative or positive because the data values greater than the mean will make this term negative and when we sum all the terms, the negatives will cancel the positives resulting in an error. Also, we want to measure the dispersion disregarding which side of the mean the value occurs.

2. Now, if there will be more deviations, then the sum will become bigger. That's why we take the average of the sum of squared differences so that the number of instances doesn't matter anymore:- `sum((Data value - mean)²)/N`.

3. At last, we take the square root of the average to bring the standard deviation to the same unit as the original data or mean. Remember, if we don't take the square root of the average, then that term is called variance. Standard deviation is the square root of the variance:- `sqrt(sum((Data value - mean)²)/N)`

So its formula comes as-

where

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