Can a sample have a standard deviation of zero?

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Oct 21, 2015

It is possible but (in my opinion) only if a sample consists of the same data.

Explanation:

It is possible but (in my opinion) only if a sample consists of the same data.
For example let there be a set of data: {5;5;5;5;5;5;5;5;5;5}.

There are ten fives in the set.
Now let's calculate mean and standard deviation.

Mean: $\overline{x} = \frac{5 \cdot 10}{10} = 5$

Standard deviation:

$\sigma = \sqrt{{\Sigma}_{i = 1}^{n} \left({x}_{i} - \overline{x}\right)} = \sqrt{{\Sigma}_{i = 1}^{10} \left(5 - 5\right)} = \sqrt{{\Sigma}_{i = 1}^{10} \left(0\right)} = \sqrt{0} = 0$

Every component of this sum is equal to zero because the mean is equal to every element in the data set. Sum of 10 zeros is also zero, and the square root of zero is zero, therefore the deviation $\sigma$ is also zero.

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