Standard Deviation of Negative Numbers

Can you find standard deviation of negative numbers?

Sure you can.

While standard deviation (the result) can’t be negative, the individual numbers that you calculate standard deviation for can reach any value, including negative.

How to calculate standard deviation of negative numbers

Exactly in the same way you calculate standard deviation for positive numbers, or any numbers.

In fact, very often you will be calculating standard deviation for data sets which contain both positive and negative numbers (and even some zeroes) at the same time. Especially in the world of finance, where standard deviation of investment returns is often calculated.

How negative numbers affect standard deviation

For standard deviation calculation it is not that important whether the individual numbers are positive or negative. It does not even matter whether the individual numbers are big or small as a whole.

For example, the data set [2, 4, 6, 8, 10] has exactly the same standard deviation as the data set [-2, -4, -6, -8, -10] and exactly the same standard deviation as the data set [-8750, -8752, -8754, -8756, -8758] or the data set [-5, -3, -1, 1, 3]. These all have standard deviation equal to 2.8284 (which is the square root of 8) – assuming we’re talking about population, not sample (in such case they would all have standard deviation of 3.1623, which is the square root of 10).

So what decides standard deviation being big or small?

The thing which does affect how big or small standard deviation will be is the diversity of the data set – how the individual numbers differ from each other, or from the average (mean) of the data set. In all the examples above, the individual numbers differ from the particular mean of the data set by [-4, -2, 0, +2, +4].

This is why standard deviation is often used together with mean (arithmetic average). The former measures diversity of a data set (how much the individual numbers differ from each other), while the latter measures the overall (average or typical) level of the data set – whether the numbers (as a whole) are big or small, positive or negative.


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