can a unit be associated with the mean and standard deviation values or these are "dimensionless" quantities?

For instance, if I am computing distance, with unit being meters (m), and I have got the following values:

Min: 11 (m)

Max: 18 (m)

Mean: 14.5 (m)

Standard deviation: 1.5 (m)

Can we use "m" with mean and standard deviation to express unit of distance?


1 Answer 1


Mean and standard deviation have dimensions, the same ones from your data. Take your example: $$ \text{mean distance } d \; [\text{in}\; m] = \mathbb{E}( d \; [\text{in}\; m]) = \cfrac{1}{n} \sum\limits_{j = 1}^{n} d_j \; [\text{in} \; m] = \mu \; [\text{in}\; m] $$

$$ \begin{aligned} \text{variance of distance } d \; [\text{in}\; m] &= \mathbb{E}(\, (d \; [\text{in}\; m] - \mu \; [\text{in}\; m])^2\,) \\ &= \cfrac{1}{n} \sum\limits_{j = 1}^{n} (d_j - \mu \; [\text{in} \; m] )^2 = \sigma^2 \; [\text{in}\; m^2] \\ &\Rightarrow \text{standard deviation } = \sqrt{\sigma^2 \; [\text{in}\; m^2]} = \sigma \; [\text{in}\; m] \end{aligned}$$

(for simplicity I've assumed that all distances in your dataset follow a uniform probability)


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