# Nonparametric Data – Can Non-Parametric Data Have Mean Value and Standard Deviation?

density functionmeannonparametricstandard deviation

I understand that for non-parametric data, the probability density function (pdf) cannot be obtained using parameters like (mean value) and (standard deviation), and I understand that we use Kernel Density Estimation to estimate pdf.
However, for any non-parametric data, can I still estimate (mean value) and (standard deviation) ?
For example, can I use Matlab commands (mean(x)) and (std(x)) to estimate mean and standard deviation?
Thanks

You have to assume that the mean and variance exist, but once you do, $$\bar X$$ is always an unbiased estimator of the mean, and $$s^2$$ is always an unbiased estimator of the variance. You don't need to make any parametric assumptions for those to hold. Those are unbiased for normal distributions, Poisson, binomial, exponential,...

EDIT

Note that, even though $$s^2$$ is an unbiased estimator of $$\sigma^2$$, $$s$$ is (amazingly) a biased estimator of $$\sigma$$.

EDIT 2

To correct a comment from a few hours ago, an unbiased estimator $$\hat\theta$$ of a parameter $$\theta$$ has the property that $$\mathbb E[\hat\theta]=\theta$$, and bias is a technical term in statistics that refers to $$\mathbb E[\hat\theta-\theta]$$.

The Wikipedia article on bias includes some additional technical details.