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In my data science textbook, it says that the variance of a variable $Y$ can be written as:

$v_y = \frac{1}{n-1} \sum_{k=1}^{n} y_k^{T} y_k$,

I have never seen variance defined like this before. Could someone provide some insights into this definition (i.e., why it is valid to define variance like this?)

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  • $\begingroup$ it's the same as the usual formula but it assumes that the mean of $y_{k} = 0$. $\endgroup$ – mlofton Dec 29 '18 at 4:31
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    $\begingroup$ This is the formula for sample covariance $\endgroup$ – Francis Dec 29 '18 at 8:57
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I think, a couple of things should be noted here:

1) This equation resembles $E[X^2]$, and it resembles the variance only if $E[X]$ is zero, since $var(X)=E[X^2]-E[X]^2$.

2) It assumes that the $y_k$'s are row vectors. So, $y_k^Ty_k$ will a matrix of $d\times d$, where $d$ is the dimension of each sample. So, we're averaging $d \times d $ matrices here, as in @Francis's sample covariance link (which assumes $y_k$'s as column vectors).

3) What you get out of this equation is a covariance matrix estimate, with entries $C_{rm}=cov(y^{r}, y^{m})$, where $y^{i}$ represents the $i$-th element of a random vector $y$.

4) $\frac{1}{n-1}$, instead of $\frac{1}{n}$ comes from unbiased estimation of sample covariance.

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