variance of a vector

Topics

• variance
• descriptive statistics

Variance measures the spread of data points around their mean. If we are given a vector $M$ and asked to find the variance, we can do the following:

from numpy import array, var
 
M = array([1,2,2,2,5,6])
n = len(M)
 
M_centered = M - M.mean()
dot_prod = M_centered @ M_centered
 
variance = dot_prod / (n - 1)
print(variance) # 4.0
 
# confirming
print(var(M, ddof=1)) # 4.0

So basically,

• center the vector (around the mean) and get $M^{\prime }$
• calc inner product: $X\leftarrow M^{\prime }\cdot M^{\prime }$
• variance: $V\leftarrow \frac{X}{n-1}$

Additionally, covariance, between 2 vectors $M$ and $N$ is found in the same way:

• center $M$ and $N$ around their respective means
• calc inner product between $M^{\prime }$ and $N^{\prime }$
• $V\leftarrow \frac{X}{n-1}$

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