If $X \sim N_k(\mu,\Sigma)$, then $Q=(X-\mu)'\Sigma^{-1}(X-\mu) \sim \chi^2_k$. Further, the level sets of $Q$ are the ellipsoids you refer to. So the 72% you mention comes from a chi-square distribution (these calcs in R):

> pchisq(1.96^2,df=3)
[1] 0.7209157

As do the other numbers:

> pchisq(1.96^2,df=10)
[1] 0.04579014

> sqrt(qchisq(0.95,df=10))
[1] 4.278672

See http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Prediction_Interval

If $X \sim N_k(\mu,\Sigma)$, then $Q=(X-\mu)'\Sigma^{-1}(X-\mu) \sim \chi^2_k$. Further, the level sets of $Q$ are the ellipsoids you refer to. So the 72% you mention comes from a chi-square distribution (these calcs in R):

> pchisq(1.96^2,df=3)
[1] 0.7209157

As do the other numbers:

> pchisq(1.96^2,df=10)
[1] 0.04579014

> sqrt(qchisq(0.95,df=10))
[1] 4.278672

See http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Prediction_Interval

If $X \sim N_k(\mu,\Sigma)$, then $Q=(X-\mu)'\Sigma^{-1}(X-\mu) \sim \chi^2_k$. Further, the level sets of $Q$ are the ellipsoids you refer to. So the 72% you mention comes from a chi-square distribution (these calcs in R):

> pchisq(1.96^2,df=3)
[1] 0.7209157

As do the other numbers:

> pchisq(1.96^2,df=10)
[1] 0.04579014

> sqrt(qchisq(0.95,df=10))
[1] 4.278672

If $X \sim N_k(\mu,\Sigma)$, then $Q=(X-\mu)'\Sigma^{-1}(X-\mu) \sim \chi^2_k$. Further, the level sets of $Q$ are the ellipsoids you refer to. So the 72% you mention comes from a chi-square distribution (these calcs in R):

> pchisq(1.96^2,df=3)
[1] 0.7209157

As do the other numbers:

> pchisq(1.96^2,df=10)
[1] 0.04579014

> sqrt(qchisq(0.95,df=10))
[1] 4.278672

See http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Prediction_Interval