The mean of two points x1 and x2 usually is assumed to sit on the line joining the two points, namely at m0=(x1+x2)/2. This is the statistically optimal solution if both points have the same covariance matrix Σ, especially if Σ=σ2I2. Then the mean has covariance matrix Σ/2. In case the points have individual covariance matrices Σ11 and Σ22 the best estimate m for the mean is m=Σmm(W11x1+W22xx1) with its covariance matrix Σmm=(W11+W22)-1. Moreover, in case one chooses the mean m0, but assumes the points have general (g) covariance matrices, then its covariance matrix is Σm0m0|g=(Σ11+Σ22)/4. The animation is meant to show (1) the mean m can lie anywhere in the plane, and (2) the difference in uncertainty of the simple mean m0 when (a) assuming general covariance matrices for the two points and (b) assuming isotropic uncertainty of the two points.
The animation allows to freely choose the two uncertain points by moving them in the plane and their covariance matrices by changing their semi-axes a and b by shifting the left red points horizontally and the direction of the major axes by rotating the blue lines around the two points.
The animation shows the resultant simple mean m0 (blue) and the weighted mean m together with their covariance matrices (red). For the simple mean the geometric mean of the four semi-axis is taken as standard deviation σ (blue circles). The covariance matrix Σm0m0|g of the simple mean m0 for the assumption of general covariance matrices for the two points is shown in white.
- Change each of the six elements (axes, directions) of the two covariance matrices of the two points individually and observe the effect onto the estimated mean.
- Change the six parameters such that the intersection point of the major axes lies in the window and inside the standard ellipse of m.
- Choose your own point with coordinates (x,y), e.g. (4,7), and adapt the six parameters such that the estimated point sits at (x,y).
- Find two covariance matrices, which are different for both points, such that the estimated mean sits on the line joining the two points.