How to fit an ellipse to 2D data
Following up on my deleted answer... If you take a filled ellipse and project all the points onto the $x$ axis, more points will be projected near the origin than on the extremes, in a circular-shaped distribution. Not a gaussian distribution, and not the uniform distribution I mentioned in the 1-D analogy in my deleted answer. The resulting distribution actually has pdf $p(x) = \sqrt{(1 - (\frac{x}{r})^2)}$, and from there you can compute that the standard deviation is $\frac{r}{2}$.

Thus, if data is uniformly distributed in the interior of an ellipse of radii $a, b$ (whose axes are the $x$ and $y$ axes), the standard deviation of the $x$ coordinate is $\frac{a}{2}$ and of the $y$ coordinate is $\frac{b}{2}$. So the correction factor you need to use is simply 2.

Here is a worked example in python recovering the center (translation matrix), rotation matrix, and radii of an ellipse from points randomly sampled from its interior:

import numpy

# Generate some points distributed uniformely along an ellipse
x = 2 * (numpy.random.rand(10000, 1) - 0.5)
y = 2 * (numpy.random.rand(10000, 1) - 0.5)
d = (x / 0.5) ** 2 + (y / 0.25) ** 2
inside = numpy.where(d &lt; 1)[0]
x = x[inside]
y = y[inside]
data = numpy.hstack((x, y)).T

# Now rotate by 0.5 rad and translate it to (4, -8)
angle = 0.5
rotation = numpy.array([
    [numpy.cos(0.4), -numpy.sin(0.4)],
    [numpy.sin(0.4), numpy.cos(0.4)]])

data = numpy.dot(rotation, data)
data[0, :] += 4
data[1, :] -= 8

# Step 1: center the data to get the translation vector.
print 'Translation', data.mean(axis=1)
data -= numpy.reshape(data.mean(axis=1), (2, 1))

# Step 2: perform PCA to find rotation matrix.
scatter = numpy.dot(data, data.T)
eigenvalues, transform = numpy.linalg.eig(scatter)
print 'Rotation matrix', transform

# Step 3: Rotate back the data and compute radii.
# You can also get the radii from smaller to bigger
# with 2 / numpy.sqrt(eigenvalues)
rotated = numpy.dot(numpy.linalg.inv(transform), data)
print 'Radii', 2 * rotated.std(axis=1)