A circle, centre (1,-2) and passing through a point (3,-1) is drawn on a cartesian plane. Determine the y intercepts of the circle.

Angle ABC is an angle at the circumference subtended by arc AC. Angle AOC is the angle at the center subtended by the same arc.

DA and DC are tangents to the circle from point D. The radii OA and OC are perpendicular to the tangents at the points of contact A and C, respectively.
Therefore, $\angle OAD = 90^\circ$ and $\angle OCD = 90^\circ$.

The sum of angles in a quadrilateral is $360^\circ$.

The value of x is $\boxed{52^\circ}$.

: Standard Deviation
The given data is 6, x, 7, 12, 13, 11, 9, 6. There are $n=8$ data points.
The mean shoe size is $\bar{x} = 9$.

The mean is the sum of all data points divided by the number of data points.

The complete data set is: 6, 8, 7, 12, 13, 11, 9, 6.

The formula for standard deviation is $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$.

The standard deviation is $\boxed{2.55}$ (to 3 significant figures).

: Circle y-intercepts
The circle has center $(h,k) = (1, -2)$ and passes through the point $(3, -1)$.

The equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$.
Substituting the center $(1, -2)$:

Substitute $x=3$ and $y=-1$ into the equation:

The equation of the circle is $(x-1)^2 + (y+2)^2 = 5$.

Y-intercepts occur when $x=0$. Substitute $x=0$ into the circle's equation:

Take the square root of both sides:

This gives two possible values for y:
Case 1: $y+2 = 2 \implies y = 2 - 2 \implies y = 0$
Case 2: $y+2 = -2 \implies y = -2 - 2 \implies y = -4$
The y-intercepts of the circle are $\boxed{0 \text{ and } -4}$.