Independent solution

How to solve this Uniform Distribution question

Setup

Setup

Let w=b-a be the positive width of the uniform support. Write the standard deviation and percentile formula in terms of w.

SD⁡(X)=w12\operatorname{SD}(X)=\frac{w}{\sqrt{12}}
Q(r)=a+rwQ(r)=a+rw

Model

Model

Subtract the lower quartile from the upper quartile to obtain the interquartile range.

IQR⁡=Q(0.75)−Q(0.25)\operatorname{IQR}=Q(0.75)-Q(0.25)
IQR⁡=(0.75−0.25)w=w2\operatorname{IQR}=(0.75-0.25)w=\frac{w}{2}

Compute

Compute

Divide the two scale measures. Both the interval width and any location shift disappear.

SD⁡(X)IQR⁡=w/12w/2\frac{\operatorname{SD}(X)}{\operatorname{IQR}}=\frac{w/\sqrt{12}}{w/2}
SD⁡(X)IQR⁡=212=13\frac{\operatorname{SD}(X)}{\operatorname{IQR}}=\frac{2}{\sqrt{12}}=\frac{1}{\sqrt3}

Answer

Answer

The invariant ratio is 1 to the square root of 3.

1:3(A)\boxed{1:\sqrt3\quad\text{(A)}}