Consider numerical observations x1,…, xn. It is frequentlyof interest to know whether the xi s are (at least approximately)symmetrically distributed about some value. If n isat least moderately large, the extent of symmetry can beassessed from a stem-and-leaf display or histogram.However, if n is not very large, such pictures are not particularlyinformative. Consider the following alternative.Let y1 denote the smallest xi, y2 the second smallest xi, andso on. Then plot the following pairs as points on atwo-dimensional coordinate system: (yn 2 x ,, x , 2 y1),(yn21 2 x , , x , 2 y2), (yn22 2 x ,, x , 2 y3),… There are n/2points when n is even and (n 2 1)y2 when n is odd.a. What does this plot look like when there is perfectsymmetry in the data? What does it look like whenobservations stretch out more above the median thanbelow it (a long upper tail)?
Consider numerical observations x1,…, xn. It is frequently
of interest to know whether the xi s are (at least approximately)
symmetrically distributed about some value. If n is
at least moderately large, the extent of symmetry can be
assessed from a stem-and-leaf display or histogram.
However, if n is not very large, such pictures are not particularly
informative. Consider the following alternative.
Let y1 denote the smallest xi, y2 the second smallest xi, and
so on. Then plot the following pairs as points on a
two-dimensional
(yn21 2 x , , x , 2 y2), (yn22 2 x ,, x , 2 y3),… There are n/2
points when n is even and (n 2 1)y2 when n is odd.
a. What does this plot look like when there is perfect
symmetry in the data? What does it look like when
observations stretch out more above the median than
below it (a long upper tail)?
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