Solution to Quiz 2
The full quiz is here. The answers appear
below in boldface. Comments, which are not part
of the answers, are italicized.
For the batch {0, 0, 6, 6} compute the following or state why
they are not defined:
- Mean: (0 + 0 + 6 + 6) / 4 = 3
- Five-letter summary: median depth is (4+1)/2 = 2h, hinge depths are
(2h+1)/2 = 1h. Therefore (min, lower hinge, median, upper hinge, max)
= (0, 0, 3, 6, 6), the h-spread is 6, and the range is 6.
- Standard deviation: The residuals with respect to the mean are (-3, -3,
3, 3), their squares are all 9, their sum is 9*4 = 36. The (usual) sd
is therefore sqrt(36/3) = sqrt(12). The mm estimate of sd is
sqrt(36/4) = 3. Either answer is acceptable, but if you
computed the mm estimator, you needed to say so.
- Variance: this is the square of the sd, so the answer is either 12 or 9
(mm estimator). It was a real waste of time to redo the
computation in #3. If you were using a calculator and got any answer
even slightly different from 12, then your answer was wrong. You
should understand the formula well enough to recognize that the variance
must be a multiple of 1/3 or 1/4 (the denominator in the formula).
- Skewness (use the method of moments estimator): The residuals are
(-3, -3, 3, 3) as before. The cubes of the -3's balance the cubes of
the 3's, so the skewness must be zero. There was no need to
compute the mm estimator of the sd in this case.
- MAD. The median is the mean, so the residuals with respect to the
median are the same as those relative to the mean; their absolute values are
all 3, so their median--the MAD--is 3 as well. Remember that
MAD measures a spread: its computation definitely should not involve
negative numbers. Remember the absolute value (the "A" part
of the acronym)!
- Geometric mean. Not defined: two of the numbers (the zeros) are
nonpositive. It is possible to compute the fourth root of
0*0*6*6 = 0, but this does not mean the GM is defined.
- 25% trimmed mean. This amount of trimming removes one zero and
one six; the trimmed mean therefore is (0+6)/2 = 3.
Show your work (the answer alone is not sufficient).
Consider the batches A = {1, 2, 3}, B = {1, 2,
4}, C = {1, 3, 4}, D = {2, 3, 4}, E = {1, 10,
100}, and F = {-1, -10, -100}.
- Order the batches from least to greatest variance.
Evidently, A and D are related by a change of location, so Var(A) =
Var(D). B and C are related by a change of location and
scale--multiply all values of B by -1, then add 5 to get the values in C--so
Var(C) = (-1)2 * Var(B) = Var(B). Similarly, Var(E) = Var(F).
It's easy to check that Var(A) < Var(B) < Var(E), so the correct order
is
A=D, B=C, E=F.
- Order the batches from least to greatest skewness.
Using the same reasoning as in #9 we see Skew(A) = Skew(D), but Skew(C)
= (-1)3 * Skew(B) = -Skew(B) and similarly Skew(F) = -Skew(E).
Clearly the batches with negative skewness are C and F. It follows
quickly that the correct order is
F, C, A=D, B, E.
In your ordering, if two or more batches have the same variance or skewness
then please indicate that.
