Answers: Skewness
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1. A box contains these five tickets:
0, 0, 0, 0, 5
a) What is the average of the box?
b) What is the standard deviation of the box?
c) Does the box have a median? If so, what is it?
d) Does the box have a mode? If so, what is it?
e) Is this set of numbers symmetrical or asymmetrical? If it
is asymmetrical, does the tail extend to the right or to the
left?
mean = 1, sd = 2 if it is
population, sqrt(5) if it is a sample; the median and mode
are both 0, it is asymmetrical, and the tail is to the right
with 5 as the extreme value.
2. Pearson's coefficient of skewness is
3*(meanmedian)/(standard deviation).
 a) Compute it for these numbers: {0, 1, 10, 7, 2, 2,
1}.
Is this distribution symmetrical, or does it have a tail
to the right or to the left?
 b) Compute it for these numbers: {6, 19, 0, 8, 1, 2,
4}.
Is this distribution symmetrical, or does it have a tail
to the right or to the left?
 c) Do the same with {17, 18, 20, 21, 40, 33, 27, 23,
21}.
 d) What is the logic behind this measure? Why
subtract the median from the mean? Why divide by the
standard deviation?
a) mean = 3, median = 2, sd =
4; skewness = .75; tail to the right;
b) mean = 4, median = 2, sd = 7.85; skewness = .25; tail to
the right;
c) mean = 24.44, median = 21, sd = 7.62; skewness = 1.35;
tail to the right.
d) Extreme values affect the mean but not the median, so if
the tail is to the right, the mean will exceed the median.
Dividing by the standard deviation eliminates the units of
measurement, which is also done when computing zscores, the
topic of the next section.
3. The U.S. Geological Survey keeps track of river flow
throughout the United States. One of its tracking stations
is just east of Rensselaer, Indiana and on a January 22 a
few years ago it reported that the flow of the mighty
Iroquois River was 445 cubic feet per second. Looking at the
past 55 years, the lowest flow was 11 cubic feet per second,
and the highest on record was 1670 cubic feet per second.
The average on January 22 for the previous 55 years was 193
cubic feet per second, while the median flow was 105 cubic
feet per second.
Based on this information, if we graphed the data from
the these 55 years on how much water was flowing in the
river, which of the following graphs would be most like the
one that we would obtain. Explain how you get your
answer.
Most of the data are a bit
below or a bit above 100. However, there are some years in
which there are floods, and those data are extremes the pull
the mean up well above the median. Hence, the distribution
will look like the first chart, with the long tail to the
right.
4. If a distribution is skewed to the left (meaning it
has a long tail to the left), which is greatest: the mean,
median, or mode? Which is smallest?
The mean is pulled by the
extreme values, the mean and mode are not. So if there are
extreme values are small, the mean will be less than the
median, and if the extreme values are large, the mean will
be greater than the median.
5. After handing back a test a professor noted that the
distribution of scores was positively skewed. On leaving
class, one of your friends who has never had statistics
turns to you and says, "Even the prof thought that this test
was rotten. He said was positively screwed up." How would
you explain to your friend what the professor really
meant?
You can tell him that
positively skewed means that there were many low scores and
only a few really high scores. Whether your friend will
understand or not depends on the intelligence of the people
you hang with.
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