Skewness
Measuring the center and the dispersion of a data set is
an attempt to simplify the information in the set to only
two numbers. Often this extreme simplification is highly
useful. However, sometimes we need to be aware of a third
characteristics, the lopsidedness of the data. The
statistical term for lopsidedness or asymmetry around the
mean is skewness.
Problems: Skewness
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?
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?
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.
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?
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?
Answers here.

