Answers: 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?
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*(mean-median)/(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 z-scores, 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.