Game Development Reference
In-Depth Information
In its purest (and most cited) form, the bell curve is symmetrical. That means
that the mean, median, and mode values are identical. They are also co-located
exactly in the middle of the range. In a normal distribution with a range of [0...100],
all three figures would be 50. Therefore, we could assert that:
The average of all the values is 50.
There are as many items
below
50 as there are
above 50
.
The number 50 is the most common value in the population.
As we will find out later, these assertions come in handy.
Skew
When mean, median, and mode are
not
aligned we call that
skew
. If the “tail� of the
curve is longer in the positive direction, we call that “positive� skew. Naturally, if
the “tail� is longer in the negative direction, we say that the distribution is “nega-
tively� skewed. We can look at the features of this a different way. In a positively
skewed distribution, the bulk of the population (and therefore the mean) is on the
left half of the range (i.e., left of the median). In a negatively skewed distribution,
the bulk of the population is on the right side.
A convenient example of a skewed distribution is national household income.
The curve has a significant skew in the positive direction. That is, the tail extending
to the positive end of the range is longer than the one to the negative side. For the
sake of example, let us place the median household income at $50,000. That means
that half the households make more than $50,000 and half make less.
The lower limit of the range is necessarily $0—there are people with
no
income
whatsoever. (For the sake of saking, ignore the fact that people can actually have
negative
income.) On the other hand, there is no limitation on how much money
you can make. That is, there is no upper limit to the range. The current upper limit
is the income of the household making the most money. Because people on that
long positive tail can make millions of dollars, the
mean
income is well above the
median $50,000—for purposes of example, let's say $70,000. The mode, on the other
hand, is below the median. Again, pulling numbers out of the air, let us assume that
the
most common
household income is $40,000.
When a distribution is skewed, the mean, median, and mode are no longer
co-located. They do have tendencies, however. In a positively skewed distribution
such as the one showed in Figure 11.5, the mode will be less than the median, which
will, in turn, be less than the mean. In a negatively skewed distribution, the oppo-
site will be true:
mean < median < mode
.
