Now that I have explained to you the ubiquity of the normal distribution, Its
regular appearance in human measurements, you may begin to hope or even expect
that all of the distributions that we will encounter will be normal curves, but
if that is your expectation, you will have to get used to disappointment.
(Princess bride reference there) Because many curves, perhaps most curves, are not
normal distributions, we need a way to talk about the shape of distributions
when they differ from normality. The first difference that we may find is
that the scores in the distribution are more spread out than we would have
expected, or we may find the scores are more closely packed together than we
expected. The name for the peaked or flatness of a curve is called kurtosis.
When the scores are very close together, then the curve becomes peaked. We call
this a "leptokurtic" curve think of the scores leaping up - leptokurtic. When the
scores are very spread out, the curve becomes flat like a plate, we call this
platykurtic. "Plat" rhymes with flat. Platykurtic is a flattened curve in
the shape of a plate. A normal curve is mesokurtic. It's kurtosis is medium. So
kurtosis can be measured as leptokurtic - tall, platykurtic - flat, or mesokurtic -
medium. Kurtosis is caused by the variability in the distribution. Another
thing that can happen to a curve is when the scores are pulled out in only one
direction.
When the scores are dragged down (or rather, out) in only one direction, this
creates a skew in our curve. Therefore, we need to talk about the skewness of
our distribution. Negatively skewed distributions have a higher than
expected frequency of high or extreme scores on the right, and the tail is
pulled out to the left end of the number line on the x-axis. For example, if we
were interested in the running speeds of football players, we might find a lot of
very fast players - high scores, but only a few slower runners - low scores. Skewness
is always caused by outliers in the direction of the tail. In a positively
skewed distribution, the higher than expected frequencies are on the low end
of the curve. The tail is pulled back on the right or positive end of the number
line. If we were measuring reaction time, we would expect to have a large number
of very quick responses - low scores, and only a few slower responses, taking more
time, further up the positive end of that scale. Skewed distributions are not
normal. How can you remember which direction is positive or negative when
we talk about skewness? Stats cow tells us that the skew is in the tail. Skewness
is caused by outliers, extreme scores in the tail of the distribution, the
direction that the tail is pulled out, (positive or negative) is the direction of
the skew. Here are two curves. This first one is positively skewed, and the second
is negatively skewed the top curve is positively skewed because the tail is
pulled out on the right, or the positive direction of the number line. The bottom
curve, is negatively skewed, the tail is pulled out on the negative, or left end
of the number line. Both of these curves show us
what happens to the mean and the median in the case of kurtosis. In both of these
curves, you can see what happens to the mean and the median in the case of
skewness. Both of them are pulled in the direction of the outlier but the mean is
pulled further. That is because the mean is more susceptible to the outlier that
is causing the skewness. Mathematically we can calculate a measure of skewness
by comparing the mean and the median and this will give us a value that we can
use to quantify the skewness of our curve. But there are other things that
can go wrong with our normal curve! Instead of having one peak sometimes we
have two peaks. This occurs when there is more than one most frequently occurring
score we call this type of curve bimodal. A curve can be bimodal when there really
are two most frequently occurring scores. For instance when is the best time to go
fishing? At what time of day will you catch the most fish? Probably early in
the morning, and then in the evening when the sun is going down. In the middle of
the day, when the Sun is at its height, you will catch fewer fish. So if we plot
the number of fish caught, we will see a peak in the morning at dawn, and another
peak in the evening at dusk. This would be a true bimodal distribution. On the
other hand, we might have a bimodal distribution when there are actually two
distributions overlying each other. When we had both males and females on the
football field and we were comparing heights, we saw that there was a
distribution for males and another distribution for females. The
distributions overlapped - some females were taller than some males - but the
average heights were taller for males. They really were two distinct
distributions that should be separated before being analyzed. A multimodal
distribution has three or more most frequently occurring scores.
You may wonder why we don't call it a trimodal distribution or a quadrimodal
distribution - four peaks - the answer is that when we start getting three, four,
five, modes, there is something very wrong in our data set. Three or more modes is
multimodal, and it's messed up. We need to figure out what is going on before we
try to analyze those data. Rectangular distributions have the same frequency
for all scores. If you roll a single die 100 times, how many times do you expect
to get a one? About one-sixth of the time, in fact you would expect to get each of
the scores, one through six, approximately one sixth of a time. That is a
rectangular distribution. Once you add a second die, however your distribution
will begin to look more normal. Rectangular distributions have exactly
the same frequency for all scores, and do not have tails. Before we conclude, there
is - one more thing - that I want to tell you about the normal curve, and that is
that the normal curve can be overlaid with a number line, and this is where
things get really interesting and quite useful. If we have a normal curve, we can
add the value of the mean right in the middle where it belongs, and in this
example we're going to imagine that our mean is 50, so then we could lay out a
number line with four point delineations. Half of our scores will always be above
the mean, or above 50. The remaining half of the scores will always be below 50.
That is what a measure of central tendency tells us. It is the point at
which half of the scores fall above and half of the scores fall below. The
next thing that we could do is measure the proportion of the scores that fall
within a certain range, above or below the mean. The next thing that we could do
is measure the proportion of the scores that fall within a certain range above
or below the mean. The proportion is the total area under the
normal curve that corresponds to the relative frequency of those scores. To
better understand this, let's return to our picture of the people standing on
the football field. Remember that everyone (100%) are
standing below the rope that represents our distribution we want to know the
proportion of people who are between five foot six and five foot nine inches
tall. We ask everyone who is in those rows, five foot six, seven, eight, and nine
to stay where they are, everyone else please leave the field. So how many
people are in these four rows? Divide the number of people in the four rows by the
total number of people and you have a proportion. This is the proportion of
people who are in that range underneath the distribution. It would also be the
relative frequency of the number of people in that range, and this is going
to become a very useful technique when we talk about z-scores. But for now, just
remember what we've learned about the frequency table, and specifically how the
relative frequency relates to what we know about the normal curve.