Dispersion, which is also called variability,
scatter or spread, is the extent to which
the data distribution is stretched or squeezed.
The common measures of
dispersion are standard deviation and variance.
If you are at a university or college,
you may have heard about the bell curve,
which looks like this.
You will often hear this is within
one standard deviation of the mean or
within two standard deviations of the mean.
Let's see what that means.
Let's look at the age of an instructor.
Let's say the average age is 52.
This means that the individual ages may differ,
some may be 48 or maybe 55 or 75.
So the average age is an estimate.
But what we also need is
an estimate for the dispersion in the data-set.
The other thing to note is the range in our data-set.
For example, the difference of the ranges from a minimum
of 29 years of age to a maximum of 73 years.
This to you refers to a distance or
the difference between the minimum and the maximum.
Unlike the difference between population and sample mean,
the difference between a variance for
the population and a sample is
that when you compute the population variance
denoted as sigma squared,
you divide it by the total number of observations.
These are the deviations between
observation and the mean squared,
then added and then divided by
the total number of observations.
For sample variance, which is denoted as S squared,
you divide it by N minus 1.
The purpose of using N minus 1 is so
that our estimate is unbiased in the long run.
That means if we take a second sample,
we'll get a different value of S squared.
If we take a third sample,
we'll get a third value of S squared and so on.
We use N minus 1 so that the average of
all these values of S squared is equal to sigma squared.
We usually talk about squares called
standard deviation rather than the variance.
Standard deviation is essentially the square root of
the variance or the variances in square units.
It's good to use the standard deviation because
it's exactly the same units as the variable.
The standard deviation of age will also
be measured in years rather than years squared.
Here you see that we just took the square root of
variance and this becomes standard deviation.
We'll return to our data-set and we'll
look at the variables that we computed before.
You can see that the standard deviation
for beauty, evaluation scores,
and age were computed with
the descriptive statistics using
the describe function in Python.
Let me explain why mean and
standard deviation have to go hand in hand.
I will use the example from
basketball about the two giants,
Michael Jordan and Wilt Chamberlain,
who preceded Michael Jordan.
If you consider their average score per game,
you would notice that they didn't differ much.
Their average was around 30 points
for both Jordan and Chamberlain.
However, when you look at
the standard deviation of their performance,
Jordan was around 4.76 compared to
Chamberlain who was at around 10.59.
If you were to plot this distribution to see
Michael Jordan scores using
the mean and standard deviation.
Assuming that their scores are normally distributed.
You would notice even though
both players had around the same mean,
the tighter distribution for Jordan suggests that he
was more consistent in his performance than Chamberlain.
The main takeaway is that
average will only paint a partial picture.
If you really want to understand
the complete picture about a variable or data-set,
it is important to compute both the average and
the standard deviation to get
insights on what the data is telling us.
So a mean with a standard deviation means
something more useful than the mean by itself.