- In this Screencast, I'm gonna cover
descriptive statistics.
This is usually in respect to samples
so samples as opposed to populations.
To kind of preface this, we usually are interested
in a stochastic quantity or property, or variable
stochastic meaning that the true values are statistical
and they behave according to a distribution function.
This population has some sort of distribution.
Obviously you cannot know all of the
values of the population.
In other words, if I wanted to calculate the
average height of all Americans
if I wanted to get that exactly
I would have to measure the height of every American
which obviously is not possible.
Instead, we take a small sample
we take measurements, and hopefully our sample
is a true representation of the population.
There arises an issue.
How big of a sample should you take such that
your sample is representative of the population?
There's always a chance that your sample is not
gonna be representative of the population.
After we take a sample
we usually generate sample statistics and diagrams.
We also then try to analyze the type of distribution
its parameters and its uncertainty.
In this Screencast, I'm just gonna cover how
we can calculate descriptive statistics for our sample.
Some of these you've already been exposed to
in this class and in others.
We have statistics for the central value
also known as the central tendency.
The most common by far is the sample mean
also known as the average, and we see
that denoted a lot of times by X bar.
The median looks similar, but on top
of the X there we have a tilde.
Be sure that you make note of that.
The mode is the most frequently occurring value.
There's also some other things that we're not gonna talk
about too much in this class.
mEstimator, this is a robust estimator and it
removes the effects of outliers.
You're also interested in looking at the dispersion
or the variability in your sample.
The most common of these by far, is a sample
standard deviation.
We denote that by S, and you guys have seen that
there's also the range, which is the maximum
minus the minimum.
Obviously if you have a really dispersed sample
you're gonna have a really big range, and if you have a very
non dispersed sample, your range is gonna be small.
Some people will use this measure of dispersion known
as M-A-D, the median absolute deviation, and you
might see some of these in homework assignments
in this class.
There's some others that I'll show you related
to how symmetric the distribution is.
That's skewness.
It shows the asymmetry of the distribution.
If a distribution is skewed one way or the other
then you're gonna get a negative or positive skewness.
For example, if we had a distribution
like this that had a long tail to the right,
that would have a positive skewness.
A long tail to the right would represent
a negative skewness.
These are relative to a normal distribution.
Kurtosis shows the peakedness and flatness
of a distribution compared to the normal distribution.
If you had something that was really sharp
and peaked like this, then that would have
a positive kurtosis.
If you had something that was relatively flat compared
to the normal distribution
then that would have a negative kurtosis.
We can compute the sample mean
or the average using the average function
in Excel or by hand using this formula.
It's just the sum of all
of your X values divided by the number in your sample.
Little N.
The sample median can be computed using
the median function in Excel.
You sort the data and then it's the very middle value.
If you have an odd number, then that's easy.
For example, if you had seven numbers
you just choose the fourth.
If it's an even number
you just take the average of those two.
For example, if we had these data
and I sorted from lowest to highest, I have an even number.
So I would just take the average of those two
and my median would be 5.5.
Sample standard deviation.
Note the division by N minus one here.
And that's important compared to population standard
deviation where you just divide by N.
In Excel, you can use the STDEV.S function
for sample standard deviation.
We can also compute the range by determining the max
and the min and the difference between those.
I want to just briefly explain the difference
between sample and population statistics.
Oftentimes you have a population of size big N
and you're just taking a small sample to estimate parameters
of the population.
Sample size is usually denoted by little N.
The mean, the actual mean of the population is generally
denoted by Mu.
The mean of your sample is X bar.
The actual standard deviation of your population is sigma.
The variance would be sigma squared.
An estimate for sigma based upon your sample is
the sample variance S squared and the sample
standard deviation, which is the square root of S squared
and that's just S.
And so we use little n, x bar, and little s to sort of
represent hopefully your much larger population that has
capital N items, has an average of mu, and it has
an actual standard deviation of sigma.
I've got a file here called Old Faithful.
This is found on canvas.
If you wanna work along.
I'm just gonna real quickly compute some of these things.
The first thing I'm gonna do is I'm gonna select B6
all the way down to the bottom by doing control shift down
and I'm just gonna call this data just to name that.
Let's compute a couple of things.
First of all, the count
you can always use the count function of data.
If you want to know the the number of items,
it's a quick way to do that.
The average, we can simply do average of our data.
There's a median function.
We can compute the median of the data.
There's also a mode function.
We can compute that using mode of our data.
The standard deviation.
We want to make sure if this is a sample
and not the entire population
it would be rare that you use the dot P version.
This is a sample standard deviation.
Again, the difference between dot S and dot P,
in dot S, you're dividing by N minus one
in dot P, you're dividing by N.
As your sample size gets bigger, it's not as big of a deal
but we can do that for our data.
We can compute the minimum of data.
We can compute the maximum of our data.
The range is the max minus the min
and let's just compute a few more things here.
The skewness and kurtosis
we can use the skew function of our data.
There's the kurt function.
This is telling us that compared to a normal distribution
if we plotted this in a histogram and I'll show a histogram
in the next screen cast for these data
it's going to have a tail slightly out to the left
and it's gonna be slightly flatter
than a normally distributed variable.
So hopefully this screen cast kind of gives you
a better idea of descriptive statistics and the difference
between population parameters and sample parameters.
Thanks for watching.