Let me illustrate how to obtain
the probability of getting
a high or low teaching evaluation score from our dataset.
First, an important concept is
the standardization of a variable such that it
returns a dataset with a mean of
zero and a standard deviation of one.
I use the formula in equation shown here,
where the standardization is taking
a variable X and
subtracting from it the average value mu,
then dividing it by the standard deviation so that if
the teaching evaluation score of
an instructor on a scale of one to five is 4.5,
we subtract the average teaching evaluation of
3.998 from it and divide it by the standard deviation,
which is 0.554, resulting in a Z score of 0.906.
If we were to just display the data as a histogram,
you would see that it has a mean around zero.
And the spread is shown on a scale where
the X axis varies from minus three to two.
In a case where you do not have access to
a computer with statistical software,
you can still compute probabilities from
a probability table using
a simple and standard normal table found in
statistics textbooks or downloaded online.
A copy of such a table is on the right.
Notice that the normal distribution graph
to the left is grayed out in some parts.
That grayed-out area represents
the probability of getting some value Z,
in this case Z.
This value of Z or less than,
we will need to first standardize
the variable to determine the probability of
a teaching evaluation score higher
than 4.5 or less than 4.5.
Let's say, we have a dataset where
the average teaching evaluation is
3.998 and the standard deviation is 0.554.
And we are interested in
determining the probability of getting
a teaching evaluation score of 4.5 or less.
So, from the table that I showed
in the last slide, we can determine this.
If we were to standardize the data,
it becomes 0.906 because
the accuracy of this table is
only good for two decimal places.
So 0.906 effectively becomes 0.91.
We get a 0.8186 value here, hence,
the probability of obtaining
a teaching evaluation score of
4.5 or less is 0.8186.
If you were to look at this graphic,
you will see that I have plotted
the area under the curve by shading it gray.
That's the area that depicts
the probability of an instructor receiving
a teaching evaluation of less than or equal to 4.5.
And that probability is 0.8176 or 81.76 percent.
Now, what will be the probability of receiving
a teaching evaluation score of greater than 4.5.
In fact, you can see from
the next graphic that the probability
is the reverse of one that we saw earlier.
And hence, the probability of obtaining
a teaching evaluation score of greater than
4.5 is 18.24 percent,
which is the area shaded in gray.
The reason for this is because the area
under the normal distribution curve is equal to one.
So one minus 0.8176 will give you
the area for evaluation scores greater than 4.5.
Let me illustrate the example of getting
a teaching evaluation score of greater than 4.5.
In Python, when we use the norm
dot cdf function in the scipy.stats package.
After finding the mean standard deviation,
we plug it into the function with the X value of 4.5.
And we will get the area to the left,
which is the less than 4.5 area.
Because we want the area to the right of 4.5, that is,
the probability of greater than 4.5,
we will remove the value from one as indicated here.