(light music)
- Hello, I am Professor Rick Jerz,
and I welcome you to this session
titled Exploring Dispersion.
In our last session, we learned about ways
to describe data
based upon its central tendency and dispersion.
In this session, we're going to learn
that there are a few more graphical and numeric ways
to describe dispersion.
And in our next session, we will see how to use Excel
for some of these additional ways.
Let's get started.
We have several goals in this session.
We want to understand and interpret a dot plot.
We want to compute and understand quartiles,
deciles, and percentiles.
We want to understand how to construct
and interpret box plots.
We want to understand the concept of skewness
and understand and interpret a scatter diagram.
A dot plot groups the data as little as possible
and the identity of an individual observation is not lost.
To develop a dot plot, each observation is simply displayed
as a dot along the horizontal number line
indicating the possible values of the data.
If there are identical observations
or the observations are too close to be shown individually,
the dots are piled on top of each other.
Let's look at an example.
Reported in this slide are the number of vehicles sold
in the last 24 months at Smith Ford Mercury Jeep,
Incorporated in Kane, Pennsylvania and Brophy Honda
Volkswagen in Greenville, Ohio.
We can construct dot plots for each set of data.
This graphic shows the dot plot for this data.
You might notice that this looks very similar
to a bar chart, except dots are used instead
of a solid bar.
Statisticians will sometimes use this dot plot
because they feel it gives you
a better visual interpretation of the data
instead of just a bar chart or frequency diagram.
The dot plot is an example of a graphic
that cannot be produced in Microsoft Excel.
You would need specialized statistical software,
such as Minitab, to produce this graph.
Quartiles, deciles, and percentiles are alternative ways
of describing spread of data,
including determining the location of values
that divide a set of observations into equal parts.
The location of a percentile can be calculated
with this equation.
Let's look at an example.
Listed below are the commissions earned last month
by a sample of 15 brokers at Smith Barney's Oakland office.
Locate the median, which is by definition
the 50th percentile, the first quartile,
and the third quartile for the commissions earned.
The first thing we do is to organize the data
lowest to largest value.
We have 15 pieces of data, so n is equal 15.
The first quartile represents the lowest 25% of data,
so P is equal to 25.
The third quartile is equivalent
to the 75th percent of data, so P is equal to 75.
The equation gives us the fourth and 12th observation
in our array, so we look at the data
and find the fourth and 12th piece of data.
You might recall from a previous session
that when we calculate the median,
which is by definition the 50th percentile,
if we have an even quantity of data values,
there will not be a middle data point.
Instead, we calculate the average of the two data values
that would be surrounding a middle data point.
So likewise, the calculation of the 25th
and 75th percentiles could get a little bit more involved.
We will see how to do this in Microsoft Excel.
A boxplot is another graphic display of our data.
A boxplot is intended to show the middle 50% quartile range,
sometimes called the interquartile range.
We need five data points to construct a boxplot.
These are the minimum value,
the median, and the maximum value,
and the first quartile, and the third quartile.
Here is an example of a boxplot.
Notice the minimum value and the maximum value,
the median, which is the middle value,
and the bar represents this middle 50% quartile range.
In other words, 50% of the data appears
between the left side of this bar and the right side.
The boxplot lets you quickly see where the middle 50%
of your data is located relative to the minimum
and maximum value.
This location of most of your data gets us into
the concept of skewness.
If most of the data is centered around the mean,
we describe this data as symmetric.
If a bigger chunk of the data is to the left of the mean,
we describe this characteristic as positively skewed.
If most of the data is to the right side of the mean,
we describe this as negatively skewed.
Another way to describe this using some
of our statistics is that if the mean is to the right
of the median, the distribution is said
to be positively skewed.
If the mean is to the left of the median,
it is said to be negatively skewed.
Karl Pearson came up with one way to calculate
the coefficient of skewness.
Microsoft Excel and Minitab do it a little bit differently.
This chapter introduces us to the concept
of a scatter diagram.
It is another important graphical technique
that shows the relationship between two variables.
One variable is plotted along the horizontal or x-axis
of a graph,
and the other variable along the vertical or y-axis.
This slide shows some examples of scatter diagrams.
Scatter diagrams are useful because they let you
visually see if there is a relationship between variables.
Consider that we have information concerning
the prices of 80 vehicles sold last month
at the Whitner Autoplex lot in Raytown, Missouri.
The data shown include the selling price of the vehicle,
as well as the age of the purchaser.
We might wonder whether or not there is a relationship
between selling price of a vehicle
and the age of the purchaser.
A scatter diagram would let us visually see
if there is a relationship.
Then, we can ask a question whether it would be reasonable
to conclude that the more expensive vehicles
are purchased by older buyers.
Later in this course, we will explore the use
of scatter diagrams.
The author simply is presenting this technique
in this chapter.
We will also learn how to produce scatter diagrams
with Microsoft Excel.
Well, that's it for these miscellaneous topics
relative to describing data.
In another session, we will explore these concepts
using Microsoft Excel.
I'd like to take a moment to summarize
what we have learned so far.
We have learned that there are two ways
that we might want to describe data,
graphically and numerically.
The types of charts that we might create for data
and the descriptive statistics do depend
upon the type of data that we have.
We have learned how to produce relative frequency diagrams
for both discrete and continuous data.
The beauty of using relative frequency diagrams
is that by interpreting the area represented
under the curve for data of interest,
we are able to determine the likelihood
of that data occurring.
Saying this just a little bit differently,
using a graphical method, we are able to calculate
the probability of certain values occurring.
We will see later on that using descriptive statistics
that we can also calculate probabilities.
With our descriptive statistics and the right equations,
we can get to probabilities even quicker
than looking at a chart and our results will also
be more accurate.
(light music)
Thanks for taking time to spend some moments with me.
I hope that you have enjoyed what I have shared with you.
If you ever have any questions or need my help,
please let me know.