Describing Quantitative Data

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Describing Quantitative Data

• Here we study ways of describing a variable that
  is quantitative.

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Quantitative variables have values that are
numbers. Remember that qualitative variables may
use numbers, but the variable really has values
that represent groups. Example of a qualitative
variable eye color 1 blue, 2 green, 3 red
(especially on Friday morning?). Our initial
method of describing a quantitative variable will
be basically the same as with a qualitative
variable, with some modification in our
understanding. Lets consider the variable age.
Consider the first 20 people you see today.
Consider yourself if you look in the mirror, but
just count yourself once. The age of these folks
could be 1 day to 110 years in Nebraska, right?
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Remember, a frequency distribution is a tabular
summary of data showing the number, or frequency,
of items in each of several nonoverlapping
classes. With a variable like eye color
(qualitative), we typically make each color a
class. But with a variable like age
(quantitative), if we make each age a class then
we could have so many classes that the
distribution is hard to interpret. The authors
suggest grouping the ages into classes and having
anywhere from 5 to 20 classes. Lets digress for
a minute and think about a data set. Say I have
data on people. Say I have social security
number, eye color, age and blood alcohol level
last Thursday night at 1130. On the next screen
I have what the data might look like in Excel, or
other computer programs. Note each column is a
variable. Each row represents a person in this
example. Thus in each row we see the values of
the variables for each person.
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The reason for my digression was to have you
begin to think about data sets. (Typically) A
variable is in a column. The values down the
column are for different people (or what ever the
subject might be). I believe it is useful to
think about data as you consider statistical
ideas. Here we are looking at how to describe a
column of data, one variable. Now, when we have a
quantitative variable like age we have to think
about how many classes to have. We want each
class to have more than a few people in it. For
now, lets not worry too much about how many
classes to have. The width of each class should
be equal. Using age as an example, we might have
classes that have 5 consecutive ages included.
The first class might be 10-14 year olds, then
15-19 year olds and so on.
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Class limits need to be considered. Each
person should be in only one class. Each class
has a lower limit and an upper limit and these
limits are exclusive to the class. On the next
screen I have an example of the frequency,
relative frequency and percent frequency
distributions for the variable age for 50
people. The frequency column is just the counting
of the number of people in each class. The
relative frequency is the frequency of each class
divided by the total number of people in the data
set. The percent frequency is the relative
frequency times 100. (Look back at the
distributions we had for the qualitative
variable. Does it look the same?)
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Do you know why we put information in columns?
Because then we can callum as we seeum.
Sorry) So, the frequency, relative frequency and
percent frequency distributions are different
ways of summarizing information about a numerical
variable. Notes about our table. 1) The total,
or sum, of the frequency column is equal to the
number of observations, n. 2) The total, or sum,
of the relative frequency column is equal to 1.
3) The total, or sum, of the percent frequency
column is equal to 100.
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Bar graphs are used for qualitative variables.
What amounts to the same thing for quantitative
variables are called histograms. Histograms just
put the the frequency, relative frequency and
percent frequency distributions into visual form.
The form is a graph with certain properties. The
variable of interest is put along the horizontal
axis. We would have the variable age on the axis.
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Imagine you have a piece of construction paper
that is blue. Do you remember way back when in
school you would cut strips of paper and then
curl the paper with the scissors? Well, we will
not need to curl the paper here! I mention this
silly example because I want you to think about
cutting strips that are of the same with and are
as wide as the class width (remember class widths
are equal). The height of each strip would then
represent the frequency, relative frequency or
percent frequency on the variable. You would
tape each strip onto the graph above each
category. So the vertical axis, or height, in the
bar graphs is either the frequency, relative
frequency or percent frequency distributions. In
constructing the histogram on a quantitative
variable THERE IS NO SPACE between each bar to
help us remember we have a quantitative variable.
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Pie Charts The authors do not mention it, but pie
charts could be made in a similar fashion to what
we saw before. Cumulative Distributions Have you
every accumulated a bunch of junk in your room?
Yea, me to. Each day more stuff just shows up.
So tomorrow I will have all the stuff I have
today and more. Cumulative distributions are kind
of like my story. When you look at the frequency
distribution we just saw, a slight modification
can make then into cumulative distributions. For
the cumulative frequency, start with the first
class in the first row. The cumulative value for
this row is the frequency.
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But the cumulative value for the second row is
the frequency for the first row plus the
frequency for the second row. So to get the
cumulative frequency for a given row, add up the
frequencies for that row and all previous
rows. The cumulative relative frequency and
cumulative percent frequency are found as before
cumulative relative frequency is cumulative
frequency divided by total and the cumulative
percent frequency is the cumulative relative
frequency times 100. Whats a henway? About 4 or
5 pounds! ? Whats an Ogive? It is what we call
a graph of a cumulative frequency distribution.
The horizontal axis has values of the variable
and the vertical axis has the appropriate
cumulative frequency.
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What is the most frequently occurring age group
in this example? How many times does it occur?
(the group is 30-39 and the frequency 17)
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This is a frequency Ogive (or polygon).