## Frequency Tables -- Discrete Values

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This page presents issues related building and interpreting
**frequency tables for discrete values.** [There is a link at the end of the page to
another page that demonstrates creating these same values in R.]
We start with an example using the data in **Table 1**
given below.

Just a quick look at the values in **Table 1** reveals
that there are not all that many different values.
We want to see how many different values are in the
table and how often each value appears.
A table that provides such information is called a **frequency table**.
Here is such a table for the values in **Table 1** above.

As you may have observed,
**relative frequency** is really the decimal form of the **percent** of
times that each value occurs in the original data table. If we multiply
the **relative frequency** by 100 (i.e., move the decimal point two places to the right)
we get that value in percent form. Thus, in **Table 3**
we see that

Another commonly computed value is the **cumulative frequency** of the values.
Remembering that the *Data Values* in **Table 2** are arranged in increasing order,
as we read from left to right across the frequency values, we could keep a running total
and record that sum as we go along.

Just as we found the **relative frequency** by dividing the **frequency**
by the **number of values in the data table**, we can find the
**relative cumulative frequency** by dividing the **cumulative frequency**
by that **total number of values**. **Table 5** displays this new value
for our example.

There are four things to note about the **relative cumulative frequency**.
- The first value is always the same as the first value in the
**relative frequency**
because the first **cumulative frequency ** is the same as the first **frequency** and
to get the **relative** values we divide each by the same number of total items.
- The final value is always 1.000, because the final
**cumulative frequency**
is always the total number of values in the data table, which is the same number that
we use in the denominator for the calculation.
- The
**relative cumulative frequency**
is also the **cumulative relative frequency**. That is, instead of doing our divisions
to find these values, we could have just kept a running total of the
**relative frequenies**. The two processes are mathematically identical.
- We can use the
**relative cumulative frequency** to answer questions that mirror
those discussed above for using the **cumulative frequency**. For example,
the question "What percent of the values is **less than or equal to** some
value?" can be read right from this new line in the table.

There is another line that we can add to the table but we should add it with
some caution. In particular, it is unfortunately common for people to
be asked to construct a **pie chart** to show the distribution of values.
The **unfortunate** part, as discussed in the page about
**pie charts** is that
people actually have a hard time comparing areas in
**pie charts** and, more importantly, **pie charts** are
easily manipulated to sway those impressions. Nonetheless,
the task of making a **pie chart** is a common assignment.

To make a **pie chart** we need to determine the number of
degrees in the central angle of each slice. We can do that by
multiplying 360 (the number of degrees in the whole circle) by
the **relative frequency** of each item. So, as long as we
are making a frequency chart, we might as well add that computation to it.
**Table 5** displays this new value
for our example.

Just so that you can see another example right here let us
consider the data in **Table 7**.

From that data we can generate a new Frequency table as

Before leaving this topic, we note that the frequency tables given so far have been
organized by row. We could do the same thing organized by column.
**Table 9** below simply repeats the information of
**Table 8** above, but organized by column.

For some people it is easier to read a vertical table such as **Table 9**
than it is to read a horizontal table such as **Table 8**.
Also, the vertical table tends to be more compact, especially when the number of distinct
items gets big. Let us say that we have 20 different values in our data.
We wouldn't want a table with 20 columns because it would be hard to fit
on a printed page. However a table with 20 rows
would not be a problem.

Computing these **frequency** values, and, in effect, constructing a **frequency table**
in R is discussed on the Frequency Tables in R page.

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©Roger M. Palay
Saline, MI 48176 November, 2015