Frequency: frequency is the raw count of the number of occurrences of a certain variable.
Relative Frequency: relative frequency is the proportion of occurrences of a certain variable.
Example: Suppose a random sample of $25$ SWOCC students were anonymously asked how many times they flossed their teeth in the past week. Four of the $25$ students answered "$7$ times."
$4$ is the frequency of students who flossed $7$ times in the past week.
$\displaystyle \frac{4}{25}=0.16=16\%$ is the relative frequency of students who flossed $7$ times in the past week.
Frequency Tables: a frequency table is a list of frequencies that a particular value of a variable occurs in a data set.
Note: all the relative frequencies should add to $1.$ This essentially says that $100\%$ of the data is represented in the table.
Example: Twenty SWOCC students were asked how many hours they worked per day. Their responses, in hours, are as follows: $$5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3.$$ Construct a frequency table for the above data set which includes both frequency and relative frequency.
$$
\begin{array}{|l|l|l|}
\hline
\mbox{Data Value} & \mbox{Freq.} & \mbox{Relative Freq.} \\ \hline
2 & 3 & 0.15 \\ \hline
3 & 5 & 0.25 \\ \hline
4 & 3 & 0.15 \\ \hline
5 & 6 & 0.30 \\ \hline
6 & 2 & 0.10 \\ \hline
7 & 1 & 0.05 \\ \hline
\end{array}
$$
Cumulative Relative Frequency: When we add up all the previous relative frequencies up to a particular data point, we get the cumulative relative frequency of that point.
Note: cumulative relative frequencies tell us the proportion of the data that falls at or below a particular data point.
Example: Consider our table of frequencies we created for our SWOCC student data. Fill in the cumulative relative frequency column $$ \begin{array}{|l|l|l|l|} \hline \mbox{Data Value} & \mbox{Freq.} & \mbox{Relative Freq.} & \mbox{Cumulative Relative Freq.} \\ \hline 2 & 3 & 0.15 & \\ \hline 3 & 5 & 0.25 & \\ \hline 4 & 3 & 0.15 & \\ \hline 5 & 6 & 0.30 & \\ \hline 6 & 2 & 0.10 & \\ \hline 7 & 1 & 0.05 & \\ \hline \end{array} $$
Example: from the our completed SWOCC student table below, answer the following questions.
a) What percentage of students work $5$ hours a day or less?
b) What percentage of students work $5$ hours a day or more?
c) What percentage of students work more than $5$ hours a day?
d) What percentage of students work between $3$ and $5$ hours a day inclusive? $$ \begin{array}{|l|l|l|l|} \hline \mbox{Data Value} & \mbox{Freq.} & \mbox{Relative Freq.} & \mbox{Cumulative Relative Freq.} \\ \hline 2 & 3 & 0.15 & 0.15 \\ \hline 3 & 5 & 0.25 & 0.40 \\ \hline 4 & 3 & 0.15 & 0.55 \\ \hline 5 & 6 & 0.30 & 0.85 \\ \hline 6 & 2 & 0.10 & 0.95 \\ \hline 7 & 1 & 0.05 & 1 \\ \hline \end{array} $$
Fact: For some data, especially continuous data, we must organize it into intervals rather than the values of the data themselves.
This should become clearer with an example.
Example: Consider the frequency table below which shows in inches, the annual rainfall in a sample of towns. $$ \begin{array}{|l|l|l|l|} \hline \mbox{Rainfall (in)} & \mbox{Freq.} & \mbox{Relative Freq.} & \mbox{Cumulative Relative Freq.} \\ \hline 3.00–4.99 & 6 & 0.12 & 0.12 \\ \hline 5.00–6.99 & 7 & 0.14 & 0.26 \\ \hline 7.00–8.99 & 15 & 0.30 & 0.56 \\ \hline 9.00–10.99 & 8 & 0.16 & 0.72 \\ \hline 11.00–12.99 & 9 & 0.18 & 0.90 \\ \hline 13.00– & 5 & 0.10 & 1.00 \\ \hline %& \mbox{Total} = 50 & \mbox{Total} = 1.00 & \\ \hline \end{array} $$ a) Find the percentage of rainfall that was less than $9.00$ inches.
b) Find the percentage of rainfall that is between $7.00$ and $12.99$ inches.
c) What percentage of towns got $13.00$ inches of rain or more?