Chapter 1. Descriptive Statistics: Frequency Distributions
Frequency Distribution Tables
Frequency measures
Definitions
The absolute frequency of a score is the number of times that particular score occurs in the dataset and is denoted by lowercase #f#.
The relative frequency of a score is the absolute frequency of that score divided by the total number of scores in the dataset and is denoted #f_{relative}#.
To transform a relative frequency into a percent frequency, multiply the relative frequency by #100\%#. Percent frequencies are denoted #\%f#.
Formulas
\[\begin{array}{rcl}
f &=& \text{number of times} \\
&&\text{a score occurs}\\
\\
f_{relative} &=& \dfrac{f}{n}\\
\\
\\
\%f &=& f_{relative} \cdot 100\%\\
\end{array}\]
Consider the following set of #n=20# scores:
\[4,\,\,5,\,\,7,\,\,5,\,\,9,\,\,8,\,\,7,\,\,7,\,\,11,\,\,6,\,\,4,\,\,9,\,\,9,\,\,11,\,\,9,\,\,10,\,\,12,\,\,12,\,\,8,\,\,7\]
The table belows displays the absolute, relative, and percent frequency of each score in the set.
\[
\begin{array}{c|c|c|c}
\text{Score }(X)&\text{Absolute frequency }(f)&\text{Relative frequency }(f_{relative})&\text{Percent frequency }(\%f)\\
\hline
4&2&0.10&10\%\\
5&2&0.10&10\%\\
6&1&0.05&5\%\\
7&4&0.20&20\%\\
8&2&0.10&10\%\\
9&4&0.20&20\%\\
10&1&0.05&5\%\\
11&2&0.10&10\%\\
12&2&0.10&10\%\\
\end{array}
\]
Cumulative frequency measures
Definitions
The cumulative frequency of a score is the number of scores that have a value at or below the score in question and is denoted #cf#.
The cumulative relative frequency is the proportion of scores that have a value less than or equal to the score in question and is denoted #cf_{relative}#.
The cumulative percent frequency is the percentage of scores that have a value less than or equal to the score in question and is denoted #c\%#.
Formulas
\[\begin{array}{rcl}
cf &=& \text{sum of all } f \text{ up to and} \\ &&\text{including that value}\\
\\
cf_{relative} &=& \dfrac{cf}{n}\\
\\
\\
c\% &=& cf_{relative} \cdot 100\%\\
\end{array}\]
Consider the following set of #n=20# scores:
\[4,\,\,5,\,\,7,\,\,5,\,\,9,\,\,8,\,\,7,\,\,7,\,\,11,\,\,6,\,\,4,\,\,9,\,\,9,\,\,11,\,\,9,\,\,10,\,\,12,\,\,12,\,\,8,\,\,7\]
The table belows displays the absolute, cumulative, cumulative relative, and cumulative percent frequency of each score in the set.
\[
\begin{array}{c|c|c|c|c}
&\text{Absolute}&\text{Cumulative}&\text{Cumulative}&\text{Cumulative}\\
\text{Score }(X)&\text{frequency }(f)&\text{frequency }(cf)&\text{relative frequency }(cf_{relative})&\text{percent frequency }(\%c)\\
\hline
4&2&2&0.10&10\%\\
5&2&4&0.20&20\%\\
6&1&5&0.25&25\%\\
7&4&9&0.45&45\%\\
8&2&11&0.55&55\%\\
9&4&15&0.75&75\%\\
10&1&16&0.80&80\%\\
11&2&18&0.90&90\%\\
12&2&20&1.00&100\%\\
\end{array}
\]
Or visit omptest.org if jou are taking an OMPT exam.
