Frequency Distribution Tables

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

$\begin{array}{rcl} f &=& \text{# times a score occurs}\\ \\ \\ f_{relative} &=& \dfrac{f}{n}\\ \\ \\ \%f &=& f_{relative} \cdot 100\%\\ \end{array}$

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

$\begin{array}{rcl} cf &=& \begin{split}\text{sum of all consecutive f} \\ \text{up to and including that value}\end{split}\\ \\ \\ cf_{relative} &=& \dfrac{cf}{n}\\ \\ \\ \\ c\% &=& cf_{relative} \cdot 100\%\\ \end{array}$

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.

$\begin{array}{c|c|c|c|c} &\text{Absolute}&\text{Cumulative}&\text{Cumulative}&\text{Cumulative}\\ \text{Score }(X)&\text{frequency }(f)&\text{frequency }(cf)&\text{relative frequency }(cf_{relative})&\text{percent frequency }(\%c)\\ \hline 4&2&2&0.10&10\%\\ 5&2&4&0.20&20\%\\ 6&1&5&0.25&25\%\\ 7&4&9&0.45&45\%\\ 8&2&11&0.55&55\%\\ 9&4&15&0.75&75\%\\ 10&1&16&0.80&80\%\\ 11&2&18&0.90&90\%\\ 12&2&20&1.00&100\%\\ \end{array}$