### 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}

\]

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