This is “Presentation of Data”, section 1.3 from the book Beginning Statistics (v. 1.0).
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In this book we will use two formats for presenting data sets. The first is a data listAn explicit listing of all the individual measurements made on a sample., which is an explicit listing of all the individual measurements, either as a display with space between the individual measurements, or in set notation with individual measurements separated by commas.
The data obtained by measuring the age of 21 randomly selected students enrolled in freshman courses at a university could be presented as the data list
$$\begin{array}{ccccccccccc}\hfill 18\hfill & \hfill 18\hfill & \hfill 19\hfill & \hfill 19\hfill & \hfill 19\hfill & \hfill 18\hfill & \hfill 22\hfill & \hfill 20\hfill & \hfill 18\hfill & \hfill 18\hfill & \hfill 17\hfill \\ \hfill 19\hfill & \hfill 18\hfill & \hfill 24\hfill & \hfill 18\hfill & \hfill 20\hfill & \hfill 18\hfill & \hfill 21\hfill & \hfill 20\hfill & \hfill 17\hfill & \hfill 19\hfill & \end{array}$$or in set notation as
$$\left\{\mathrm{18,18,19,19,19,18,22,20,18,18,17,19,18,24,18,20,18,21,20,17,19}\right\}$$A data set can also be presented by means of a data frequency tableA table listing each distinct value x and its frequency f., a table in which each distinct value x is listed in the first row and its frequencyHow often a value x appears in a data set. f, which is the number of times the value x appears in the data set, is listed below it in the second row.
The data set of the previous example is represented by the data frequency table
$$\begin{array}{cccccccc}\hfill x\hfill & \hfill 17\hfill & \hfill 18\hfill & \hfill 19\hfill & \hfill 20\hfill & \hfill 21\hfill & \hfill 22\hfill & \hfill 24\hfill \\ \hfill f\hfill & \hfill 2\hfill & \hfill 8\hfill & \hfill 5\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}$$The data frequency table is especially convenient when data sets are large and the number of distinct values is not too large.
List all the measurements for the data set represented by the following data frequency table.
$$\begin{array}{cccccc}\hfill x\hfill & \hfill 31\hfill & \hfill 32\hfill & \hfill 33\hfill & \hfill 34\hfill & \hfill 35\hfill \\ \hfill f\hfill & \hfill 1\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 4\hfill & \hfill 2\hfill \end{array}$$List all the measurements for the data set represented by the following data frequency table.
$$\begin{array}{ccccccccc}\hfill x\hfill & \hfill 97\hfill & \hfill 98\hfill & \hfill 99\hfill & \hfill 100\hfill & \hfill 101\hfill & \hfill 102\hfill & \hfill 103\hfill & \hfill 105\hfill \\ \hfill f\hfill & \hfill 7\hfill & \hfill 5\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}$$Construct the data frequency table for the following data set.
$$\begin{array}{cccccc}22& 25& 22& 27& 24& 23\\ 26& 24& 22& 24& 26& \end{array}$$Construct the data frequency table for the following data set.
$$\begin{array}{c}\{\mathrm{1,5,2,3,5,1,4,4,4,3,2,5,1,3,2},\\ \mathrm{1,1,1,2}\}\end{array}$${31,32,32,32,32,32,33,33,33,33,33,33,34,34,34,34,35,35}.
$\begin{array}{ccccccc}\hfill x\hfill & \hfill 22\hfill & \hfill 23\hfill & \hfill 24\hfill & \hfill 25\hfill & \hfill 26\hfill & \hfill 27\hfill \\ \hfill f\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}.$