Raising the bar

Chapter 3. Describing Data: Numerical

Unit 3.1. Three representations of data: raw, ordered and frequency

In this chapter we consider numerical characteristics of sets of data. As you think about the definitions given here, keep in mind three possible representations of observations on a real random variable.

Three representations of data.

1. Raw representation. Write down the data in the order you observe them:
   $latex {x_1},\,...,\,{x_n}. \qquad (\textrm{Eq. 3.1})$
2. Ordered representation. Put the observations on the real line and renumber them in an ascending order:
   $latex {y_1} \le ... \le {y_n}. \qquad (\textrm{Eq. 3.2})$
   These are the same points as in Eq. (3.1), just the order is different, so the number of points is the same.
3. Frequency representation. This is a table similar to Table 2.1 (see Frequencies and distributions). That is, you see how many distinct values there are among the observed points and denote them $latex {z_1} \le ... \le {z_m}$ (obviously, $latex m$ does not exceed $latex n$). For each $latex {z_i}$ you find its absolute frequency $latex {n_i}$ and relative frequency $latex {r_i}$. The result will be Table 3.1.

Table 3.1. Absolute and relative frequencies distribution.

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