# Statistical Sequences and Frequencies

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## Statistical Sequence

In recording data we generate a statistical sequence. The original, unprocessed sequence is called raw data. Given an appropriate scale level (i.e. at least an ordinal scale), we can sort the raw data, thus creating an ordered sequence. Data collected at the same point in time or for the same period of time on different elements are called cross-section data. Data collected at different points in time or for different periods of time on the same element are called time-series data. The sequence of observations is ordered along the time axis.

## Frequency

The number of observations falling into a given class is called the frequency. Classes are constructed to summarize continuous or quasi-continuous data by means of frequencies. In discrete data one regularly encounters so-called ties, i.e. two or more observations taking on the same value. Thus, discrete data may not require grouping in order to calculate frequencies. Absolute frequency Counting the number of observations taking on a specific value yields the absolute frequency: ${\displaystyle h\left(X=x_{j}\right)=h\left(x_{j}\right)=h_{j}}$ When data are grouped, the absolute frequencies of classes are calculated as follows: ${\displaystyle h\left(x_{j}\right)=h\left(x_{j}^{l}\leq X Properties:${\displaystyle 0\leq h\left(x_{j}\right)\leq n}$${\displaystyle \sum _{j}h\left(x_{j}\right)=n}$ Relative frequency The proportion of observations taking on a specific value or falling into a specific class is called the relative frequency, the absolute frequency standardized by the total number of observations. ${\displaystyle f\left(x_{j}\right)={\frac {h\left(x_{j}\right)}{n}}}$

Frequency distribution By standardizing class frequencies for grouped data by their respective class widths, frequencies for differently sized classes are made comparable. The resulting frequencies can be compiled to form a frequency distribution. {\displaystyle {\begin{aligned}{\widehat {h}}\left(x_{j}\right)&={\frac {h\left(x_{j}\right)}{x_{j}^{u}-x_{j}^{l}}}\\{\widehat {f}}\left(x_{j}\right)&={\frac {f\left(x_{j}\right)}{x_{j}^{u}-x_{j}^{l}}},\end{aligned}}} where ${\displaystyle x_{j}^{l},x_{j}^{u}}$ are the upper and lower class boundaries with ${\displaystyle x_{j}^{l}.

150 persons have been asked for their marital status: 88 of them are married, 41 single and 21 divorced. The four conceivable responses have been assigned categories as follows:

• single: ${\displaystyle x_{1}}$
• married: ${\displaystyle x_{2}}$
• divorced: ${\displaystyle x_{3}}$
• widowed: ${\displaystyle x_{4}}$

The number of statistical elements is ${\displaystyle n=150}$. The absolute frequencies given above are:

• ${\displaystyle h\left(x_{1}\right)=41}$
• ${\displaystyle h\left(x_{2}\right)=88}$
• ${\displaystyle h\left(x_{3}\right)=21}$
• ${\displaystyle h\left(x_{4}\right)=0}$

Dividing by the sample size ${\displaystyle n=150}$ yields the relative frequencies:

• ${\displaystyle f\left(x_{1}\right)=41/150=0.27}$
• ${\displaystyle f\left(x_{2}\right)=88/150=0.59}$
• ${\displaystyle f\left(x_{3}\right)=21/150=0.14}$
• ${\displaystyle f\left(x_{4}\right)=0/150=0.00}$

Thus, 59 per cent of the persons surveyed are married, 27 per cent are single and 15 per cent divorced. No one is widowed.