Data Representation

1. Core Concept Overview & Intuition

Data representation is the systematic organization and visual display of collected observations so that patterns, concentrations, and comparisons become immediately intelligible. Raw data, however carefully gathered, is rarely informative in its original form: a long list of numbers or categories conceals the structure that statistical analysis requires. Representation converts such lists into compact, ordered structures.

Two complementary modes of representation are used. The first is tabular: observations are sorted into classes and counted, producing a frequency distribution. The second is graphical: the information in the table is rendered as a picture — a bar chart, histogram, pie chart, or line graph — so that relative magnitudes can be compared at a glance.

The organizing idea in both modes is the frequency, the number of observations falling into a given category or class. A frequency distribution answers the question "how many observations have each value (or fall in each interval)?" A graph answers the same question visually, trading numerical precision for immediate comparability.

The choice of representation depends on the type of data, a distinction established earlier in the course:

• Qualitative (categorical) data — nominal or ordinal — consists of labels or categories. Such data is represented by a frequency table, a bar chart, or a pie chart.
• Quantitative (numerical) data — discrete or continuous — consists of counts or measurements. Such data is represented by a frequency table, a histogram, a frequency polygon, or a line graph.

A second distinction governs the form of the frequency table itself. A discrete frequency distribution lists each distinct observed value against its frequency; it is appropriate when the variable takes isolated values. A grouped (continuous) frequency distribution collects values into class intervals and records the number of observations in each interval; it is necessary when the variable is continuous or takes many distinct values.

Graphs are not decoration. Each graph type encodes a specific comparison: bar charts compare category totals, histograms display the shape of a continuous distribution, pie charts display parts of a whole, and line graphs display change across an ordered sequence. Selecting the graph that matches the comparison of interest is the central practical skill of this topic.

2. Mathematical Formalization & Core Definitions

> Definition (Frequency). For a variable $X$ and a value or class $c$, the frequency of $c$, denoted $f$, is the number of observations in the data set that take the value $c$ or fall within the class $c$.

> Definition (Frequency Distribution). A frequency distribution is a table that pairs each distinct value or class interval of a variable with its corresponding frequency. The sum of all frequencies equals the total number of observations, denoted $N$ or $n$:

$$
\sum f = N.
$$

> Definition (Discrete Frequency Distribution). A discrete frequency distribution lists each distinct observed value $x$ of a discrete variable alongside its frequency $f$. No class intervals are used.

> Definition (Grouped Frequency Distribution). A grouped frequency distribution partitions the range of a continuous variable into non-overlapping class intervals and records the frequency of each interval. Each interval has a lower class limit and an upper class limit; the class width is the difference between consecutive lower class limits.

> Definition (Class Mark). The class mark (class midpoint) of a class interval is the average of its lower and upper class limits. For an interval with limits $L$ and $U$,

$$
\text{class mark} = \frac{L + U}{2}.
$$

> Definition (Cumulative Frequency). The cumulative frequency up to and including a given class interval is the total frequency of all values less than the upper class boundary of that interval. It is obtained by successively adding the frequencies of the classes up to that point.

> Definition (Bar Chart). A bar chart is a graph in which each category is represented by a rectangular bar whose height (or length) is proportional to the frequency of that category. Bars are separated by gaps, reflecting the discrete, non-continuous nature of the categories.

> Definition (Histogram). A histogram, or frequency histogram, consists of a set of rectangles whose bases are the class intervals and whose heights are proportional to the class frequencies. Adjacent rectangles touch, reflecting the continuity of the underlying variable.

> Definition (Frequency Polygon). A frequency polygon is a line graph of class frequency plotted against class mark. It is obtained by connecting the midpoints of the tops of the rectangles in the histogram.

> Definition (Pie Chart). A pie chart is a circular graph divided into sectors whose angles (and hence areas) are proportional to the frequencies of the categories. The whole circle represents the total frequency $N$.

> Definition (Line Graph). A line graph plots a variable against an ordered sequence (such as time or position) and joins consecutive plotted points with straight line segments, displaying trend and change.

The pie chart sector angle for a category with frequency $f$ out of total $N$ is determined by proportionality of the full circle:

$$
\text{sector angle} = \frac{f}{N} \times 360^{\circ}.
$$

This relation follows directly from the requirement that the sectors partition the complete circle of $360^{\circ}$ in the same proportion as the frequencies partition $N$.