Understanding Frequency Polygons and Discrete Variables

A frequency polygon is a graphical representation of a frequency distribution. It is typically constructed by plotting points corresponding to the frequencies of classes or values against the respective class marks (for grouped data) or variable values (for discrete data), and then joining these points with line segments. For a discrete variable, the frequency distribution lists the frequency of each distinct value or group of values.

Let's examine each given pair in the context of a frequency polygon derived from the frequency distribution of a discrete variable:

Analyzing Each Pair

1. Base line of the polygon - X-axis

   When drawing a frequency polygon, the variable values (or class marks for grouped data) are plotted along the X-axis, and the frequencies along the Y-axis. The polygon is usually closed by connecting the first and last points to the X-axis at appropriate locations (often the midpoints of the preceding and succeeding class intervals). Therefore, the X-axis serves as the base line for the polygon.

   This pair is generally considered correctly matched.

2. Ordinates of the vertices of the polygon - Class frequencies

   The vertices (points) of the frequency polygon are plotted at heights corresponding to the frequencies. The ordinate refers to the y-coordinate of a point. Thus, the height of each vertex above the X-axis represents the frequency of the corresponding value or class. For a discrete variable, if points are plotted at distinct values, the height is the frequency of that value. If grouped into classes, the height is the class frequency.

   This pair is correctly matched.

3. Abscissa of the vertices of the polygon - Class marks of the frequency distribution

   The abscissa refers to the x-coordinate of a point. For a frequency polygon representing grouped frequency distribution (which can be for discrete or continuous data), the vertices are plotted against the class marks (midpoints) of the respective classes on the X-axis. If it's ungrouped discrete data, the points would be plotted at the discrete variable values themselves. However, if the discrete variable data is presented in a grouped frequency distribution, using class marks is standard practice for constructing a frequency polygon. Assuming a grouped distribution context for the polygon, this statement holds true.

   This pair is generally correctly matched, especially for grouped discrete data.

4. Area of the polygon - Total frequency of the distribution

   The area under a frequency polygon is constructed to approximate the area under the corresponding histogram. The area of a histogram is equal to the total frequency of the distribution (when frequency density is used on the Y-axis or when class widths are uniform and frequency is on Y-axis). While the frequency polygon is designed to have approximately the same area as the corresponding histogram, it is not guaranteed to be exactly equal to the total frequency just by stating "Area of the polygon". The equality of areas holds when the polygon is closed by connecting the first and last points to the X-axis at points corresponding to the class marks (or midpoints of intervals) one step before the first class and one step after the last class, assuming uniform class widths. Simply stating "Area of the polygon" without this condition does not guarantee it equals the total frequency.

   This pair is not correctly matched as a general property without specific constructions.

Identifying the Incorrect Match

Based on the analysis, the pair that is not correctly matched is the one relating the area of the polygon to the total frequency. While the polygon's area is related to the total frequency and designed to approximate the histogram's area (which equals total frequency under certain conditions), stating the area of the polygon *is* the total frequency is not universally correct without specifying how the polygon is closed on the X-axis.

Therefore, the incorrect pair is:

Area of the polygon - Total frequency of the distribution

Revision Table: Frequency Polygon Components

Additional Information: Frequency Polygon vs. Histogram

A frequency polygon is closely related to a histogram. A frequency polygon can be obtained by connecting the midpoints of the tops of the bars of a histogram. For grouped frequency distributions, both represent the distribution of data. However, a frequency polygon provides a smooth curve-like representation and is useful for comparing multiple distributions on the same graph. A histogram uses bars to show the frequency of each class interval.

For discrete variables, a frequency polygon might connect points plotted at the specific values with their frequencies on the X-axis, or it might be constructed from a grouped frequency distribution of the discrete variable using class marks.