12.1.C. Frequency Polygon

There is yet another visual way of representing quantitative data and its frequencies. This is a polygon. To see what we mean, consider the histogram represented by Fig. 12.3. Let us join the mid-points of the upper sides of the adjacent rectangles of this histogram by means of line segments. Let us call these mid-points B, C, D, E, F and G. When joined by line segments, we obtain the figure BCDEFG (see Fig. 12.6).

A frequency polygon is a graphical representation that connects the midpoints of the top edges of bars in a histogram with straight lines. It effectively summarizes the data distribution represented by the histogram, helping visualize trends and patterns.

Imagine the frequency polygon as a line graph that tracks the movement of a roller coaster. Each peak represents a high frequency in the dataset, showing that a lot of data points fall within that category.

To complete the polygon, we assume that there is a class interval with frequency zero before 30.5 - 35.5, and one after 55.5 - 60.5, and their mid-points are A and H, respectively. ABCDEFGH is the frequency polygon corresponding to the data shown in Fig. 12.3. We have shown this in Fig. 12.6.

To ensure the frequency polygon represents the total area of the histogram, we add two hypothetical intervals at the beginning and end with a frequency of zero. This allows the polygon to extend smoothly and reflect the entire range of data.

Think of this step like placing end caps on a tunnel. If your tunnel (graph) doesn't have secure ends, people might think it leads nowhere. Including zero-frequency intervals keeps the graph consistent and complete, representing reality accurately.

Frequency polygons can also be drawn independently without drawing histograms. For this, we require the mid-points of the class-intervals used in the data. These mid-points of the class-intervals are called class-marks.

A frequency polygon can be created directly using the midpoints (class-marks) of the class intervals and their corresponding frequencies. This method is useful when histograms are not needed or desired, allowing for a simpler representation.

Picture trying to create a map of your favorite restaurants without drawing the actual streets. You can still show where the concentrated areas of your favorite spots are located, much like plotting points for the frequency polygon.

To find the class-mark of a class interval, we find the sum of the upper limit and lower limit of a class and divide it by 2. Thus, Class-mark = Upper limit + Lower limit / 2.

To determine the midpoint of a class interval, simply add the lower and upper values and divide by two. This gives us a single point that represents where most of the data within that interval lies.

It’s like finding the average age of a group of friends. If one friend is 20 and another is 30, saying they 'average the age of 25' helps you capture the essence of their ages without listing everyone.

We can now draw a frequency polygon by plotting the class-marks along the horizontal axis, the frequencies along the vertical axis, and then plotting and joining the points B(145, 5), C(155, 10), D(165, 20), E(175, 9), F(185, 6) and G(195, 2) by line segments.

To create the frequency polygon, you plot the calculated midpoints on the x-axis and the corresponding frequencies on the y-axis. Then connect these points with straight lines, forming the frequency polygon.

This process is akin to connecting the dots in a drawing. Each dot corresponds to critical data points, and when you connect them, a clear picture emerges, showing trends and relationships between pieces of information.

Frequency polygons are used when the data is continuous and very large. It is very useful for comparing two different sets of data of the same nature, for example, comparing the performance of two different sections of the same class.

Frequency polygons are particularly advantageous for visualizing large datasets, especially when comparing different groups. They allow for an easy visual comparison of distributions, showing where one group may outperform another.

Think of frequency polygons as colorful ribbons draped over park benches during a festival. Each ribbon (dataset) can be seen side by side, allowing attendees to easily note which bench (data group) is the most popular for sitting.

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