Figure 1. How to create a frequency polygon
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An old definition for a frequency polygon is: "A distribution of discrete variates may be represented graphically by plotting the points, and drawing a broken line through them." (Kenney, 1939). This mentions the frequency polygon is used for discrete variates. However the frequency polygon is often used for grouped tables. A more modern definition is shown below.
| Frequency polygon “A diagram used to display graphically the values in a frequency distribution. The frequencies are graphed as ordinate against the class mid-points as abscissae. The points are then joined by a series of straight lines.” (Everitt, 2004, p. 152) |
A frequency polygon is like a histogram that uses the same bin size, and therefore the frequency (and not the frequency density), but instead of columns it places a dot at the center of the top of each column and then connects these dots with straight lines. Figure 1 shows the process of creating a frequency polygon.
Figure 1. How to create a frequency polygon
On the top left in Figure 1 we start with a histogram that uses equal bin sizes and has the frequency on the vertical axis. We then place a dot at the bin midpoint (this is the upper bound + lower bound, then divided by 2) for each bar (top right), then we connect the dots by straight lines (bottom left) and finally remove the bars and dots (bottom right).
You don't have to draw a histogram first, you can also simply use the data from a frequency table. The only additional calculation needed is for the midpoints. Suppose we are given frequency and bins of the ages from another survey as shown in Table 1.
| Age | Frequency | lower bound | upper bound | midpoint |
|---|---|---|---|---|
| 0 < 10 | 5 | 0 | 10 | (0 + 10) / 2 = 5 |
| 10 < 20 | 7 | 10 | 20 | (10 + 20) / 2 = 15 |
| 20 < 30 | 10 | 20 | 30 | (20 + 30) / 2 = 25 |
| 30 < 40 | 8 | 30 | 40 | (30 + 40) / 2 = 35 |
| 40 < 50 | 4 | 40 | 50 | (40 + 50) / 2 = 45 |
The midpoints will be used for the position on the horizontal axis, and the frequency for the vertical axis. From table 1 we can then create the frequency polygon as shown in Figure 2.
Figure 2. Frequency polygon from Table 1
A line indicates that there is a continuous movement. A frequency polygon should therefor be used for scale variables that are binned, but sometimes a frequency polygon is also used for ordinal variables. Instead of starting with a histogram, you then start with a bar-chart (or a Cleveland dot plot). Some authors (Lane, n.d.) would then call this a line graph, but others call a frequency polygon a line graph.
My personal problem with a frequency polygon is that it can be misleading. If we look at the lower right figure in Figure 1, we might think there are more than 10,000 people with the age of 29, but actually the bin 10 < 30 only has 10,000 people.
Instead of the absolute frequencies, the relative frequencies can also be used. If the cumulative frequencies are used, the graph is called an ogive (oh-jive), which will be discussed in the next segment.
Note that I've only draw line segments between midpoints of bins that we actually have. Some authors will argue to anchor the line on the horizontal axis by adding another bin at the beginning and end, each with a frequency of 0.