# Statistics Frequency Polygons

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### Graphing Distribution : Frequency Polygons

#### The Objectives:

• Frequency polygons - creation and interpretation.

• Cumulative frequency polygons - creation and interpretation.

• Overlaid frequency polygons - creation and interpretation.

### Frequency Polygons:

Graphical display of the frequency table can also be achieved through a frequency polygon. To create a frequency polygon the intervals are labeled on the X-axis and the Y axis represents the height of a point in the middle of the interval. The points are then joined are connected to the X-axis and thus a polygon is formed. So, frequency polygon is a graph that is obtained by connecting the middle points of the intervals. We can create a frequency polygon from a histogram also. If the middle top points of the bars of the histogram are joined, a frequency polygon is formed.

Frequency polygon and histogram fulfills the same purpose. However, the former one is useful in comparison of different datasets. In addition to that frequency polygon can be used to display cumulative frequency distributions.

#### How to Create a Frequency Polygon?

As already mentioned, histogram can be used for creating frequency polygon. The X-axis represents the scores of the dataset and the Y-axis represents the frequency for each of the classes. Now, mark the mid top points of each bar of the created histogram for each class interval. One generally uses a dot for marking. Now join all the dots by straight lines and connect it with the X-axis on both sides. For creating a frequency polygon without a histogram, you just need to consider the midpoint of the class intervals, such that it corresponds to the frequencies. Then connect the points as stated above.

The following table is the frequency table of the marks obtained by 50 students in the pre-test examination.

Table 1. Frequency Distribution of the marks obtained by 50 students in the pre-test examination.

#### Cumulative frequency (Less than type)

30.5-40.5

1

1

40.5-50.5

14

20

50.5-60.5

20

40

60.5-70.5

7

47

70.5-80.5

3

50

Total

50

The labels of the X-axis are the midpoints of the class intervals. So the first label on the X-axis will be 35.5, next 45.5, followed by 55.5, 65.5 and lastly 75.5. The corresponding frequencies are then considered to create the frequency polygon. The shape of the distribution can be determined from the created frequency polygon. The frequency polygon is shown in the following figure.

Fig 1: Frequency polygon of the distribution of the marks obtained by 50 students in the pre-test examination.

From the above figure we can observe that the curve is asymmetric and is right skewed.

#### Cumulative Frequency Polygon:

Cumulative frequency polygon is similar to a frequency polygon. The difference is that in creating a cumulative frequency polygon we consider cumulative frequencies instead of actual frequencies. Cumulative frequency of less than type is obtained by adding the frequency of each class interval to the sum of all frequencies in the lower intervals. In table 1 for example, the cumulative frequency for the class interval 30.5-40.5 is 6 since the sum of all frequencies in the lower intervals is 0. Again the cumulative frequency for the class interval 40.5-50.5 is 20 since the sum of all frequencies in the lower intervals is 14, i.e, 6+14=20, so for the next interval it will be 6+14+20=40 and so on.

The following is the cumulative frequency polygon

Fig2: Cumulative Frequency polygon of the marks obtained by 50 students in the pre-test examination.

#### Overlaid Frequency Polygon:

Also to compare the distributions of different data sets, frequency polygon can be used. In such case frequency polygons of different data are drawn on the same graph. The above thing can be made clear through illustrations.

The following is an example of dice where the distribution of observed frequencies and the distribution of expected frequencies are compared for different scores of two dice. The frequency curves of the two distributions are used for comparison.

Fig3: Overlaid Frequency polygon of the distributions of rolling two dice

The observed curve overlaps expected curve. The expected curve is smooth while the observed curve is not smooth.

Also cumulative frequency polygon can also be plotted in the same graph. The following figure shows such plot. The marks of two papers are compared through cumulative frequency polygon.

Fig4: Overlaid cumulative frequency polygon

Fig5: Frequency polygon drawn over the histogram

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