Frequency Polygons

Definition: Frequency polygons are graphical devices utilized to communicate the shapes of distributions of a dataset. They are similar to histograms but provide an effective means for comparing distributions and displaying cumulative frequency distributions.

Initial Steps:
- Start by selecting a class interval, similar to the process followed when creating histograms.
- Construct an x-axis that represents the values of the scores in the data set.

Data Set: Consider a frequency distribution generated from scores on a psychology test, previously discussed in the histogram section.
Class Interval Choice: A class interval of 10 is chosen.
- First Interval: Ranges from 29.5 to 39.5 with a midpoint of 35.
- Observation: This interval contains no test scores, which is also true for the last interval ranging from 169.5 to 179.5.
Justification for Interval Selection: The selection of the first and last intervals that contain no scores allows the resulting line on the frequency polygon to touch the x-axis.
Y-Axis Creation: Draw a y-axis to indicate the frequency of each class interval.
Plotting Points: Use the frequency data from the frequency table:
- Place a point at the midpoint of each class interval representing its frequency on the y-axis.
- Note that the graph will not show the upper or lower limits of the class intervals.
Connecting Points: Connect the plotted points to form the frequency polygon.

Graph Characteristics: There are intervals below and above the bounds of the data, leading to the graph touching the x-axis on both ends.
Distribution Shape: Most scores range from 65 to 115. The distribution is noted as not symmetric; there is a gradual decline in the frequency of higher, good scores to the right, and a steeper decline for lower, poor scores to the left.
- Terminology: This phenomenon is referred to as a positive skew (or skewed to the right) where the tail on the right side of the distribution extends longer compared to the left.

Definition: A cumulative frequency distribution indicates the cumulative frequency for each class interval.
First Interval: Frequency and cumulative frequency are both zero.
Subsequent Intervals:
- The second interval has a frequency of 3; cumulative frequency thus also is 3.
- Third interval: Its frequency is 10, leading to a cumulative frequency calculation as follows:
- 0 + 3 + 10 = 13.
Final Interval Cumulative Frequency: For the last interval, which has a total of 642 students tested, its cumulative frequency equals 642.

Graph Similarity: The cumulative frequency polygon graph resembles the frequency polygon but uses cumulative frequency as the y-value for each plotted point.
- For instance, with intervals labeled:
- Interval 35: Frequency = 0
- Interval 45: Frequency = 3
- Interval 55: Frequency = 10
- Cumulative frequency for 55: 0 + 3 + 10 = 13.

Comparative Analysis: Frequency polygons facilitate direct comparison between different distributions. This is accomplished by overlaying multiple frequency polygons for varied datasets.
- Illustrative Example: A study involving movement time to targets using two shapes: a small rectangle and a large rectangle. Data is recorded for 20 trials for each target shape.
- Observation: The graph reflects that, generally, it took a longer time to reach the small target compared to the larger one, despite some overlap in times.
Cumulative Frequency Graphs: Similar comparisons can be made through cumulative frequency distributions, clearly evidencing the differences in movement times between the two targets.