Cumulative Frequency Diagrams
are very closely related to Frequency Polygons. They both provide a way of displaying univariate grouped data. Note that you may be required to produce frequency polygons/cumulative frequency diagrams for a given dataset but you may also be asked to do it for a dataset that you are already familiar with. See more on this.

## Frequency Polygons

Frequency Polygons are a common form of chart used for presenting continuous grouped data. Frequencies are plotted on the graph as points (or crosses etc.) and joined up by straight lines. They are called frequency polygons because polygons are shapes where the vertices are joined up by straight lines. However, the first and final points are not joined up so it should be thought of as more of an open shape. The polygon should never touch the bottom axis (unless the data point is 0). See more on Frequency Polygons.

You may be asked to add a Frequency Polygon to a Histogram. See Histograms Example 1.

## Cumulative Frequency Diagrams

‘Cumulative’ is the adjective for accumulate. It means to add up successively. It follows that cumulative frequency is where the grouped frequencies are added up at each step. Cumulative Frequency Diagrams look very similar to Frequency Polygons but since frequency is added up each time, the graph should trend upwards. Note that for a frequency polygon, the points align with the middle of the interval. For a cumulative frequency diagram, the points align with the end of the intervals. Cumulative frequency diagrams always start from the x-axis.
The lower quartile, the median and the upper quartile can be estimated from a cumulative frequency table and diagram. See the example below for a demonstration.

Frequency Polygons and Cumulative Frequency Diagrams are useful for finding estimates for the quartiles and percentiles but are not so good at showing the ‘spread’ of the data (see Measures of Variation). Box Plots and Histograms are good for this. See the example in Box Plots for more information.

## Exam-Style Example

The following table shows the prices in millions of pounds of some houses in a given postcode:

 House Price, P (£m) Frequency $0.2\leq P\hspace{2pt}\textless\hspace{2pt} 0.3$ $21$ $0.3\leq P\hspace{2pt}\textless\hspace{2pt} 0.4$ $37$ $0.4\leq P\hspace{2pt}\textless\hspace{2pt} 0.5$ $24$ $0.5\leq P\hspace{2pt}\textless \hspace{2pt}0.6$ $11$ $0.6\leq P\hspace{2pt}\textless\hspace{2pt} 0.7$ $7$
1. Create a cumulative frequency diagram to show this data.
2. Find an estimate for the median, the interquartile range and the 10th to 90th percentile range for house prices in this area.
3. Show this data in a box plot.

Click here for more statistical analysis using this example (Ranges – Example 2).

Alternatively, click here to practice exam questions from Past Applied Maths Papers.

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