Cumulative frequency diagrams
A cumulative frequency diagram creates a running total of the amounts within a table.
Example
The table below shows the lengths of 40 babies at birth.
To calculate the cumulative frequencies, add the frequencies together.
| Length (cm) | Frequency | Cumulative frequency |
| \(30 \textless l \leq 35\) | 4 | 4 |
| \(35 \textless l \leq 40\) | 10 | 14 (\(4 + 10 = 14\)) |
| \(40 \textless l \leq 45\) | 11 | 25 (\(14 + 11 = 25\)) |
| \(45 \textless l \leq 50\) | 12 | 37 (\(25 + 12 = 37\)) |
| \(50 \textless l \leq 55\) | 3 | 40 (\(37 + 3 = 40\)) |
| Length (cm) | \(30 \textless l \leq 35\) |
|---|---|
| Frequency | 4 |
| Cumulative frequency | 4 |
| Length (cm) | \(35 \textless l \leq 40\) |
|---|---|
| Frequency | 10 |
| Cumulative frequency | 14 (\(4 + 10 = 14\)) |
| Length (cm) | \(40 \textless l \leq 45\) |
|---|---|
| Frequency | 11 |
| Cumulative frequency | 25 (\(14 + 11 = 25\)) |
| Length (cm) | \(45 \textless l \leq 50\) |
|---|---|
| Frequency | 12 |
| Cumulative frequency | 37 (\(25 + 12 = 37\)) |
| Length (cm) | \(50 \textless l \leq 55\) |
|---|---|
| Frequency | 3 |
| Cumulative frequency | 40 (\(37 + 3 = 40\)) |
A cumulative frequency diagram is drawn by plotting the upper class boundary with the cumulative frequency. The upper class boundaries for this table are 35, 40, 45, 50 and 55.
Cumulative frequency is plotted on the vertical axis and length is plotted on the horizontal axis.
Finding averages from a cumulative frequency
A cumulative frequency diagram is a good way to represent data to find the The median is the value of the middle item of data when all the data is arranged in order., which is the middle value.
To find the median value, draw a line across from the middle value of the table. In the example above, there are 40 babies in the table. The middle of these 40 values is the 20th value, so go across from this value and find the median length.
Finding the interquartile range
A cumulative frequency diagram is also a good way to find the interquartile range, which is the difference between the The upper quartile (Q3) is three quarters of the way through the data, after the data have been arranged in order of size. and The lower quartile (Q1) is one quarter of the way through the data, after the data have been arranged in order of size..
The interquartile range is a measure of how spread out the data is. It is more reliable than the range because it does not include extreme values. A high value for the interquartile range shows that the data is spread out. A low value for the interquartile range means the data is closer together or more consistent.
Example
There are 40 babies in the table, so to find the lower quartile, find \(\frac{1}{4}\) of 40, which is the 10th value. Reading from the graph, the lower quartile is 38.
To find the upper quartile, find \(\frac{3}{4}\) of 40, which is the 30th value. Reading from the graph, the upper quartile is 47.
The interquartile range is the upper quartile – the lower quartile, so for this data the interquartile range is 47 – 38 = 9.