Cumulative frequency diagrams – Higher tier
A cumulative frequency table shows a running total of the frequencies. A cumulative frequency diagram reproduces this table as a graph.
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 cumulative frequency against the upper class boundary of the respective group. 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 the median and interquartile range from a cumulative frequency diagram
Using a cumulative frequency diagram is a good way to find an estimate of the The median is the value of the middle item of data when all the data is arranged in order. average, or middle, value, and interquartile range.
To find the median, work out \(\frac{1}{2}\) of the total frequency. Find this value on the vertical axis (the cumulative frequency axis). Draw a line across until it meets the curve. Draw a vertical line from that intersection to meet the horizontal axis. This will be the median.
The interquartile range 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.
To find the lower quartile, use the same method as for the median, except use \(\frac{1}{4}\) of the total frequency rather than \(\frac{1}{2}\). To find the upper quartile, use \(\frac{3}{4}\) instead.
Example
In the earlier example, there were 40 babies.
For the median:
\(\frac{1}{2}\) of 40 = 20. The median length is about 43 cm.
For the lower quartile:
\(\frac{1}{4}\) of 40 = 10. Read across at 10 and down. The lower quartile is about 38 cm.
For the upper quartile:
\(\frac{3}{4}\) of 40 = 30. Read across at 30 and down. The upper quartile is about 47 cm.
The interquartile range is the upper quartile – the lower quartile, so for this data the interquartile range is \(47 - 38 = 9\).