0580
Histograms
Statistics
The formula both ways
- The syllabus gives the definition: frequency density = frequency ÷ class width
- Rearranged, that gives the reverse reading: frequency = frequency density × class width
- The class width is the upper bound minus the lower bound of the interval
- Set the work out in a table with two extra columns, one for the class width and one for the frequency density
- Write the division in each cell rather than only the answer, since that is what shows the method
Worked example
Frequency densities from a table
The table records the mass m grams of 110 items: 24 with 0 < m ⩽ 20, 35 with 20 < m ⩽ 30, 30 with 30 < m ⩽ 50 and 21 with 50 < m ⩽ 80. Find the frequency density of every class.
Solution:
- Find each class width first, then divide the frequency by it
| Mass (m g) | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0 < m ⩽ 20 | 24 | 20 | 24 ÷ 20 = 1.2 |
| 20 < m ⩽ 30 | 35 | 10 | 35 ÷ 10 = 3.5 |
| 30 < m ⩽ 50 | 30 | 20 | 30 ÷ 20 = 1.5 |
| 50 < m ⩽ 80 | 21 | 30 | 21 ÷ 30 = 0.7 |
- Compare the second and third classes: 35 items against 30, almost the same count, yet the second bar is more than twice as tall because those 35 are squeezed into a width of 10 while the 30 are spread across 20