How to Find Class Boundaries
In grouped quantitative data, class boundaries (also called real limits) convert discrete-looking class limits into truly continuous intervals. This prevents gaps between adjacent classes and makes histograms, frequency polygons, and ogives mathematically consistent.
Key Rule Using the Measurement Unit
If measurements are recorded to the nearest unit \(U\), then half a unit is \(\dfrac{U}{2}\). The class boundaries for a class with lower limit \(L\) and upper limit \(H\) are:
For integer-valued data (recorded to the nearest 1), \(U = 1\) and \(\dfrac{U}{2} = 0.5\). For values recorded to the nearest 0.1, \(U = 0.1\) and \(\dfrac{U}{2} = 0.05\).
Equivalent Rule Using Adjacent Class Limits
When classes are consecutive (for example, 10–19 followed by 20–29), the boundary between them is the midpoint between the upper limit of the first class and the lower limit of the next class:
This midpoint becomes the upper boundary of the first class and the lower boundary of the next class, ensuring there is no gap and no overlap.
Worked Example: 10–19, 20–29, 30–39
Suppose the data are whole-number measurements (nearest 1), so \(U = 1\) and \(\dfrac{U}{2} = 0.5\).
| Class limits | Lower boundary | Upper boundary | Class width |
|---|---|---|---|
| 10–19 | \(10 - 0.5 = 9.5\) | \(19 + 0.5 = 19.5\) | \(19.5 - 9.5 = 10\) |
| 20–29 | \(20 - 0.5 = 19.5\) | \(29 + 0.5 = 29.5\) | \(29.5 - 19.5 = 10\) |
| 30–39 | \(30 - 0.5 = 29.5\) | \(39 + 0.5 = 39.5\) | \(39.5 - 29.5 = 10\) |
The shared values \(19.5\) and \(29.5\) show why class boundaries matter: the interval for 10–19 ends exactly where the interval for 20–29 begins, which is required when drawing a histogram with adjacent bars.
Visualization: Class Limits vs Class Boundaries on a Number Line
Practical Checklist
- Identify the measurement unit \(U\) implied by the recorded values (nearest 1, nearest 0.1, etc.).
- Compute \(\dfrac{U}{2}\).
- For each class \(L\)–\(H\), use \(L - \dfrac{U}{2}\) and \(H + \dfrac{U}{2}\) as the class boundaries.
- Confirm adjacent classes share a boundary (the upper boundary of one equals the lower boundary of the next).
Correctly computed class boundaries ensure that grouped-data graphs represent a continuous scale and that class widths and cumulative frequencies are computed consistently.