Class Boundaries — Definition, Formula & Examples

Class boundaries are the values that lie exactly halfway between the upper limit of one class and the lower limit of the next class in a frequency distribution. They eliminate the gaps between consecutive classes so every possible data value falls within a class.

Given adjacent classes with an upper limit $U_k$ for class $k$ and a lower limit $L_{k+1}$ for class $k+1$ , the shared class boundary is $\frac{U_k + L_{k+1}}{2}$ . The lower boundary of a class equals its lower limit minus half the gap, and the upper boundary equals its upper limit plus half the gap.

Key Formula

\text{Lower boundary} = L - \frac{d}{2}, \quad \text{Upper boundary} = U + \frac{d}{2}

Where:

$L$ = Lower class limit of the class
$U$ = Upper class limit of the class
$d$ = Gap between the upper limit of one class and the lower limit of the next (typically 1 for whole-number data)

How It Works

When data is grouped into classes like 10–19, 20–29, 30–39, there is a gap of 1 unit between each pair of consecutive classes. Class boundaries split that gap evenly. You subtract half the gap from each lower class limit to get the lower boundary, and add half the gap to each upper class limit to get the upper boundary. These boundaries are essential for drawing histograms correctly, since histogram bars touch each other with no space between them.

Worked Example

Problem: A frequency distribution has the classes 10–19, 20–29, and 30–39. Find the class boundaries for each class.

Find the gap: The gap between 19 (upper limit of class 1) and 20 (lower limit of class 2) is 1, so half the gap is 0.5.

d = 20 - 19 = 1, \quad \frac{d}{2} = 0.5

Compute boundaries: Subtract 0.5 from each lower limit and add 0.5 to each upper limit.

10\text{–}19 \rightarrow 9.5\text{–}19.5, \quad 20\text{–}29 \rightarrow 19.5\text{–}29.5, \quad 30\text{–}39 \rightarrow 29.5\text{–}39.5

Answer: The class boundaries are 9.5–19.5, 19.5–29.5, and 29.5–39.5. Notice that the upper boundary of each class equals the lower boundary of the next, leaving no gaps.

Why It Matters

Class boundaries are required when constructing histograms, since the bars must cover the entire number line with no gaps. They also appear when calculating the class midpoint or class width for grouped-data statistics, which you will encounter in AP Statistics and introductory college statistics courses.

Common Mistakes

Mistake: Confusing class limits with class boundaries.

Correction: Class limits are the smallest and largest values that belong to a class (e.g., 10 and 19). Class boundaries extend half a gap unit beyond the limits (e.g., 9.5 and 19.5). Use limits to describe data membership and boundaries to draw histograms.