01 · Read a class
For 4.5 ≤ m < 5.1 kg, find the boundaries, width and midpoint.
Hint
The decimals are already the true interval cutoffs.
Worked solution
Boundaries 4.5 and 5.1 kg; width 0.6 kg; midpoint 4.8 kg.
Understand · explore · practise
Class boundaries and midpoints
Read grouped classes correctly, account for recording precision, and calculate widths and midpoints. Original examples, a manual interval model and worked practice.
Before you startContinuous and discrete data, decimals and inequalities.
01 / A class is an interval
Understand what the labels represent before calculating.
Width = upper boundary − lower boundary. Midpoint = (lower boundary + upper boundary)/2.
Boundaries are the interval’s cutoffs, not necessarily observed values.
A grouped frequency table counts observations in classes. It normally hides the exact individual values. A class written with inequalities already specifies its boundaries. A class written as rounded labels may require adjustment for the stated recording precision.
Read the recording precisionExplore
Width = 24.5 − 19.5 = 5; midpoint = (19.5 + 24.5)/2 = 22.
The possible rounded labels are 20, 21, 22, 23, 24. Extend by half a centimetre at each outer edge.
The diagram rescales each interval to fit; screen lengths cannot be used to compare their numerical widths. For these positive measurements, nearest-step rounding sends exact halfway values upwards, giving a closed lower and open upper endpoint.
02 / Explicit inequalities
Use the boundaries as written.
A class is 12 ≤ t < 18 seconds.Worked example
Lower boundary 12; upper boundary 18
The inequalities specify the interval directly.
Width = 18 − 12 = 6 seconds
Do not add an extra unit.
Midpoint = (12 + 18)/2 = 15 seconds
Halfway between the boundaries.
12 belongs; 18 does not
An open upper cutoff is still the upper boundary.
02 · Which class?
Classes are 0 ≤ t < 5, 5 ≤ t < 10 and 10 ≤ t < 15. Which contains t = 10?
Hint
Check the endpoint signs.
Worked solution
The third class, 10 ≤ t < 15. The middle class excludes 10.
03 / Rounded whole-unit labels
Extend by half the recording step at the outside edges.
Lengths are rounded to the nearest centimetre. A class is labelled 20–24 cm inclusive.Worked example
Smallest rounded label 20 represents 19.5 ≤ x < 20.5
Use the stated half-up tie rule for positive lengths.
Largest rounded label 24 represents 23.5 ≤ x < 24.5
The intermediate labels fill the intervals in between.
Combined interval: 19.5 ≤ x < 24.5
The true boundaries are 19.5 and 24.5 cm.
Width = 5 cm; midpoint = 22 cm
Using 24 − 20 would incorrectly give width 4.
Watch: rounded labels cover intervals
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03 · A different rounded group
Times are rounded to the nearest second and grouped as recorded values 8–12 inclusive. Find the boundaries, width and midpoint.
Hint
Extend by 0.5 s at each outer edge.
Worked solution
Boundaries 7.5 and 12.5 s; width 5 s; midpoint 10 s.
04 / Other recording steps
Half a unit means half the stated unit of precision.
Rounded labels a to b, step h: boundaries a − h/2 and b + h/2
This assumes all consecutive rounded labels from a to b are included.
Masses are rounded to the nearest 0.1 kg; the recorded group is 1.6–1.9 kg.Worked example
h/2 = 0.05 kg
Half of 0.1 is 0.05, not 0.5.
Boundaries = 1.55 and 1.95 kg
Extend each outer edge by 0.05.
Width = 0.40 kg; midpoint = 1.75 kg
Retain the original unit.
04 · Hundredths
Measurements rounded to the nearest 0.01 m are grouped as 0.32–0.36 m inclusive. Find width and midpoint.
Hint
Use half-step 0.005 m.
Worked solution
Boundaries 0.315 and 0.365 m; width 0.050 m; midpoint 0.340 m.
05 · Five-unit recording
A positive measurement is rounded to the nearest 5 units. A group includes recorded labels 20, 25 and 30. Find its boundaries.
Hint
The half-step is 2.5.
Worked solution
17.5 and 32.5 units. The width is 15 units and the midpoint is 25.
05 / Make neighbouring classes meet
Avoid gaps and double counting.
For lengths rounded to the nearest centimetre, groups 10–14 and 15–19 correspond to boundaries 9.5–14.5 and 14.5–19.5. They meet at 14.5. With the stated rounding rule, 14.5 enters the upper group. Explicit half-open intervals give an unambiguous allocation.
Unequal class widths are allowed. Later, histograms will use frequency density to handle different widths.
06 · A gap in coverage
What is wrong if continuous classes are written 0 ≤ x < 5 and 6 ≤ x < 10 and intended to cover all values from 0 to 10?
Hint
Consider x = 5.5.
Worked solution
The interval 5 ≤ x < 6 is missing. Use adjoining boundaries appropriate to the intended grouping.
07 · Unequal widths
Classes are 0 ≤ x < 4, 4 ≤ x < 10 and 10 ≤ x < 20. Find their widths.
Hint
Subtract each lower boundary from its own upper boundary.
Worked solution
The widths are 4, 6 and 10. They are not all equal.
06 / What grouping loses
A midpoint is a representative value, not a recovered observation.
If seven observations fall in 20 ≤ x < 30, their exact values and their exact mean are unknown. The midpoint 25 can represent the class when estimating a grouped mean, but every observation need not equal 25. An open-ended class such as x ≥ 50 has no finite upper boundary, so its midpoint cannot be calculated without an additional assumption.
08 · An unknown mean
Four values lie in 10 ≤ x < 20. Is their mean necessarily 15?
Hint
The values could cluster near either side.
Worked solution
No. Fifteen is the class midpoint, not the known mean of the four values.
09 · An open-ended class
Can you calculate the midpoint of “at least 80 minutes” from this description alone?
Hint
There is no stated upper cutoff.
Worked solution
No. A finite upper boundary or a justified extra assumption is required.
07 / Read, adjust, subtract and average
Recording precision decides whether adjustment is needed.
- Identify whether boundaries are already explicit.
- If labels are rounded measurements, use half the recording step.
- Calculate width by subtraction.
- Calculate midpoint by averaging the boundaries.
- Check adjacent intervals and endpoint conventions.
- Do not treat a midpoint as an exact observation.
10 · Compare two definitions
Compare the width of 30 ≤ x < 34 with the width of recorded labels 30–34 for measurements rounded to the nearest unit.
Hint
The definitions describe different intervals.
Worked solution
The explicit class has width 4. The rounded-label class has boundaries 29.5 and 34.5, so width 5. The same-looking labels can describe different intervals.
Section 1 of 7 · A class is an interval