Frequency Tables: Class Boundaries, Counts and Midpoints

A frequency is a count, not a measurement

Twenty fictional learners complete a practice task. Their completion times are grouped below. The unit for time is minutes; the unit for frequency is learners.

The boundary rule is explicit: include the lower endpoint and exclude the upper endpoint. Thus 10 minutes belongs to the second class, not the first. All recorded times satisfy 0 ≤ t < 40.

Completion time t, in minutes	Frequency: learners	Class midpoint, in minutes
0 ≤ t < 10	4	5
10 ≤ t < 20	6	15
20 ≤ t < 30	8	25
30 ≤ t < 40	2	35

Every class has width 10 minutes. The intervals do not overlap and leave no gap within the stated range.

Count learners by adding frequencies

Total learners = 4 + 6 + 8 + 2 = 20. Do not add the midpoints to count people: 5, 15, 25 and 35 are time values, not numbers of learners.

Fewer than 20 minutes includes the first two classes. Count = 4 + 6 = 10 learners, or 10/20 × 100 = 50% of the group. The value 20 itself is excluded from these first two classes and belongs to 20 ≤ t < 30.

Find the midpoint and understand its limit

For a class from a to b, midpoint = (a + b)/2. For 20 ≤ t < 30, midpoint = (20 + 30)/2 = 25 minutes.

This marks the centre of that class. It does not say all eight learners finished in 25 minutes. The table does not reveal their exact individual times. Do not claim an exact individual result or an exact mean from the midpoints alone.

Link the table to chart data

For an equal-width frequency histogram using frequency on the vertical axis, these four adjacent classes would have bar heights 4, 6, 8 and 2 on a common scale. The third class contains twice as many learners as the first: 8/4 = 2. This lesson uses equal-width classes only.

A frequency polygon uses class midpoint on the horizontal axis and frequency on the vertical axis. The data points for these four classes are (5, 4), (15, 6), (25, 8) and (35, 2). In the point (25, 8), 25 is a time in minutes and 8 is a count of learners. It does not mean a learner finished in 8 minutes.

These are explicit data for the charts; the table is enough to check the coordinates and counts. Reading a drawn axis or estimating an unlabelled height is a separate skill.

Check yourself

1. Which classes contain an exact time of 10 minutes and an exact time of 30 minutes?
2. How many learners took at least 20 minutes? What percentage of the group is that?
3. Give the frequency-polygon point for 30 ≤ t < 40. Explain both coordinates.

Worked answers

1. The lower endpoint is included. 10 belongs to 10 ≤ t < 20, and 30 belongs to 30 ≤ t < 40. Assigning either boundary to the previous class would count it under the wrong rule. A time of 40 minutes is outside this table's stated range.
2. At least 20 minutes means the last two classes. Count = 8 + 2 = 10 learners. Total = 20, so percentage = 10/20 × 100 = 50%. The first two classes account for the other 10, checking the total.
3. Midpoint = (30 + 40)/2 = 35 minutes; frequency = 2 learners. The point is (35, 2). The first coordinate locates the class centre; the second reports its count. It does not assert that either learner's exact time was 35 minutes.

The grouped data and practice questions are original. They are not attributed to an SSC paper.