Cumulative Frequency

El método, paso a paso

1. 1

   Build the running total

   frequencies 4, 7, 10, 6 → cumulative 4, 11, 21, 27

   Each entry is the sum of all frequencies up to and including that class. The final value must equal the total frequency, which is a free check that the addition was done correctly.

2. 2

   Plot against the upper bound of each class

   class 10 < x ≤ 20, cumulative 11 → plot (20, 11)

   The upper bound, never the midpoint. The cumulative figure counts everything up to the top of the class, so that is the only x value it is true at. Plotting at midpoints is the most common error and shifts the whole curve left.

3. 3

   Join with a smooth curve

   an S-shaped curve rising to the total

   The curve rises and never falls, since a running total cannot decrease. Its characteristic S shape reflects data clustering in the middle classes, and a curve that is not S-shaped is worth re-checking.

4. 4

   Read the median at half the total

   total 27 → read across at 13.5 → median ≈ 22

   Go up the vertical axis to half the total frequency, across to the curve, then down to the value. For grouped data this gives an estimate, and questions expect that word in the answer.

5. 5

   Quartiles at a quarter and three quarters

   LQ at 6.75, UQ at 20.25 → IQR = UQ − LQ

   The lower quartile sits at a quarter of the total and the upper at three quarters. The interquartile range is their difference, and it measures the spread of the middle half of the data while ignoring the extremes entirely.

Always the upper bound

The single fact that decides whether a cumulative frequency question is right is where the points are plotted. A cumulative figure of 11 for the class 10 < x ≤ 20 means eleven values are at or below 20 — it says nothing about how many are below 15, so the point belongs at 20.

Students who have just come from estimating a mean, where midpoints are correct, carry the midpoint habit across. Naming the difference explicitly when the topic starts prevents a term of consistent errors.

• Cumulative frequency = running total
• Plot at the upper class bound
• Final value = total frequency
• Median at ½ the total, quartiles at ¼ and ¾
• IQR = upper quartile − lower quartile

Why the interquartile range beats the range

The range is the largest minus the smallest, so a single extreme value determines it entirely. One unusually high score can double the range while telling you nothing about the rest of the data.

The interquartile range covers the middle half and ignores the top and bottom quarters, so outliers cannot affect it. When a question asks which measure of spread is more appropriate for data with extremes, the IQR is the answer and this is the reason.

From curve to box plot

A box plot displays five values: minimum, lower quartile, median, upper quartile and maximum. Three of those come straight off a cumulative frequency curve, which is why the two topics are almost always taught and examined together.

Comparing two box plots is a standard question, and the expected answer compares both a measure of average — usually the medians — and a measure of spread, usually the IQRs. Comparing only one of the two loses half the marks.

How we teach cumulative frequency

We check the last cumulative value against the total frequency every time. It costs nothing, and a mismatch means an addition error that would otherwise corrupt every reading taken from the curve.

We also teach the median and quartile readings as a single procedure with three different heights, rather than as three separate techniques. It is one skill applied at ¼, ½ and ¾ of the total, and framing it that way makes it considerably faster to learn.