A cumulative frequency graph answers “how many values are below this boundary?” Its horizontal coordinates are class boundaries, not class marks. Once that distinction is correct, the median, quartiles and interval counts become readings from a running-total graph. The answers inside a class are estimates because the original individual values are no longer available.
These cumulative frequency worked examples use three invented datasets. None represents a real investigation. We use a less-than ogive, with the measured quantity on the horizontal axis and cumulative frequency on the vertical axis. For numerical checks, straight-line interpolation assumes the observations are spread uniformly within each class; that is a model, not information recovered from grouping.
Class limits, boundaries and marks are different
Suppose a duration was recorded to the nearest minute and placed in class 20–29 minutes. The displayed limits are 20 and 29. Its continuous class boundaries are 19.5 and 29.5 minutes, since values from 19.5 up to 29.5 round into that group, using the usual half-up convention. Its class mark is the midpoint, (20 + 29)/2 = 24.5 minutes.
The boundary 29.5 is where the entire first class has accumulated. The class mark 24.5 represents the class’s centre and is useful in other grouped-data calculations; it does not mark the end of that class. A less-than ogive therefore plots the running total at 29.5, not at 24.5.
Do not automatically add and subtract 0.5. The half-unit correction depends on recording precision. Measurements rounded to the nearest 10 mL use a 5 mL correction. Classes already written as 10 ≤ x < 15 supply their boundaries directly and need no correction.
Worked case 1: build and plot the running totals
Givens: Sixty fictional parcel-handling durations were recorded to the nearest minute. All recorded values are between 20 and 79 minutes. The six class frequencies are shown below.
| Recorded minutes | Continuous boundaries | Frequency | Cumulative frequency |
|---|---|---|---|
| 20–29 | 19.5–29.5 | 4 | 4 |
| 30–39 | 29.5–39.5 | 8 | 12 |
| 40–49 | 39.5–49.5 | 14 | 26 |
| 50–59 | 49.5–59.5 | 18 | 44 |
| 60–69 | 59.5–69.5 | 10 | 54 |
| 70–79 | 69.5–79.5 | 6 | 60 |
Add frequencies successively: 4, then 4 + 8 = 12, then 12 + 14 = 26, continuing to 60. The last total must equal the sum of all class frequencies. Plot the starting point (19.5, 0), followed by (29.5, 4), (39.5, 12), (49.5, 26), (59.5, 44), (69.5, 54) and (79.5, 60).
Each horizontal interval has width 10 minutes, while each vertical grid interval represents 10 observations. The graph cannot decrease: adding observations never reduces a running total. The supplied ogive uses straight segments so its readings match the interpolation calculations below. If a question requests a smooth curve, draw a sensible non-decreasing curve through the plotted points and accept appropriate graph-reading variation.
Worked case 2: estimate the median and quartiles
For this grouped ogive use cumulative positions N/4, N/2 and 3N/4. Here N = 60, so the lower quartile Q₁ is at 15, the median at 30, and the upper quartile Q₃ at 45. These are positions on the frequency axis, not duration values. Read horizontally to the graph, then vertically down to the duration axis.
For a numerical check, an estimate at cumulative position r is L + [(r − F)/f]w. Here L is the lower boundary of the containing class, F is the cumulative frequency before it, f is that class’s frequency, and w is its width. The fraction tells you how far through that class the required position lies.
- Median: position 30 is between cumulative totals 26 and 44. Estimate = 49.5 + [(30 − 26)/18] × 10 ≈ 51.7 minutes.
- Q₁: position 15 is between 12 and 26. Estimate = 39.5 + [(15 − 12)/14] × 10 ≈ 41.6 minutes.
- Q₃: position 45 is between 44 and 54. Estimate = 59.5 + [(45 − 44)/10] × 10 = 60.5 minutes.
The estimated interquartile range is Q₃ − Q₁ ≈ 60.5 − 41.642857 = 18.9 minutes. Use the unrounded quartiles for subtraction, then round. The order Q₁ < median < Q₃ is a useful check. Do not claim the exact raw-data median is 51.7: many different individual datasets share this same grouped table.
Worked case 3: subtract two cumulative readings
Estimate how many durations lie from 44.5 minutes up to, but not including, 64.5 minutes. At 44.5, halfway through boundaries 39.5–49.5, the estimated cumulative frequency is 12 + (5/10) × 14 = 19. At 64.5, halfway through 59.5–69.5, it is 44 + (5/10) × 10 = 49.
The interval count is therefore approximately 49 − 19 = 30 observations. The count at or above 64.5 is approximately 60 − 49 = 11. A cumulative reading includes everything below its boundary, so reporting 49 alone would answer a different question from the interval count.
Compare a boundary-to-boundary question: the number from 39.5 up to 59.5 is exactly 44 − 12 = 32 in this grouped dataset. That interval contains two whole classes. The total is known from the table; it does not require an assumption about positions inside a class.
Three practice tasks with solved answers
Practice 1: classes with boundaries already stated
Dataset B gives invented ribbon lengths x in centimetres, grouped using the inequalities shown. Estimate its median and interquartile range using linear interpolation.
| Length x, cm | Frequency | Cumulative frequency |
|---|---|---|
| 10 ≤ x < 15 | 3 | 3 |
| 15 ≤ x < 20 | 7 | 10 |
| 20 ≤ x < 25 | 12 | 22 |
| 25 ≤ x < 30 | 10 | 32 |
| 30 ≤ x < 35 | 6 | 38 |
| 35 ≤ x < 40 | 2 | 40 |
Solution: Start at (10, 0), then use upper boundaries 15, 20, 25, 30, 35 and 40. With N = 40, median position 20 gives 20 + [(20 − 10)/12] × 5 ≈ 24.2 cm. Q₁ is at cumulative 10, exactly on boundary 20 cm. Q₃ is at 30, giving 25 + [(30 − 22)/10] × 5 = 29 cm. Estimated IQR = 29 − 20 = 9 cm.
Practice 2: change the boundary correction with precision
Dataset C gives invented jar volumes recorded to the nearest 10 mL. Find the continuous boundaries and estimate the median.
| Recorded mL | Continuous boundaries, mL | Frequency | Cumulative frequency |
|---|---|---|---|
| 100–140 | 95–145 | 2 | 2 |
| 150–190 | 145–195 | 5 | 7 |
| 200–240 | 195–245 | 9 | 16 |
| 250–290 | 245–295 | 8 | 24 |
| 300–340 | 295–345 | 4 | 28 |
| 350–390 | 345–395 | 2 | 30 |
Solution: The correction is 5 mL, half the recording unit. Begin the ogive at (95, 0). Its other coordinates are (145, 2), (195, 7), (245, 16), (295, 24), (345, 28) and (395, 30). Median position 15 lies in boundaries 195–245. Estimate = 195 + [(15 − 7)/9] × 50 ≈ 239.4 mL. Using class limit 200 as L would introduce a 5 mL error.
Practice 3: repair a midpoint plot
A learner plots Dataset A’s first cumulative point at (24.5, 4) and the next at (34.5, 12). Explain and correct the error. Solution: Those x-values are class marks. The complete first class accumulates at 29.5, and the complete second at 39.5. Replace the points with (29.5, 4) and (39.5, 12), and include starting point (19.5, 0). Keep the correct cumulative totals.
Frequently asked questions
Is cumulative frequency the same as frequency?
No. Frequency counts one class; cumulative frequency counts that class and all earlier classes.
Why must the graph begin with zero?
At the lower boundary of the first class, none of the dataset has yet accumulated. This anchors readings within that class.
Should I use N/2 or (N + 1)/2?
This grouped-ogive method uses N/2. Raw-data ordering conventions are a different procedure; follow the question’s stated convention.
Are quartiles exact if interpolation gives a whole number?
No. A neat number can still depend on the uniform-within-class assumption. Exact-looking notation does not restore missing data.
Can a cumulative graph have a flat section?
Yes, across a class with zero frequency. It still must not slope downward.
What changes for unequal class widths?
Use the actual boundaries on the horizontal scale and the actual width in interpolation. Cumulative totals still add ordinary frequencies.
Must an estimated count be a whole number?
The actual count is whole, but interpolation may return a fraction. Report a sensible approximate count and retain its estimated status.
Check the question before reading the graph
Identify whether you need a value at a cumulative position, a cumulative count at a value, or the difference between two counts. Then use the existing Mathematics topical revision collection for further resource navigation. Its contents may require membership; no specific statistics download is promised here.
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