Running totals, plotted at the top of each class: Cumulative frequency is a running total of the frequencies.
Plot each running total against the upper boundary of its class and join the points into a smooth S-shaped curve.
Read the median and quartiles off the cumulative-frequency curve (ogive): up from the x-axis to the curve, then across.
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IB-style question — build the running totals
Travel times (minutes) for 80 commuters have class frequencies 6, 16, 28, 20, 10 for 0–10, 10–20, 20–30, 30–40, 40–50.
Find the cumulative frequencies.
Step by step
- Add each frequency to the running total.
- Plot against the upper boundaries.
Final answer
Cumulative frequencies 6, 22, 50, 70, 80, plotted at 10, 20, 30, 40, 50.
Plot at the UPPER boundary: Each running total means 'this many values are less than the upper boundary', so plot it at the top of the class — not the midpoint.
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Read across at n/2, n/4 and 3n/4: For n values: the median is read across from n/2, the lower quartile from n/4, and the upper quartile from 3n/4.
Go up the cumulative axis to that level, across to the curve, then down to the data axis.
IB-style question — median from the curve
Using the curve through (10,6), (20,22), (30,50), (40,70), (50,80), estimate the median travel time.
Step by step
- Median level for n = 80.
- Read across from 40 to the curve, then down.
Final answer
Median ≈ 26 minutes (read across from a cumulative frequency of 40).
Quartiles the same way: Q₁ from n/4 = 20 → ≈ 19 min; Q₃ from 3n/4 = 60 → ≈ 35 min.
Then IQR = Q₃ − Q₁ ≈ 16 min.
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Read up from a value, or subtract two reads: To find how many values are below a given value, read up from the data axis to the curve.
For values between a and b, subtract the cumulative frequency at a from the one at b.
A percentile is read across from that percentage of n.
IB-style question — how many in a range
Using the same curve (80 commuters), estimate how many took between 15 and 35 minutes.
Step by step
- Read the cumulative frequency at each value.
- Subtract.
Final answer
About 46 commuters took between 15 and 35 minutes.
Percentile = read across from p% of n: The 90th percentile is read across from 0.9 × 80 = 72 → ≈ 42 min: 90% of commuters took less than that.
Turn the percentage into a cumulative frequency, then read: If the top X% exceed an unknown value, then (100 − X)% are below it.
Convert that to a cumulative frequency, read across to the curve and down to the value.
IB-style question — the slowest 10%
For the 80 commuters, the slowest 10% take longer than t minutes.
Estimate t.
Step by step
- Top 10% slowest ⇒ 90% are below t.
- Read across from 72 to the curve, then down.
Final answer
t ≈ 42 minutes (the 90th percentile).
Top X% → use 100 − X: The curve counts values below a level, so always convert a 'top X%' into the percentage below before reading across.
IB-style question — frequencies from the curve
A cumulative frequency curve for 80 students reads: CF = 20 at 7 hours, CF = 56 at 15 hours, CF = 72 at 21 hours.
Find the frequencies of the classes 7 < h ≤ 15 and 15 < h ≤ 21.
Step by step
- A class frequency is the DIFFERENCE of the cumulative totals at its boundaries.
- Likewise for the next class.
Final answer
36 students, then 16 students.
Read the cumulative frequency at each class boundary off the curve, then subtract consecutive totals to get each class frequency.
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