IB Maths AI SLTopic 4 — Statistics ToolkitPaper 1 & 2CF curves & percentiles~8 min read
Cumulative Frequency Graphs
A box plot collapses grouped data to five numbers. A cumulative frequency graph keeps every class — you plot the running total at each upper boundary and join the points with a smooth curve. From the curve you can read the median, quartiles, any percentile, and the number of values above or below any threshold.
📘 What you need to know
Cumulative frequency = running total of the frequencies up to the upper boundary of each class.
Plotting points: (upper boundary, cumulative frequency). Also plot (lower bound of first class, 0).
Curve shape: a smooth, increasing S-curve (sometimes called an ogive) joining the points.
Read down for x, across for y: draw a horizontal line at the target cum freq, drop down to the x-axis.
Quartiles: Q1 at 14n · median at 12n · Q3 at 34n. The pth percentile is at p100n.
For “how many are above x?”: go up from x, read the CF, then subtract from n.
Reading a cumulative frequency graph
The graph below shows homework times (min) for 40 students. The dashed orange lines show the three quartile reads.
The S-shaped CF curve for 40 students’ homework times. Dashed orange lines at y = 10, 20, 30 (= n4, n2, 3n4) hit the curve and drop down to x = 35, 50, 65 — the three quartiles.
Reading values from a CF graph
to find a percentile p%: draw horizontal at y = p100n, drop to x-axis
to find # values below threshold t: go up at x = t, read off y
# values above t = n − (value read off)
🧭 Recipe — build & use a CF graph
Build the CF column in the frequency table — running total of frequencies.
List plotting points: pair each upper boundary with its cumulative frequency. Add (lower bound of class 1, 0).
Draw the curve: plot the points and join smoothly — always increasing, never going down.
For a percentile p: horizontal at y = pn/100, then vertical drop — the x-value is your answer.
For “above / below a threshold”: vertical at the threshold, horizontal to the y-axis — read the CF, then subtract from n if “above”.
The curve must be increasing: cumulative frequency can never go down. If yours decreases, you’ve mis-added.
Worked examples
WE 1
Build the cumulative frequency table
The number of hours per week 30 students spend watching TV:
hours h
0≤h<2
2≤h<4
4≤h<6
6≤h<8
8≤h<10
freq f
4
8
11
5
2
Complete the cumulative frequency column and list the points used to plot the CF graph.
Running totals at each upper boundaryat 2: 4at 4: 4+8 = 12at 6: 12+11 = 23at 8: 23+5 = 28at 10: 28+2 = 30Plotting points (include (0, 0))(0, 0), (2, 4), (4, 12), (6, 23), (8, 28), (10, 30)CF: 4, 12, 23, 28, 30 · plot 6 pointsalways include (lower bound of first class, 0). The curve has to start somewhere — otherwise the lower portion of the graph isn’t anchored.
WE 2
Find the median from a CF curve
The CF curve for the commute times of 50 employees passes through these points:
n = 50, so median at cum = n/2 = 25Find where the curve has y = 2525 lies between (20, 15) and (30, 35)it sits exactly halfway between cum = 15 and 35so x is halfway: (20+30)/2 = 25median ≈ 25 minwhen the half-line lands midway between two plotted points, the read is midway in x too. Otherwise interpolate or just read off the graph.
WE 3
Find Q1, Q3 and the IQR
Using the cumulative frequency graph shown above (homework times for 40 students), find Q1, Q3 and the IQR.
n = 40Q₁: horizontal at y = n/4 = 10read off Q₁ ≈ 35 minQ₃: horizontal at y = 3n/4 = 30read off Q₃ ≈ 65 minIQR = Q₃ − Q₁IQR = 65 − 35 = 30Q₁ ≈ 35 · Q₃ ≈ 65 · IQR ≈ 30 minalways state CF graph reads with “≈” — they come from a hand-drawn curve, not exact arithmetic.
WE 4
Find a percentile (P₈₀)
Using the same CF graph (40 students, homework times), estimate the 80th percentile.
P₈₀ → horizontal at y = 80% × ny = 0.8 × 40 = 32Read off x where the curve has y = 3232 lies between (60, 28) and (80, 36)midway between 28 and 36 → x midway = 70P₈₀ ≈ 70 min80% of students spend less than about 70 min on homework; equivalently, the top 20% spend more.
WE 5
Percentage above a threshold
Using the same CF graph, estimate the percentage of students who spent more than 70 minutes on homework.
Read CF at x = 70 (number below 70)at x = 70: y ≈ 32Above 70 = n − belowabove = 40 − 32 = 8 studentsAs a percentage8 / 40 = 0.2 → 20%≈ 20%CF graphs read “less than” directly. For “more than” or “above”, subtract from n. This matches WE 4 — the 80th percentile is 70, so 100 − 80 = 20% lie above it.
WE 6
Construct a box plot from the CF graph
Using the CF graph for 40 students’ homework times, state the five-number summary and hence describe the box plot you would draw.
Read quartiles (from WE 3)Q₁ ≈ 35, median ≈ 50, Q₃ ≈ 65Min and max — class boundarieslowest class is 0 ≤ t < 20 → min ≈ 0highest class is 80 ≤ t < 100 → max ≈ 100Five-number summary0, 35, 50, 65, 100box plot: 0 — 35 — 50 — 65 — 100for grouped data we don’t know the exact min and max — best estimates are the outer class boundaries. CF graphs are the bridge between grouped frequency tables and box plots.
💡 Top tips
Always plot at the UPPER boundary, not the midpoint. CF means “≤ this value”.
Compute n/2, n/4, 3n/4 first: these are the y-axis levels you’ll read at.
Draw faint construction lines for every read — horizontal then vertical — so the examiner can see your method.
“Less than”: read down; “more than”: subtract from n.
CF graphs are for grouped data — answers are estimates. Write “≈” and round sensibly.
⚠ Common mistakes
Plotting at the midpoint of each class. Use the UPPER boundary — cumulative is “up to this point”.
Forgetting (0, 0): the curve must start at the lower bound of the first class with zero cumulative frequency.
Connecting points with straight lines: IB wants a smooth curve (or polyline is OK if specified).
Reading from the wrong axis: “median” is an x-value; “how many above 70” is a y-related value.
Treating “more than” as the direct read: the CF graph gives “less than or equal to”. Subtract from n for the complement.
Next up: Histograms. Frequency tables get a visual partner — bars with no gaps, equal class widths, frequency on the y-axis. Useful for spotting the shape of the distribution (symmetric vs skewed) at a glance and confirming whether a normal-distribution model is plausible.
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