MathsICSE Class 10 · Histograms and Ogive

Drawing a Less Than Ogive (Cumulative Frequency Curve) Step by Step (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

To draw a less than ogive, add a cumulative frequency column, then plot each cumulative frequency against the upper boundary of its class. Start at (lowest lower boundary, 0), join the points with a smooth free-hand curve, use the given scale and draw a kink if the x-axis does not start at 0.

Plot each cumulative frequency at its upper boundary, start at 0 on the lower boundary, and see the curve climb steepest over the biggest class.

Illustration drawn for QuestionBuddy.

Understand Drawing a Less Than Ogive

The CISCE 2027 syllabus names the less than ogive (only this type) and uses it to read the median, quartiles and other values. In 2024 Q9(iii) (4 marks), a correctly plotted curve was itself one of the expected points, before any reading.

What the ogive shows. The point $(x, c)$ on a less than ogive means ' $c$ observations are less than $x$ '. That is why each cumulative frequency is plotted at the upper boundary of its class: by the end of the class, all of its observations have been counted.

Steps.

1. Cumulative frequency (c.f.) column: add the frequencies down the table. The last c.f. is the total $N$ .
2. Boundaries: for discontinuous classes (21–30, 31–40, …) use 20.5, 30.5, …; for continuous classes use the limits as given.
3. First point: (lower boundary of the first class, 0), because nothing lies below it.
4. Other points: (upper boundary, c.f.) for each class.
5. Axes and scale: boundaries on the x-axis, c.f. on the y-axis, both labelled, with the scale given in the question. Draw a kink if the x-axis starts away from 0.
6. Curve: join the points with a smooth free-hand curve, not with a ruler.

Shape. A less than ogive never falls (c.f. only adds frequencies). It is steepest over the class with the highest frequency and flattens where frequencies are small, giving an S-shape for bell-shaped data.

Reading counts from the ogive. For a value $x$ on the x-axis, draw a perpendicular up to the curve and across to the y-axis.

Less than $x$ : the c.f. read.
More than $x$ : $N -$ c.f. $(x)$ .
Between $a$ and $b$ : c.f. $(b) -$ c.f. $(a)$ .
A percentage: count $\div N \times 100$ .

The reverse also works: from a c.f. on the y-axis, go across to the curve and down to read a value (for example, the score that the top 20% exceed).

Estimates. On a good graph, reading between plotted points is close to a straight-line estimate inside each class; your graph reading may differ slightly, and examiners accept a small range.

Drawing a Less Than Ogive: formulas to remember

Points of a less than ogive: $(upper boundary, c.f.)$; Plus the first point (lowest boundary, 0).
More than a value: $N - c.f. (x)$; Read c.f. $(x)$ from the curve.
Between two values: $c.f. (b) - c.f. (a)$; Two read-offs.
Frequencies from c.f.: $f_{k} = c.f._{k} - c.f._{k - 1}$; Rebuilding a table from a printed ogive.

How to solve Drawing a Less Than Ogive questions

Table first, graph second, read-offs last; each stage carries its own marks.

1Write the c.f. column and check that the last entry equals $N$ .
2Change discontinuous classes to boundaries.
3Draw the axes with the given scale, labels and a kink if needed.
4Plot (first lower boundary, 0) and every (upper boundary, c.f.).
5Join the points with one smooth free-hand curve.
6For each value asked, draw the read-off lines and write the reading on the graph.

Drawing a Less Than Ogive in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Waiting times at a clinic

Suppose a clinic records the waiting time of 50 patients: 0–10 minutes (12), 10–20 (18), 20–30 (15), 30–40 (5).

What should the ogive show at 30 minutes, and what percentage waited less than 30 minutes?

Try it first, then show the solutionSolution

c.f.: 12, 30, 45, 50.
The point at 30 minutes is $(30, 45)$ : 45 patients waited less than 30 minutes.
$\frac{45}{50} \times 100 = 90%$ .

Answer: (30, 45); 90%

Memory hook: Each point says 'this many are below me'.

Battery life above a value

Imagine a phone shop tests 50 batteries: 8–10 hours (5), 10–12 (15), 12–14 (20), 14–16 (10).

Estimate from the ogive how many batteries last more than 13 hours, and the percentage.

Try it first, then show the solutionSolution

c.f.: 5, 20, 40, 50. At 13 hours (half-way through 12–14), the curve reads about $20 + 10 = 30$ .
More than 13 hours: $50 - 30 = 20$ batteries, i.e. 40%.

Answer: About 20 batteries (40%)

Memory hook: More than = total minus less than.

Tricks to remember Drawing a Less Than Ogive

Short memory aids. Say each one aloud once, then use it on a question the same day.

Less than goes with Upper
'Less than 40' is counted at the top of the class, so plot at the upper boundary.
Use it when: Avoiding plots at lower boundaries or class marks.
Start on the floor
The curve starts on the x-axis at the first lower boundary.
Use it when: Not forgetting the first point.
Hand, not ruler
An ogive is a curve: free-hand, smooth, one stroke.
Use it when: Keeping the curve mark.

Drawing a Less Than Ogive: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: One cumulative frequency · 1 mark · MCQBoard level

The frequencies of the classes 10–20, 20–30 and 30–40 are 3, 8 and 12. On the less than ogive, the point plotted at $x = 40$ has y-coordinate:
(a) 12
(b) 20
(c) 23
(d) 35

Try it first, then show the solutionSolution

Less than 40 includes all three classes: $3 + 8 + 12 = 23$ .

Answer: (c)

Example 2: Table and points · 3 marks · Short answerBoard level

The speeds (km/h) of 100 vehicles were recorded. Make the cumulative frequency table and list the points to plot for a less than ogive.

Speed	30–40	40–50	50–60	60–70	70–80	80–90
Vehicles	6	14	22	28	18	12

Try it first, then show the solutionSolution

c.f.: 6, 20, 42, 70, 88, 100 ( $N = 100$ ✓).
Points: (30, 0), (40, 6), (50, 20), (60, 42), (70, 70), (80, 88), (90, 100).
x-axis starts at 30, so draw a kink; join the points free-hand.

Answer: Points (30, 0), (40, 6), (50, 20), (60, 42), (70, 70), (80, 88), (90, 100)

Example 3: Discontinuous classes · 4 marks · Short answerHarder board level

Draw a less than ogive for the ages of 50 members of a club, and use it to estimate how many members are younger than 45.5 years.

Age	21–30	31–40	41–50	51–60	61–70
Members	5	9	16	12	8

Try it first, then show the solutionSolution

Boundaries: 20.5, 30.5, 40.5, 50.5, 60.5, 70.5. c.f.: 5, 14, 30, 42, 50.
Points: (20.5, 0), (30.5, 5), (40.5, 14), (50.5, 30), (60.5, 42), (70.5, 50); kink on the x-axis.
At 45.5 (half-way through 40.5–50.5), the curve reads about $14 + 8 = 22$ .

Answer: About 22 members

Example 4: More than and between · 3 marks · Short answerHOTS / competency

Using the ogive of the 100 vehicle speeds (30–90 km/h, c.f. 6, 20, 42, 70, 88, 100), estimate:
(a) the number of vehicles faster than 75 km/h;
(b) the percentage with speeds between 50 and 70 km/h.

Try it first, then show the solutionSolution

(a) At 75 (half-way through 70–80), c.f. $\approx 70 + 9 = 79$ . Faster than 75: $100 - 79 = 21$ vehicles.
(b) c.f. $(70) -$ c.f. $(50) = 70 - 20 = 50$ vehicles, i.e. 50%.

Answer: (a) About 21 (b) 50%

Example 5: Frequency table from an ogive · 3 marks · Short answerHarder board level

A less than ogive passes through the plotted points (0, 0), (10, 4), (20, 13), (30, 28), (40, 40) and (50, 46). Write the frequency table and state the total frequency and the class with the most observations.

Try it first, then show the solutionSolution

Frequencies are differences of consecutive c.f.: $4 - 0$ , $13 - 4$ , $28 - 13$ , $40 - 28$ , $46 - 40$ .
0–10: 4, 10–20: 9, 20–30: 15, 30–40: 12, 40–50: 6.
Total $N = 46$ ; the most observations are in 20–30 (the steepest part of the curve).

Answer: 4, 9, 15, 12, 6; $N = 46$ ; 20–30

Example 6: Case study: race times · 5 marks · Case-basedHOTS / competency

The 100 m times (seconds) of 80 athletes are:

Time	11–12	12–13	13–14	14–15	15–16
Athletes	4	14	30	22	10

(a) List the points for a less than ogive.
(b) Estimate how many ran 100 m in under 13.5 s.
(c) The fastest 25% qualify. Estimate the qualifying time, to 2 decimal places.
(d) What percentage took more than 14.5 s?

Try it first, then show the solutionSolution

(a) c.f. 4, 18, 48, 70, 80: points (11, 0), (12, 4), (13, 18), (14, 48), (15, 70), (16, 80).
(b) At 13.5 s, c.f. $\approx 18 + 15 = 33$ athletes.
(c) Fastest 25% = 20 athletes. Across from c.f. 20 to the curve: in 13–14, $13 + \frac{20 - 18}{30} \times 1 \approx 13.07$ s.
(d) c.f. $(14.5) \approx 48 + 11 = 59$ , so $80 - 59 = 21$ took longer: $\frac{21}{80} \times 100 = 26.25%$ .

Answer: (b) About 33 (c) About 13.07 s (d) About 26.25%

Where students lose marks in Drawing a Less Than Ogive

The slips examiners see most often, and what to write instead.

Mistake: Plotting at lower boundaries or class marks
A less than c.f. counts everything up to the end of the class.
Wrong working
Plotting c.f. 6 at $x = 30$ or $x = 35$ for the class 30–40
Right working
Plot (40, 6), after the first point (30, 0)
Do this: Write the 'upper boundary' column next to the c.f. column before plotting.
Usually costs: Reported in the CISCE examiners' comments on 2024 Q9(iii): most of the graph marks.
Mistake: Joining the points with a ruler
An ogive is a smooth curve.
Wrong working
Straight segments between points
Right working
One smooth free-hand curve through all the points
Do this: Sketch lightly first, then go over it in one stroke.
Usually costs: Reported in the CISCE examiners' comments on 2024 Q9(iii): 1 mark.
Mistake: No kink and not using the given scale
The examiner checks the scale stated in the question.
Wrong working
x-axis starting at 20 without a kink; own scale
Right working
Kink near the origin; exactly the scale given
Do this: Copy the scale onto the graph sheet first.
Usually costs: Reported in the CISCE examiners' comments on 2024 Q9(iii).
Mistake: Drawing a histogram instead
Bars do not show cumulative frequency.
Wrong working
Bars of heights 2, 4, 12, …
Right working
A rising curve through (upper boundary, c.f.)
Do this: Read the question word: 'ogive' means cumulative frequency curve.
Usually costs: Reported in the CISCE examiners' comments on 2024 Q9(iii): the whole graph.

Exam tips

Asked in 2024 Q9(iii) (4 marks): ogive of the ages of 100 policy holders with a given scale; the expected answers were a median of about 42 years and about 9 policy holders above 52.
The CISCE examiners suggest plotting at upper boundaries, a free-hand curve, the given scale with a kink, and drawing lines for every value read.
For a printed ogive (2025 Q10(iii)), read the total from the last point, and write classes in full, such as ₹6000–₹8000.
Keep the c.f. table on the answer sheet; it earns marks even if the graph has a slip.

Practice questions on Drawing a Less Than Ogive

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
On a less than ogive, the cumulative frequency at the last upper boundary equals:
(a) the largest frequency
(b) the total frequency $N$
(c) $\frac{N}{2}$
(d) 0
Hint
Everything is below the last boundary.
Answer
(b)
Q2 · 3 marks · Short answerBoard level
Make the c.f. table and list the points for a less than ogive.
Marks 0–10 10–20 20–30 30–40 40–50 50–60
Students 2 6 13 16 9 4
Hint
Start at (0, 0).
Answer
c.f. 2, 8, 21, 37, 46, 50; points (0, 0), (10, 2), (20, 8), (30, 21), (40, 37), (50, 46), (60, 50).
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): A less than ogive can never slope downwards.
Reason (R): Each cumulative frequency is obtained by adding a non-negative frequency to the previous one.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Can a running total fall?
Answer
(c)
Q4 · 3 marks · Short answerHarder board level
Write the points for a less than ogive of the following data.
Class 5–9 10–14 15–19 20–24 25–29
Frequency 3 8 14 10 5
Hint
Boundaries 4.5, 9.5, ….
Answer
c.f. 3, 11, 25, 35, 40; points (4.5, 0), (9.5, 3), (14.5, 11), (19.5, 25), (24.5, 35), (29.5, 40), with a kink on the x-axis.
Q5 · 2 marks · Short answerHarder board level
Using the ogive from the marks table above ( $N = 50$ ; c.f. 2, 8, 21, 37, 46, 50), estimate how many students scored more than 35, and what percentage that is.
Hint
c.f. at 35 is half-way between 21 and 37.
Answer
c.f. $(35) \approx 29$ ; more than 35: $50 - 29 = 21$ students, 42%.
Q6 · 3 marks · Short answerHarder board level
A less than ogive passes through (100, 0), (120, 5), (140, 17), (160, 36), (180, 48) and (200, 55). Write the frequency table and state the modal class.
Hint
Subtract consecutive c.f. values.
Answer
100–120: 5, 120–140: 12, 140–160: 19, 160–180: 12, 180–200: 7; modal class 140–160.
Q7 · 3 marks · Short answerHOTS / competency
From the marks ogive above ( $N = 50$ ), a prize goes to the top 20% of students. Estimate the least mark needed, to 1 decimal place.
Hint
Top 20% = 10 students, so look across from c.f. 40.
Answer
c.f. 40 lies in 40–50 (c.f. 37 to 46): $40 + \frac{40 - 37}{9} \times 10 \approx 43.3$ marks.

Marks	0–10	10–20	20–30	30–40	40–50	50–60
Students	2	6	13	16	9	4

Class	5–9	10–14	15–19	20–24	25–29
Frequency	3	8	14	10	5

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Less than ogive: x-coordinate of each point?Show answer
The upper class boundary.
2.First point?Show answer
(Lower boundary of the first class, 0).
3.More than $x$ from the ogive?Show answer
$N -$ c.f. $(x)$
4.How do you join the points?Show answer
With a smooth free-hand curve.
5.Frequency of a class from c.f. values?Show answer
Its c.f. minus the previous c.f.

Drawing a Less Than Ogive: frequently asked questions

How do you draw a less than ogive?

Make a cumulative frequency column, plot each cumulative frequency against the upper boundary of its class, start from (first lower boundary, 0), and join the points with a smooth free-hand curve using the given scale.

Why is a less than ogive plotted at upper boundaries?

Each point means 'this many observations are less than this value'. All observations of a class are below its upper boundary, so the cumulative frequency belongs there.

How do I find the number above a value using an ogive?

Read the cumulative frequency at that value from the curve and subtract it from the total frequency N.