NCERT Solutions for Class 10 Maths Chapter 14 Ex 14.4 — Statistics Ogives | 2026-27

1. Q: What type of curve is a less than ogive? A: An S-shaped (ogive) curve that rises from left to right.
2. Q: In a more than ogive, which class limits are plotted on the x-axis? A: The lower class limits.
3. Q: If N = 50, what value do you locate on the y-axis to find the median from a less than ogive? A: \( \frac{N}{2} = 25 \).

Q1: The following data gives the marks of 30 students. Convert to a less than cumulative frequency distribution and list the points for the ogive.
Marks: 0–10 (f=2), 10–20 (f=5), 20–30 (f=8), 30–40 (f=10), 40–50 (f=5)

Answer: Less than cf: Less than 10 → 2; Less than 20 → 7; Less than 30 → 15; Less than 40 → 25; Less than 50 → 30. Points: (0,0), (10,2), (20,7), (30,15), (40,25), (50,30). Plot these and join with a smooth curve to get the less than ogive.

Q2: Explain how the intersection of a less than ogive and a more than ogive gives the median.

Answer: The less than ogive gives cumulative frequencies from below; the more than ogive gives cumulative frequencies from above. At the intersection point, the less than cf and more than cf are equal, which happens at exactly \( \frac{N}{2} \). Therefore, the x-coordinate of the intersection is the median of the distribution.

Q: The ages of 80 patients in a hospital are given below. Draw a less than ogive and find the median age from the graph. Verify using the formula.
Age (years): 10–20 (f=6), 20–30 (f=11), 30–40 (f=21), 40–50 (f=23), 50–60 (f=14), 60–70 (f=5)

Answer:

Step 1 — Cumulative Frequency Table:

Step 2 — Ogive Points: (10,0), (20,6), (30,17), (40,38), (50,61), (60,75), (70,80). Plot and join smoothly.

Step 3 — Median from Graph: \( \frac{N}{2} = 40 \). From y = 40, draw horizontal to ogive, drop vertical → median ≈ 40 years.

Step 4 — Formula Verification: Median class = 40–50 (cf just exceeds 40 at cf=61). \( l=40, cf=38, f=23, h=10 \).

\[ M = 40 + \left(\frac{40 – 38}{23}\right) \times 10 = 40 + \frac{20}{23} \approx 40 + 0.87 = 40.87 \text{ years} \]

The graphical estimate (~40) and formula value (~40.87) are very close, confirming the result. (Small difference is due to reading precision on graph paper.)

Mistake 1: Using lower class limits for a less than ogive.

Why it’s wrong: Less than ogive requires upper class limits on the x-axis. Using lower limits gives a completely wrong graph.

Correct approach: Always use upper class limits for less than ogive; use lower class limits for more than ogive.

Mistake 2: Joining ogive points with straight lines instead of a smooth curve.

Why it’s wrong: An ogive is a cumulative frequency curve, not a polygon. Straight lines give a cumulative frequency polygon, which is a different graph.

Correct approach: Join all points with a smooth freehand S-shaped curve.

Mistake 3: Forgetting to start the less than ogive from (lower boundary of first class, 0).

Why it’s wrong: Without the starting point, the curve is incomplete and you lose marks.

Correct approach: Always add the point (lower boundary of first class, 0) before plotting other points.

Mistake 4: Not verifying the median using the formula when the question asks for it.

Why it’s wrong: Question 2 of Ex 14.4 specifically says “verify the result by using the formula.” Skipping this step loses 2–3 marks.

Correct approach: After reading the median from the graph, always apply \( M = l + \left(\frac{\frac{N}{2} – cf}{f}\right) \times h \) and show the calculation.

Mistake 5: Using the wrong class for the median formula — picking the class where cf equals N/2 exactly instead of the class where cf first exceeds N/2.

Why it’s wrong: The median class is the one whose cumulative frequency first becomes greater than or equal to N/2.

Correct approach: Scan the cf column and find the first row where cf ≥ N/2. That row’s class interval is the median class.

• Graph paper is essential: In board exams, ogive questions require neat graph paper work. Use a sharp pencil, ruler for axes, and freehand for the curve. Label axes with units.
• Show the cumulative frequency table: Even if the question only asks for the ogive, writing the cumulative frequency table earns you step marks in the CBSE 2026-27 marking scheme.
• Mark N/2 clearly on graph: When finding the median from an ogive, draw the horizontal and vertical construction lines clearly and label the median value on the x-axis. Examiners look for this.
• Statistics chapter weightage: The Statistics chapter (Chapter 14) carries approximately 11 marks in the CBSE Class 10 Maths board exam 2026-27. Ogive questions from Ex 14.4 typically carry 3–5 marks each.
• Both ogives on one graph: If asked to draw both less than and more than ogives on the same graph, their intersection gives the median directly. Label the intersection point clearly.
• Revision checklist:
  • ✅ Know how to build less than and more than cumulative frequency tables
  • ✅ Know which class limits to use for each type of ogive
  • ✅ Practise reading N/2 on the y-axis and dropping perpendicular to x-axis
  • ✅ Memorise the median formula and practise identifying l, cf, f, h correctly
  • ✅ Practise all 3 questions in Ex 14.4 on actual graph paper before the board exam