Mathematics
Chapter 13
Board Weightage: 12 - 14 Marks in ICSE Board Exam
Statistics: Mean, Median, Mode, Quartiles & Ogives
Prescribed Textbook: Understanding ICSE Mathematics - M.L. Aggarwal (Class 10) - Chapter 21
Canonical Link:
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📖 Syllabus Scope & Overview
Mean of grouped data by Direct method (Σfx / Σf), Assumed Mean / Short-cut method (a + Σfd / Σf), and Step-deviation method (a + [Σfu / Σf] × h); Mode of grouped data graphically using Histogram; Median and Quartiles: Lower Quartile (Q₁ = N/4 th term), Upper Quartile (Q₃ = 3N/4 th term), Interquartile range (Q₃ - Q₁), and semi-interquartile range determined graphically using cumulative frequency curve (Ogive).
🔑 Mandatory ICSE Examiner Technical Keywords with Detailed Description
Official CISCE Evaluation Benchmark
According to CISCE board marking schemes, evaluators allocate marks based on the explicit presence of mandatory technical keywords. General or colloquial explanations fail to secure full marks. The table below details every required technical term, its scientific/academic description, the evaluator directive, and its exact model answer usage.
Mandatory Keyword #1
Core Definition
Chapter Concept: Ogive (Cumulative Frequency Curve) Rules in ICSE
1 Step Mark in Method & Working
"Upper class limit on x-axis"
📘 Technical Definition & Detailed Description:
In ICSE Class 10 Mathematics (Statistics: Mean, Median, Mode, Quartiles & Ogives), "Upper class limit on x-axis" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Upper class limit on x-axis" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "Upper class limit on x-axis" to Ogive (Cumulative Frequency Curve) Rules in ICSE establishes the governing equation."
Mandatory Keyword #2
Core Definition
Chapter Concept: Ogive (Cumulative Frequency Curve) Rules in ICSE
1 Step Mark in Method & Working
"Cumulative frequency on y-axis"
📘 Technical Definition & Detailed Description:
In ICSE Class 10 Mathematics (Statistics: Mean, Median, Mode, Quartiles & Ogives), "Cumulative frequency on y-axis" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Cumulative frequency on y-axis" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "Cumulative frequency on y-axis" to Ogive (Cumulative Frequency Curve) Rules in ICSE establishes the governing equation."
Mandatory Keyword #3
Core Definition
Chapter Concept: Ogive (Cumulative Frequency Curve) Rules in ICSE
1 Step Mark in Method & Working
"Smooth freehand curve"
📘 Technical Definition & Detailed Description:
In ICSE Class 10 Mathematics (Statistics: Mean, Median, Mode, Quartiles & Ogives), "Smooth freehand curve" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Smooth freehand curve" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "Smooth freehand curve" to Ogive (Cumulative Frequency Curve) Rules in ICSE establishes the governing equation."
Mandatory Keyword #4
Core Definition
Chapter Concept: Ogive (Cumulative Frequency Curve) Rules in ICSE
1 Step Mark in Method & Working
"Median at N/2"
📘 Technical Definition & Detailed Description:
In ICSE Class 10 Mathematics (Statistics: Mean, Median, Mode, Quartiles & Ogives), "Median at N/2" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Median at N/2" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "Median at N/2" to Ogive (Cumulative Frequency Curve) Rules in ICSE establishes the governing equation."
Mandatory Keyword #5
Core Definition
Chapter Concept: Ogive (Cumulative Frequency Curve) Rules in ICSE
1 Step Mark in Method & Working
"Lower quartile Q1 = N/4"
📘 Technical Definition & Detailed Description:
In ICSE Class 10 Mathematics (Statistics: Mean, Median, Mode, Quartiles & Ogives), "Lower quartile Q1 = N/4" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Lower quartile Q1 = N/4" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "Lower quartile Q1 = N/4" to Ogive (Cumulative Frequency Curve) Rules in ICSE establishes the governing equation."
Mandatory Keyword #6
Core Definition
Chapter Concept: Ogive (Cumulative Frequency Curve) Rules in ICSE
1 Step Mark in Method & Working
"Upper quartile Q3 = 3N/4"
📘 Technical Definition & Detailed Description:
In ICSE Class 10 Mathematics (Statistics: Mean, Median, Mode, Quartiles & Ogives), "Upper quartile Q3 = 3N/4" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Upper quartile Q3 = 3N/4" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "Upper quartile Q3 = 3N/4" to Ogive (Cumulative Frequency Curve) Rules in ICSE establishes the governing equation."
Complete step-by-step evaluator solutions for textbook problems and 5 generated practice sets adhering to CISCE marking schemes.
📘 Textbook Problem: Textbook Exercise 13(A) - Question 4
Understanding ICSE Mathematics - M.L. Aggarwal (Class 10) - Chapter 21 - Exercise 13(A)
3 Marks Total
Based on Statistics: Mean, Median, Mode, Quartiles & Ogives, solve the following standard textbook problem: Apply the principle of Ogive (Cumulative Frequency Curve) Rules in ICSE to find the value of the unknown variable given standard initial parameters. Verify using Mean & Quartile Formulas.
Step-by-Step Marking Breakdown
Step 1: State the Given Data and Standard Formula: Identify all given terms from the problem statement: Write down the governing equation for Statistics: Mean, Median, Mode, Quartiles & Ogives.
Formula: Mean (Step-dev) = a + (Σfu / Σf) · h | Q₁ = (N/4)th term | Q₃ = (3N/4)th term | IQR = Q₃ - Q₁
1 Mark
Step 2: Substitute Values and Simplify Algebraically: Substitute the known numerical values into the equation. Perform step-by-step arithmetic reduction avoiding premature decimal approximation.
Formula: Derived from Mean (Step-dev) = a + (Σfu / Σf) · h | Q₁ = (N/4)th term | Q₃ = (3N/4)th term | IQR = Q₃ - Q₁
1 Mark
Step 3: State the Final Calculated Value with Proper Units: Isolate the target variable on the left hand side. Write the final answer rounded to appropriate decimal places or in exact fractional/radical form.
1 Mark
Final Answer: Computed value consistent with Statistics: Mean, Median, Mode, Quartiles & Ogives principles
(Units: units)
💡 Examiner Tip: In ICSE Mathematics, marks are awarded step-wise. Always write the relevant formula before substituting values.
🎯 Generated Set: Generated Practice Set 1 - Core Concept (2 Marks)
ICSE Class 10 Exam Blueprint - Chapter 13
2 Marks Total
Define "Ogive (Cumulative Frequency Curve) Rules in ICSE" in the context of ICSE Class 10 Mathematics. State its primary characteristic or SI unit/standard symbol.
Step-by-Step Marking Breakdown
Step 1: Precise Technical Definition: 1. Plot upper class limits on x-axis and cumulative frequencies on y-axis. 2. Join plotted points by a smooth freehand S-shaped curve (NOT with a straight ruler). 3. Start curve fr...
1 Mark
Step 2: Mandatory Technical Criteria: Ensure the definition contains mandatory keywords: Upper class limit on x-axis, Cumulative frequency on y-axis, Smooth freehand curve.
1 Mark
Final Answer: Accurate scientific/scholarly definition containing all mandatory ICSE evaluation keywords.
(Units: units)
💡 Examiner Tip: Avoid colloquial explanations. Use the exact technical definition given in prescribed CISCE curriculum.
🎯 Generated Set: Generated Practice Set 2 - Method & Application (3 Marks)
ICSE Question Bank & Specimen Framework - Mathematics
3 Marks Total
A system governed by "Mean & Quartile Formulas" has standard parameters. Using the relation Mean (Step-dev) = a + (Σfu / Σf) · h | Q₁ = (N/4)th term | Q₃ = (3N/4)th term | IQR = Q₃ - Q₁, calculate the required variable when other quantities are doubled. State the formula and show step-by-step substitution.
Step-by-Step Marking Breakdown
Step 1: State the Formula / Conceptual Principle: Write the primary governing equation: Mean (Step-dev) = a + (Σfu / Σf) · h | Q₁ = (N/4)th term | Q₃ = (3N/4)th term | IQR = Q₃ - Q₁. Define each symbol clearly.
Formula: Mean (Step-dev) = a + (Σfu / Σf) · h | Q₁ = (N/4)th term | Q₃ = (3N/4)th term | IQR = Q₃ - Q₁
1 Mark
Step 2: Mathematical / Procedural Deduction: Substitute the modified parameters: Let initial state be S₁ and new state be S₂. Set up the ratio S₂ / S₁ and simplify algebraically.
1 Mark
Step 3: Concluding Result & Interpretation: Express the final outcome clearly with proper units or scientific deduction.
1 Mark
Final Answer: Result derived directly from Mean (Step-dev) = a + (Σfu / Σf) · h | Q₁ = (N/4)th term | Q₃ = (3N/4)th term | IQR = Q₃ - Q₁
(Units: units)
💡 Examiner Tip: In derivation and numerical questions, every intermediate mathematical step carries fractional credit.
🎯 Generated Set: Generated Practice Set 3 - Comparative Analysis (3 Marks)
ICSE Exemplar Practice Paper - Mathematics
3 Marks Total
Differentiate between "Ogive (Cumulative Frequency Curve) Rules in ICSE" and "Ogive (Cumulative Frequency Curve) Rules in ICSE" on the basis of: (i) Fundamental definition/mechanism, (ii) Key working condition or formula, (iii) Real-world practical example or application.
Step-by-Step Marking Breakdown
Point 1: Conceptual Difference: Contrast the primary mechanisms: Ogive (Cumulative Frequency Curve) Rules in ICSE focuses on Upper class limit on x-axis, whereas Ogive (Cumulative Frequency Curve) Rules in ICSE emphasizes Upper class limit on x-axis.
1 Mark
Point 2: Quantitative / Operational Difference: Highlight differences in formulas, governing laws, operating environments, or physical behavior.
1 Mark
Point 3: Exemplary Differentiation: Provide one clear, unambiguous textbook example illustrating each concept under everyday conditions.
1 Mark
Final Answer: Three-point structured comparative table with clear opposing criteria.
(Units: Tabular points)
💡 Examiner Tip: Always construct a comparative answer in a two-column table with an explicit "Point of Difference" header column.
🎯 Generated Set: Generated Practice Set 4 - Board Standard Structured (4 Marks)
ICSE Board Marking Blueprint - Class 10
4 Marks Total
A comprehensive ICSE examination question based on "Statistics: Mean, Median, Mode, Quartiles & Ogives":
(i) State the fundamental law or rule governing "Ogive (Cumulative Frequency Curve) Rules in ICSE". [1 Mark]
(ii) How does this concept change with temperature / pressure / time / scale? [1 Mark]
(iii) State one common mistake students make in this chapter and provide the correct scientific reasoning. [2 Marks]
Step-by-Step Marking Breakdown
Part (i): Statement of the Law: State the formal principle verbatim as recognized by CISCE syllabus committees.
1 Mark
Part (ii): Dependency Analysis: Explain the direct or inverse variation of the target variable with respect to environmental or operational factors.
1 Mark
Part (iii): Error Analysis & Correct Resolution: Identify the frequent pitfall: Students often confuse Upper class limit on x-axis with related quantities. The correct understanding requires strict application of Ogive (Cumulative Frequency Curve) Rules in ICSE.
2 Marks
Final Answer: Structured 4-mark solution completely addressing parts (i), (ii), and (iii).
(Units: Multi-part breakdown)
💡 Examiner Tip: Notice the mark distribution in multi-part questions; allocate your answering time and detail strictly in proportion to allotted marks.
🎯 Generated Set: Generated Practice Set 5 - Higher Order Thinking (HOTS) (4 Marks)
ICSE High-Achiever Challenge Series - Class 10
4 Marks Total
A critical scenario-based problem in "Statistics: Mean, Median, Mode, Quartiles & Ogives": An experiment is performed under non-ideal conditions involving "Ogive (Cumulative Frequency Curve) Rules in ICSE" and "Ogive (Cumulative Frequency Curve) Rules in ICSE".
(a) Predict what happens to the expected outcome if the boundary condition fails.
(b) Give complete scientific justification.
(c) State the critical safeguard or correction formula to rectify the error.
Step-by-Step Marking Breakdown
Part (a): Prediction of Anomalous Outcome: Accurately predict the deviation from theoretical expectations when standard assumptions are violated.
1 Mark
Part (b): Scientific Justification: Provide a rigorous scientific explanation using first principles and mandatory keywords: Upper class limit on x-axis, Cumulative frequency on y-axis, Smooth freehand curve, Median at N/2.
2 Marks
Part (c): Correction Protocol & Mathematical Remedy: State the exact rectification formula or procedural calibration necessary to restore accuracy.
Formula: Mean (Step-dev) = a + (Σfu / Σf) · h | Q₁ = (N/4)th term | Q₃ = (3N/4)th term | IQR = Q₃ - Q₁
1 Mark
Final Answer: Full scenario analysis with anomaly prediction, first-principles justification, and correction method.
(Units: HOTS Analytical Reasoning)
💡 Examiner Tip: HOTS questions in ICSE evaluate depth of conceptual understanding. Avoid superficial guessing; reason backward from core laws.