Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).
Comprehensive Concepts & Principles: Measures of Dispersion
Dispersion refers to the extent to which data values in a distribution are scattered, spread out, or clustered around a central value (mean or median). Two distributions may share identical arithmetic averages while exhibiting radically different risks, reliabilities, or volatilities. In CBSE Class 11 Applied Mathematics, dispersion provides the quantitative foundation for risk modeling, quality control, and business analytics.
Absolute vs. Relative Measures of Dispersion:
Absolute Measures: Expressed in the same physical units as the original data observations (e.g., ₹, kg, cm). Examples include Range, Quartile Deviation, Mean Deviation, and Standard Deviation. They cannot be used to compare two series having different units or vastly different means.
Relative Measures (Coefficients of Dispersion): Pure, unitless ratios or percentages derived by dividing an absolute measure by an appropriate central value. Examples include Coefficient of Range, Coefficient of Quartile Deviation, and Coefficient of Variation. They enable direct comparison of variability across completely different datasets.
Mathematical Framework: Range & Quartile Deviation
1. The Range
The difference between the maximum and minimum observations:
• $\text{Range} = L - S$
• $\text{Coefficient of Range} = \frac{L - S}{L + S}$ Where $L$ is largest value, $S$ is smallest value.
2. Quartile Deviation (QD)
Semi-interquartile range measuring the spread of the middle 50% of data:
The closing prices (in ₹) of a stock over 8 consecutive trading sessions are:
$$142, \quad 158, \quad 135, \quad 170, \quad 162, \quad 139, \quad 185, \quad 149$$
Compute the Range and the Coefficient of Range.
Show Step-by-Step Solution
• Identify the extremes:
- Largest value $L = 185$
- Smallest value $S = 135$
• Absolute Range: $\text{Range} = L - S = 185 - 135 = \mathbf{₹50}$.
• Coefficient of Range:
$$\text{Coeff of Range} = \frac{L - S}{L + S} = \frac{185 - 135}{185 + 135} = \frac{50}{320} = \mathbf{0.15625} \quad (\text{or } 15.63\%)$$ Answer: The Range is ₹50 and the Coefficient of Range is 0.156.
Example 2 (Quartile Deviation of Individual Series):
Find the Quartile Deviation and Coefficient of Quartile Deviation for the following daily sales figures (in thousands of rupees):
$$22, \quad 26, \quad 14, \quad 30, \quad 18, \quad 11, \quad 35, \quad 41, \quad 16$$
Graded textbook exercises covering Range, Interquartile Range, and Quartile Deviation based on CBSE Class 11 Applied Mathematics.
Problem 1Range Drill
The marks scored by 10 students in an Applied Mathematics test out of 50 are:
$$28, \quad 45, \quad 12, \quad 34, \quad 40, \quad 22, \quad 49, \quad 18, \quad 31, \quad 25$$
Calculate the Range and its relative measure.
Show Step-by-Step Solution
• Largest value $L = 49$, Smallest value $S = 12$.
• $\text{Range} = L - S = 49 - 12 = \mathbf{37} \text{ marks}$.
• $\text{Coefficient of Range} = \frac{49 - 12}{49 + 12} = \frac{37}{61} \approx \mathbf{0.6066}$.
Problem 2Continuous Class Range
Calculate the Range and Coefficient of Range for the following frequency distribution:
Classes: $10-20, \quad 20-30, \quad 30-40, \quad 40-50, \quad 50-60$.
Show Step-by-Step Solution
• For continuous grouped data:
- Upper boundary of the highest class $L = 60$
- Lower boundary of the lowest class $S = 10$
• $\text{Range} = L - S = 60 - 10 = \mathbf{50}$.
• $\text{Coefficient of Range} = \frac{60 - 10}{60 + 10} = \frac{50}{70} = \frac{5}{7} \approx \mathbf{0.7143}$.
Problem 3Quartile Deviation Computation
If for a symmetrical distribution, the first quartile $Q_1$ is 30 and the third quartile $Q_3$ is 50, find:
(a) The Interquartile Range (IQR).
(b) The Quartile Deviation (QD).
(c) The Median of the distribution.
Consider the dataset: $10, 12, 14, 15, 17, 18, 20$. An error enters the system replacing the value 20 with 200.
(i) How does the Range respond to this extreme value?
(ii) How does the Quartile Deviation respond? State why QD is preferred in outlier-rich environments.
Show Step-by-Step Solution
• (i) Range: Originally $20 - 10 = 10$. With the outlier, Range $= 200 - 10 = 190$ (increases 19-fold!).
• (ii) Quartile Deviation: $Q_1$ is 12 and $Q_3$ is 18 in both cases because $Q_3$ depends on the upper 25% boundary, unaffected by whether the extreme value is 20 or 200. Thus, $QD = (18 - 12)/2 = 3$ remains entirely unchanged.
• Conclusion: QD is highly robust because it focuses strictly on the central 50% of observations and ignores extreme tail anomalies.
Problem 5Relative Variability Comparison
Two industrial machines A and B produce steel rods. Machine A has $Q_1 = 20$ mm, $Q_3 = 28$ mm. Machine B has $Q_1 = 120$ mm, $Q_3 = 132$ mm.
(a) Find the QD of both machines.
(b) Which machine exhibits greater relative dispersion in its middle 50% production?
Q1: Why cannot absolute measures of dispersion be used to compare the variability of two series measured in different units?
Show Answer & Scheme
Reason [1 Mark]: Absolute measures retain the physical units of measurement (e.g. ₹ vs kg). Quantities with different dimensions cannot be compared meaningfully without normalizing into a dimensionless ratio (relative measure).
Q2: State the algebraic relationship between the Semi-Interquartile Range and the Interquartile Range.
Show Answer & Scheme
$$\text{Semi-Interquartile Range (QD)} = \frac{1}{2} \times \text{Interquartile Range (IQR)} \quad [1 \text{ Mark}]$$
SECTION B: 2-Mark Short Answer Problems
Q3: The coefficient of quartile deviation for a distribution is 0.25 and its sum of quartiles $(Q_3 + Q_1)$ is 160. Determine $Q_1$ and $Q_3$.
Q4: State two distinct merits and two limitations of 'Range' as a measure of statistical dispersion.
Show Answer & Scheme
• Merits [1 Mark]: (1) Easiest measure to compute and understand; (2) Useful for quick statistical quality control charting.
• Limitations [1 Mark]: (1) Highly sensitive to extreme outliers; (2) Based on only 2 extreme values, completely ignoring the remaining data points.
SECTION C: 4-Mark Comprehensive Numerical Drill
Q5: The following table shows the hourly wages (in ₹) of 50 industrial workers:
Wages: 100, 110, 120, 130, 140, 150
No. of Workers: 4, 8, 15, 12, 7, 4
(i) Construct the cumulative frequency distribution.
(ii) Find the First Quartile $Q_1$ and Third Quartile $Q_3$.
(iii) Calculate the Quartile Deviation and the Coefficient of Quartile Deviation.
Summary of formulas, properties, and comparison matrix for Chapter 10.1.
Summary Comparison of Range & Quartile Deviation
Measure
Absolute Formula
Relative Formula (Coefficient)
Sensitivity to Outliers
Range
$L - S$
$\frac{L - S}{L + S}$
Extremely Sensitive
Quartile Deviation
$QD = \frac{Q_3 - Q_1}{2}$
$\frac{Q_3 - Q_1}{Q_3 + Q_1}$
Highly Robust
Examination Key Points
Open-End Classes: When a distribution has open-ended classes (e.g. "Below 20" or "Above 80"), Range cannot be computed. However, Quartile Deviation can still be calculated as long as $Q_1$ and $Q_3$ do not lie in the open-ended intervals!
Units Rule: Absolute measures carry units ($cm, ₹$). Relative coefficients are purely numeric (unitless).
10-Question Diagnostic Board Mock Examination: Measures of Dispersion
Standard timed test: 4 Objective MCQs / Assertion-Reason with instant automatic grading, plus 6 multi-mark subjective problems with CBSE marking rubrics.
PART 1: Objective Section (4 Marks)
Q1: Which of the following is an absolute measure of dispersion?
Solution: (B) Quartile Deviation is expressed in the original units of the data and is thus an absolute measure; the others are relative coefficients.
Q2: If the minimum and maximum values of a series are 25 and 75 respectively, the Coefficient of Range is:
Q4 (Assertion-Reason): Assertion (A): Quartile deviation is suitable for measuring the dispersion of distributions having open-ended classes. Reason (R): Quartile deviation depends only on the central 50% of observations and does not require the numerical values of the extreme tail intervals.
Solution: (A) Both Assertion and Reason are true and Reason explains why open-ended classes do not hinder QD calculation.
PART 2: Subjective Section (Q5 - Q10)
Q52 Marks
Explain the conceptual difference between 'Central Tendency' and 'Dispersion' with an everyday business example.
Show Step-by-Step Solution & Marking Scheme
• Central Tendency [1 Mark]: Identifies the central, typical, or average value representative of the dataset (e.g., average monthly revenue of ₹10 Lakhs).
• Dispersion [1 Mark]: Quantifies the scatter, deviation, or risk around that average (e.g., whether monthly revenue fluctuates tightly between ₹9-11 Lakhs or wildly between ₹2-25 Lakhs). Identical averages can have completely different financial risk profiles.
Q62 Marks
Calculate the Range and Coefficient of Range for the data: $14, 18, 11, 25, 30, 16, 21, 35$.
In a frequency distribution, the third quartile $Q_3$ is 72 and the quartile deviation is 14.
(a) Find the first quartile $Q_1$.
(b) Calculate the Coefficient of Quartile Deviation.
Show that if every observation in a series is multiplied by a positive constant $k$, the Range increases by a factor of $k$, but the Coefficient of Range remains completely unchanged.
Show Step-by-Step Solution & Marking Scheme
• Let original largest and smallest values be $L$ and $S$. Original Range $= L - S$, Coeff $= \frac{L-S}{L+S}$ [1 Mark].
• When each value is multiplied by $k > 0$, new values are $L' = kL$ and $S' = kS$.
• New Range $= kL - kS = k(L - S) = k \times \text{Original Range}$ [1 Mark].
• New Coeff of Range $= \frac{kL - kS}{kL + kS} = \frac{k(L - S)}{k(L + S)} = \frac{L - S}{L + S} = \text{Original Coeff of Range}$ [1 Mark].
• Hence, the relative coefficient is invariant under uniform scale transformations.
Q9 & Q10Case Study (4 Marks Total)
Case Context: Logistics Delivery Performance
An e-commerce platform monitors the delivery times (in hours) of two courier partners, Falcon Express and Cheetah Logistics. Falcon Express recorded $Q_1 = 18$ hrs and $Q_3 = 26$ hrs. Cheetah Logistics recorded $Q_1 = 30$ hrs and $Q_3 = 42$ hrs.
Q9 (2 Marks): Calculate the Quartile Deviation for both courier partners. Q10 (2 Marks): Calculate the Coefficient of Quartile Deviation for both partners, and determine which courier exhibits higher relative consistency in delivery turnaround.