Introduction:
Measures of dispersion MCQs are an essential part of competitive exam preparation for CSS, FPSC, PMS, and other tests. These questions assess how well a student understands the spread of data around a central value. While averages describe the center, dispersion explains how consistent or scattered the observations are.
In this collection, you will practice important concepts such as standard deviation, variance, and coefficient of variation. The focus is on building both conceptual clarity and problem-solving ability, which are crucial for success in exam-based statistics questions.
Figure: Key statistical measures of dispersion frequently tested in CSS, FPSC, PMS, and other competitive exams.
📑 Quick Navigation
- Important Definitions
- Formulas Cheat Sheet
- Concept Overview
- Key Types
- Examiner Traps
- MCQs 1–10
- MCQs 11–20
- MCQs 21–30
- MCQs 31–40
- MCQs 41–50
- MCQs 51–60 (Advanced)
- MCQs 61–70 (Advanced)
- 1-Minute Revision
- Comparison Table
- Key Differences
- Key Concepts
- Concept Reminder
- Flashcards
- Key Takeaways
- Conclusion
- FAQs
- Related Links
📘 Important Definitions (Measures of Dispersion)
📏 Range
The simplest measure of dispersion, calculated as the difference between maximum and minimum values.
📊 Variance
The average of squared deviations from the mean, expressed in squared units.
📉 Standard Deviation
The square root of variance, showing dispersion in original units.
📐 Coefficient of Variation
A relative measure calculated as (SD / Mean) × 100, useful for comparison.
📦 Interquartile Range (IQR)
The difference between Q3 and Q1, representing the middle 50% of data.
🎯 Exam Tip:
👉 Range = simplest but weakest
👉 SD = most commonly used
👉 CV = best for comparison
👉 IQR = best for skewed data
📐 Formulas Cheat Sheet (Measures of Dispersion)
📏 Range
Range = Maximum − Minimum
📊 Variance
σ² = Σ (x − μ)² / N
📉 Standard Deviation
σ = √Variance
📐 Coefficient of Variation
CV = (σ / μ) × 100
📦 Interquartile Range
IQR = Q3 − Q1
🎯 Exam Shortcut:
👉 Variance = square of SD
👉 SD = √Variance
👉 CV = comparison tool
👉 IQR = middle 50% only
💡 Memorize these = 70% dispersion MCQs solved!
Overview of Measures of Dispersion
This MCQ collection systematically covers both basic and advanced aspects of measures of dispersion. It includes conceptual definitions, computational formulas, transformation properties, comparison techniques, and inferential applications. Students will encounter questions related to:
- Range and its limitations
- Variance and its algebraic properties
- Standard deviation and empirical rule (68%–95%–99.7%)
- Coefficient of variation for relative comparison
- Effect of linear transformations on dispersion
- Population vs sample variance (n vs n − 1)
- Interquartile range for skewed distributions
By practicing these Measures of Dispersion MCQs, aspirants can strengthen problem-solving speed, improve conceptual clarity, and prepare confidently for statistics-related questions in competitive exams. Each question is accompanied by a clear explanation to reinforce theoretical understanding and exam-oriented preparation.
📊 Key Types of Measures of Dispersion
📏 Absolute Measures
Measure dispersion in actual units.
👉 Range, Variance, Standard Deviation
📐 Relative Measures
Used for comparison between datasets.
👉 Coefficient of Variation (CV)
📦 Robust Measures
Ignore extreme values (outliers).
👉 Interquartile Range (IQR)
📊 Graph-Based Insight
Dispersion can be visualized using
👉 histograms and box plots
🎯 Exam Shortcut:
👉 Absolute → actual spread
👉 Relative → comparison
👉 Robust → skewed data
💡 Identify question type first, then choose the correct measure!
⚠️ Examiner Trap Concepts
➕ Adding a constant does NOT change variance or SD.
✖️ Multiplying data changes variance by square of the constant (a²).
📏 Range is highly sensitive to outliers.
📉 Standard deviation can never be negative.
📐 Coefficient of variation fails when mean = 0.
🎯 Exam Strategy:
👉 Look for keywords like “added” vs “multiplied”
👉 Check if mean = 0 before using CV
👉 Watch for outliers in range-based questions
💡 Most tricky MCQs are built on these small traps!
👨🏫 Teacher Tip (Exam Strategy)
📌 Don’t just memorize—understand the demand of the question.
❌ Common Classroom Issue:
Students remember formulas but struggle in exams because they fail to interpret what is actually being asked.
🧠 Smart Thinking Approach:
Before solving, quickly identify:
• Is it about absolute dispersion? (Range, Variance, Standard Deviation)
• Or relative dispersion? (Coefficient of Variation)
💡 Golden Rule:
"First identify the type → then apply the formula 🎯"
📌 Don’t just memorize—understand the demand of the question.
❌ Common Classroom Issue:
Students remember formulas but struggle in exams because they fail to interpret what is actually being asked.
🧠 Smart Thinking Approach:
Before solving, quickly identify:
• Is it about absolute dispersion? (Range, Variance, Standard Deviation)
• Or relative dispersion? (Coefficient of Variation)
💡 Golden Rule:
"First identify the type → then apply the formula 🎯"
PART-1 (MCQs 1–10)
1. The range of the dataset 6, 11, 19, 25 is:
A. 19
B. 14
C. 17
D. 13
Explanation:
Concept: Range measures total spread using extreme values only.
Trap: Students sometimes use average instead of extremes—this is incorrect.
Exam Use: In FPSC/CSS, range questions are often direct but may include hidden minimum/maximum values.
Concept: Range measures total spread using extreme values only.
Trap: Students sometimes use average instead of extremes—this is incorrect.
Exam Use: In FPSC/CSS, range questions are often direct but may include hidden minimum/maximum values.
2. Which measure of dispersion is expressed in squared units?
A. Variance
B. Standard deviation
C. Range
D. Coefficient of variation
Explanation:
Key Idea: Variance is based on squared deviations from the mean.
Mistake: Confusing it with standard deviation, which is in original units.
Application: Squared units make variance useful in advanced statistical modeling and theory.
Key Idea: Variance is based on squared deviations from the mean.
Mistake: Confusing it with standard deviation, which is in original units.
Application: Squared units make variance useful in advanced statistical modeling and theory.
3. The standard deviation of a dataset with zero variance is:
A. Zero
B. One
C. Equal to mean
D. Undefined
Explanation:
Standard deviation equals the square root of variance. If variance is zero, standard deviation is also zero in measures of dispersion.
Standard deviation equals the square root of variance. If variance is zero, standard deviation is also zero in measures of dispersion.
4. Variance is minimized when deviations are taken from:
A. Mean
B. Median
C. Mode
D. Maximum value
Explanation:
The sum of squared deviations is least when measured from the arithmetic mean. This property defines variance in measures of dispersion.
The sum of squared deviations is least when measured from the arithmetic mean. This property defines variance in measures of dispersion.
5. If each observation in a dataset is increased by 5, the variance will:
A. Remain unchanged
B. Increase by 5
C. Double
D. Decrease
Explanation:
Logic: Adding a constant shifts all values equally without changing spread.
Why Correct: Deviations from the mean remain unchanged after addition.
Exam Use: Frequently tested transformation rule—addition does NOT affect variance.
Logic: Adding a constant shifts all values equally without changing spread.
Why Correct: Deviations from the mean remain unchanged after addition.
Exam Use: Frequently tested transformation rule—addition does NOT affect variance.
Figure: Standard deviation representing variability around the mean.
🎯 Real Exam Insight
In CSS and FPSC exams, questions rarely test direct formulas. Instead, they focus on how data behaves under changes.
When all values increase equally, the mean changes but dispersion remains unchanged.
⚠️ Exam Trap:
Equal addition shifts data but does NOT affect variance or standard deviation.
6. Which measure of dispersion uses all observations in its calculation?
A. Standard deviation
B. Range
C. Interquartile range
D. Quartile deviation
Explanation:
Concept: Standard deviation considers every observation, making it a complete measure of spread.
Trap: Students confuse it with range, which ignores most data points.
Exam Use: Questions often test which measure uses full dataset—standard deviation is the correct answer.
Concept: Standard deviation considers every observation, making it a complete measure of spread.
Trap: Students confuse it with range, which ignores most data points.
Exam Use: Questions often test which measure uses full dataset—standard deviation is the correct answer.
7. The range is considered a weak measure of dispersion because:
A. It depends only on extreme values
B. It uses squared deviations
C. It requires mean
D. It is unit-free
Explanation:
Key Idea: Range depends only on minimum and maximum values.
Mistake: Assuming it reflects overall distribution—this is incorrect.
Application: Examiners test this limitation to check conceptual understanding.
Key Idea: Range depends only on minimum and maximum values.
Mistake: Assuming it reflects overall distribution—this is incorrect.
Application: Examiners test this limitation to check conceptual understanding.
8. If variance of a dataset is 49, its standard deviation is:
A. 7
B. 14
C. 24
D. 98
Explanation:
Logic: Standard deviation is the square root of variance.
Why Correct: √49 = 7 restores original units.
Exam Use: Very common direct numerical in FPSC and GAT exams.
Logic: Standard deviation is the square root of variance.
Why Correct: √49 = 7 restores original units.
Exam Use: Very common direct numerical in FPSC and GAT exams.
9. A smaller standard deviation indicates:
A. Greater consistency
B. Higher variability
C. Larger range
D. Skewed distribution
Explanation:
Concept: Smaller standard deviation means values are closer to the mean.
Trap: Confusing it with higher variability—opposite is true.
Exam Use: Used to interpret consistency in datasets.
Concept: Smaller standard deviation means values are closer to the mean.
Trap: Confusing it with higher variability—opposite is true.
Exam Use: Used to interpret consistency in datasets.
10. If each observation is multiplied by 3, the variance will:
A. Be multiplied by 9
B. Be multiplied by 3
C. Remain unchanged
D. Increase by 3
Explanation:
Concept: Variance depends on squared deviations, so scaling affects it quadratically.
Trap: Many students think it multiplies by 3 instead of 9.
Exam Use: Transformation-based MCQs are very common in CSS and PMS exams.
Concept: Variance depends on squared deviations, so scaling affects it quadratically.
Trap: Many students think it multiplies by 3 instead of 9.
Exam Use: Transformation-based MCQs are very common in CSS and PMS exams.
👨🏫 Classroom Scenario
Consider two classes with the same average (mean) marks.
In one class, students score nearly the same marks, while in the other, scores vary widely between high and low performers.
Although the mean is identical, their consistency is different.
This difference in spread is captured by standard deviation.
🎯 Exam Connection:
Same mean + different spread → dispersion-based question
PART-2 (MCQs 11–20)
11. The variance of the first n natural numbers depends primarily on:
A. The value of n
B. The square of mean only
C. The range alone
D. The median only
Explanation:
Concept: Variance increases as the number of observations (n) increases.
Trap: Thinking mean alone determines dispersion—incorrect.
Exam Use: Frequently appears in theoretical MCQs.
Concept: Variance increases as the number of observations (n) increases.
Trap: Thinking mean alone determines dispersion—incorrect.
Exam Use: Frequently appears in theoretical MCQs.
12. If the standard deviation of a dataset is 5, the variance is:
A. 25
B. 10
C. 5
D. 20
Explanation:
Key Idea: Variance is the square of standard deviation.
Mistake: Students sometimes multiply instead of squaring.
Application: Core formula tested in every competitive exam.
Key Idea: Variance is the square of standard deviation.
Mistake: Students sometimes multiply instead of squaring.
Application: Core formula tested in every competitive exam.
13. Which measure of dispersion is most suitable for comparing variability between datasets with different units?
A. Coefficient of variation
B. Range
C. Variance
D. Standard deviation
Explanation:
Concept: Coefficient of variation standardizes dispersion relative to mean.
Trap: Using standard deviation for cross-unit comparison—incorrect.
Exam Use: Essential for comparing datasets with different units.
Concept: Coefficient of variation standardizes dispersion relative to mean.
Trap: Using standard deviation for cross-unit comparison—incorrect.
Exam Use: Essential for comparing datasets with different units.
14. If the mean of a dataset is 40 and each observation is increased by 10, the standard deviation will:
A. Remain unchanged
B. Increase by 10
C. Double
D. Decrease
Explanation:
Logic: Adding a constant shifts all values equally.
Why Correct: Spread around mean remains unchanged.
Exam Use: Classic transformation MCQ in exams.
Logic: Adding a constant shifts all values equally.
Why Correct: Spread around mean remains unchanged.
Exam Use: Classic transformation MCQ in exams.
15. The range of a dataset increases when:
A. The difference between maximum and minimum increases
B. The mean increases
C. The median changes
D. Frequencies double
Explanation:
Concept: Range depends on the gap between extreme values.
Trap: Confusing it with mean or median changes.
Exam Use: Often asked in conceptual questions.
Concept: Range depends on the gap between extreme values.
Trap: Confusing it with mean or median changes.
Exam Use: Often asked in conceptual questions.
💡 Concept Insight
Range is the simplest but weakest measure of dispersion because it considers only the maximum and minimum values, ignoring the rest of the dataset.
⚠️ Examiner Trap:
A single outlier can drastically change the range, making it unreliable for accurate data analysis.
16. Which statement about standard deviation is correct?
A. It is always non-negative
B. It can be negative
C. It equals variance
D. It is unit-free
Explanation:
Key Idea: Standard deviation cannot be negative as it is derived from squared values.
Mistake: Assuming negative values are possible.
Application: Fundamental property tested frequently.
Key Idea: Standard deviation cannot be negative as it is derived from squared values.
Mistake: Assuming negative values are possible.
Application: Fundamental property tested frequently.
17. If all frequencies in a distribution are doubled, the variance will:
A. Remain unchanged
B. Double
C. Halve
D. Become zero
Explanation:
Logic: Doubling frequencies scales numerator and denominator equally.
Why Correct: Relative spread remains unchanged.
Exam Use: Common conceptual trap in grouped data questions.
Logic: Doubling frequencies scales numerator and denominator equally.
Why Correct: Relative spread remains unchanged.
Exam Use: Common conceptual trap in grouped data questions.
18. The square root transformation applied to variance produces:
A. Standard deviation
B. Range
C. Mean deviation
D. Coefficient of variation
Explanation:
Concept: Standard deviation is defined as square root of variance.
Trap: Confusing it with mean deviation.
Exam Use: Basic but repeatedly tested concept.
Concept: Standard deviation is defined as square root of variance.
Trap: Confusing it with mean deviation.
Exam Use: Basic but repeatedly tested concept.
19. A dataset with larger standard deviation is considered:
A. More variable
B. More consistent
C. Symmetrical
D. Uniform
Explanation:
Key Idea: Larger standard deviation indicates greater spread.
Mistake: Assuming it means consistency—incorrect.
Application: Used to compare variability between datasets.
Key Idea: Larger standard deviation indicates greater spread.
Mistake: Assuming it means consistency—incorrect.
Application: Used to compare variability between datasets.
20. If each observation is multiplied by 4, the standard deviation will:
A. Be multiplied by 4
B. Be multiplied by 16
C. Remain unchanged
D. Increase by 4
Explanation:
Concept: Standard deviation scales directly with multiplication.
Trap: Thinking it squares like variance.
Exam Use: Important transformation rule.
Concept: Standard deviation scales directly with multiplication.
Trap: Thinking it squares like variance.
Exam Use: Important transformation rule.
⚡ Quick Decision Rule (MCQ Shortcut)
⏱️ Under exam pressure, don’t calculate—identify!
🔍 Use this smart selection guide:
📊 Comparing two datasets?
→ Go for Coefficient of Variation (CV) (best for relative comparison)
📈 Spread around the mean?
→ Choose Standard Deviation (most commonly tested)
📦 Outliers or extreme values mentioned?
→ Switch to Interquartile Range (IQR) (robust measure)
💡 Exam Hack:
"Compare → CV | Spread → SD | Outliers → IQR"
⏱️ Under exam pressure, don’t calculate—identify!
🔍 Use this smart selection guide:
📊 Comparing two datasets?
→ Go for Coefficient of Variation (CV) (best for relative comparison)
📈 Spread around the mean?
→ Choose Standard Deviation (most commonly tested)
📦 Outliers or extreme values mentioned?
→ Switch to Interquartile Range (IQR) (robust measure)
💡 Exam Hack:
"Compare → CV | Spread → SD | Outliers → IQR"
PART-3 (MCQs 21–30)
21. If the range of a dataset is 40 and the minimum value is 12, the maximum value is:
A. 52
B. 48
C. 28
D. 60
Explanation:
Concept: Range links minimum and maximum values directly.
Trap: Students sometimes subtract incorrectly or reverse values.
Exam Use: Often tested with missing max/min values like this question.
Concept: Range links minimum and maximum values directly.
Trap: Students sometimes subtract incorrectly or reverse values.
Exam Use: Often tested with missing max/min values like this question.
22. The variance of a population is calculated by dividing the sum of squared deviations by:
A. N
B. N − 1
C. √N
D. Σx
Explanation:
Key Idea: Population variance uses total number of observations (N).
Mistake: Confusing it with sample variance formula (n − 1).
Application: Important distinction in inferential statistics MCQs.
Key Idea: Population variance uses total number of observations (N).
Mistake: Confusing it with sample variance formula (n − 1).
Application: Important distinction in inferential statistics MCQs.
23. Sample variance differs from population variance because it is divided by:
A. n − 1
B. n
C. n²
D. Mean
Explanation:
Key Idea: Dividing by (n − 1) corrects bias in sample estimation.
Mistake: Using n instead of (n − 1) leads to underestimation of variance.
Application: Essential concept in inferential statistics and hypothesis testing.
Key Idea: Dividing by (n − 1) corrects bias in sample estimation.
Mistake: Using n instead of (n − 1) leads to underestimation of variance.
Application: Essential concept in inferential statistics and hypothesis testing.
24. If two datasets have the same mean but different standard deviations, the dataset with larger standard deviation is:
A. More spread out
B. More symmetrical
C. Less consistent
D. Having smaller range
Explanation:
Logic: Standard deviation measures spread around the mean.
Why Correct: Larger SD means values are more widely dispersed.
Exam Use: Frequently used in comparison-based MCQs.
Logic: Standard deviation measures spread around the mean.
Why Correct: Larger SD means values are more widely dispersed.
Exam Use: Frequently used in comparison-based MCQs.
25. If variance is 81, the standard deviation is:
A. 9
B. 18
C. 27
D. 40.5
Explanation:
Concept: Standard deviation is the square root of variance.
Trap: Students sometimes forget to take square root.
Exam Use: Direct calculation questions are very common.
Concept: Standard deviation is the square root of variance.
Trap: Students sometimes forget to take square root.
Exam Use: Direct calculation questions are very common.
💡 Concept Insight
Variance measures spread in squared units, while
standard deviation is the square root of variance, bringing the measure back to original units for easier interpretation.
⚠️ Examiner Trap:
Students often forget to take the square root when converting variance into standard deviation.
Figure: Visualization of data dispersion around the mean using standard deviation in a normal distribution.
26. The unit of standard deviation is:
A. Same as the original data
B. Squared units
C. Unit-free
D. Percentage only
Explanation:
Key Idea: Standard deviation retains original units of measurement.
Mistake: Confusing it with variance, which uses squared units.
Application: Helps interpret dispersion in real-world terms.
Key Idea: Standard deviation retains original units of measurement.
Mistake: Confusing it with variance, which uses squared units.
Application: Helps interpret dispersion in real-world terms.
27. A dataset with zero range must:
A. Contain identical values
B. Have zero mean
C. Have negative variance
D. Be symmetrical
Explanation:
Concept: Zero range means no difference between max and min.
Trap: Assuming distribution properties like symmetry.
Exam Use: Indicates no variability—important theoretical case.
Concept: Zero range means no difference between max and min.
Trap: Assuming distribution properties like symmetry.
Exam Use: Indicates no variability—important theoretical case.
28. Which measure of dispersion is most mathematically tractable for algebraic manipulation?
A. Variance
B. Range
C. Interquartile range
D. Quartile deviation
Explanation:
Logic: Variance allows algebraic manipulation due to squared terms.
Why Correct: It integrates well with mathematical models.
Exam Use: Used heavily in advanced statistics and derivations.
Logic: Variance allows algebraic manipulation due to squared terms.
Why Correct: It integrates well with mathematical models.
Exam Use: Used heavily in advanced statistics and derivations.
29. If standard deviation is small relative to mean, the dataset is considered:
A. Highly consistent
B. Highly skewed
C. Extremely variable
D. Having wide range
Explanation:
Concept: Small SD relative to mean implies tight clustering.
Trap: Confusing it with skewness or distribution shape.
Exam Use: Used to assess consistency in datasets.
Concept: Small SD relative to mean implies tight clustering.
Trap: Confusing it with skewness or distribution shape.
Exam Use: Used to assess consistency in datasets.
30. The coefficient of variation is calculated as:
A. (Standard deviation / Mean) × 100
B. Variance / Mean
C. Mean / Standard deviation
D. Range / Mean
Explanation:
Key Idea: Coefficient of variation expresses relative dispersion.
Mistake: Forgetting to multiply by 100.
Application: Essential for comparing datasets across different scales.
Key Idea: Coefficient of variation expresses relative dispersion.
Mistake: Forgetting to multiply by 100.
Application: Essential for comparing datasets across different scales.
💡 Concept Insight
Coefficient of Variation (CV) is a relative measure of dispersion, expressed as a percentage, making it ideal for comparing datasets with different units or scales.
⚠️ Examiner Trap:
If the mean is zero or very small, CV becomes undefined or misleading, so comparisons may not be valid.
PART-4 (MCQs 31–40)
31. If each observation in a dataset is increased by 5, the variance will:
A. Remain unchanged
B. Increase by 5
C. Double
D. Become zero
Explanation:
Concept: Adding a constant shifts data without changing spread.
Trap: Thinking variance increases with addition.
Exam Use: Highly repeated transformation rule.
Concept: Adding a constant shifts data without changing spread.
Trap: Thinking variance increases with addition.
Exam Use: Highly repeated transformation rule.
32. If each observation is multiplied by 3, the variance will:
A. Be multiplied by 9
B. Be multiplied by 3
C. Remain unchanged
D. Be divided by 3
Explanation:
Logic: Variance depends on squared deviations.
Why Correct: Multiplying by 3 increases variance by 9.
Exam Use: Common conceptual MCQ in exams.
Logic: Variance depends on squared deviations.
Why Correct: Multiplying by 3 increases variance by 9.
Exam Use: Common conceptual MCQ in exams.
33. The square root of the average of squared deviations from the mean is called:
A. Standard deviation
B. Range
C. Mean deviation
D. Quartile deviation
Explanation:
Key Idea: Standard deviation measures average spread in original units.
Mistake: Confusing it with variance or mean deviation.
Application: Most widely used dispersion measure.
Key Idea: Standard deviation measures average spread in original units.
Mistake: Confusing it with variance or mean deviation.
Application: Most widely used dispersion measure.
34. Which measure of dispersion is most sensitive to extreme values?
A. Range
B. Quartile deviation
C. Interquartile range
D. Coefficient of variation
Explanation:
Concept: Range reacts strongly to extreme values.
Trap: Assuming all measures respond equally to outliers.
Exam Use: Used to test understanding of sensitivity.
Concept: Range reacts strongly to extreme values.
Trap: Assuming all measures respond equally to outliers.
Exam Use: Used to test understanding of sensitivity.
35. The interquartile range (IQR) is defined as:
A. Q3 − Q1
B. Q1 − Q3
C. Maximum − Minimum
D. Mean − Median
Explanation:
Logic: IQR focuses on middle 50% of data.
Why Correct: Q3 − Q1 captures central spread.
Exam Use: Important for skewed distributions.
Logic: IQR focuses on middle 50% of data.
Why Correct: Q3 − Q1 captures central spread.
Exam Use: Important for skewed distributions.
💡 Concept Insight
Interquartile Range (IQR) measures the spread of the middle 50% of data and ignores extreme values, making it a robust measure for skewed distributions.
⚠️ Examiner Trap:
IQR is not the total range—it only covers the middle portion between Q1 and Q3.
36. If standard deviation equals zero, the dataset must:
A. Contain identical observations
B. Have zero mean
C. Be negatively skewed
D. Have negative variance
Explanation:
Concept: Zero SD means no variability at all.
Trap: Assuming other distribution properties apply.
Exam Use: Indicates identical observations.
Concept: Zero SD means no variability at all.
Trap: Assuming other distribution properties apply.
Exam Use: Indicates identical observations.
37. Which measure of dispersion is most appropriate for skewed distributions?
A. Interquartile range
B. Variance
C. Standard deviation
D. Range
Explanation:
Key Idea: IQR resists influence of extreme values.
Mistake: Using standard deviation for skewed data.
Application: Preferred measure for non-normal distributions.
Key Idea: IQR resists influence of extreme values.
Mistake: Using standard deviation for skewed data.
Application: Preferred measure for non-normal distributions.
38. The coefficient of variation is especially useful when:
A. Comparing variability between datasets with different units
B. Data are nominal
C. Mean is zero
D. Only range is known
Explanation:
Concept: CV standardizes variability across datasets.
Trap: Using it when mean is zero.
Exam Use: Useful for comparing performance or consistency.
Concept: CV standardizes variability across datasets.
Trap: Using it when mean is zero.
Exam Use: Useful for comparing performance or consistency.
39. A large variance indicates:
A. Greater spread around the mean
B. Smaller mean
C. Zero dispersion
D. Symmetry
Explanation:
Logic: Variance measures average squared spread.
Why Correct: Larger value means wider dispersion.
Exam Use: Core interpretation-based MCQ.
Logic: Variance measures average squared spread.
Why Correct: Larger value means wider dispersion.
Exam Use: Core interpretation-based MCQ.
40. Among range, variance, and standard deviation, the most widely used measure in inferential statistics is:
A. Standard deviation
B. Range
C. Quartile deviation
D. Mid-range
Explanation:
Key Idea: Standard deviation is widely used in inferential statistics.
Mistake: Assuming range is sufficient for analysis.
Application: Used in confidence intervals and hypothesis testing.
Key Idea: Standard deviation is widely used in inferential statistics.
Mistake: Assuming range is sufficient for analysis.
Application: Used in confidence intervals and hypothesis testing.
PART-5 (MCQs 41–50)
41. The variance of the first n natural numbers is:
A. (n² − 1) / 12
B. n(n + 1) / 2
C. (n + 1) / 2
D. n² / 2
Explanation:
Concept: Variance of first n natural numbers follows a standard formula.
Trap: Confusing it with sum formula n(n+1)/2.
Exam Use: Frequently appears in advanced numerical MCQs.
Concept: Variance of first n natural numbers follows a standard formula.
Trap: Confusing it with sum formula n(n+1)/2.
Exam Use: Frequently appears in advanced numerical MCQs.
42. If the standard deviation of a dataset is 4, the variance is:
A. 2
B. 16
C. 8
D. 12
Explanation:
Logic: Variance is square of standard deviation.
Why Correct: 4² = 16 gives dispersion in squared units.
Exam Use: Direct formula-based MCQ.
Logic: Variance is square of standard deviation.
Why Correct: 4² = 16 gives dispersion in squared units.
Exam Use: Direct formula-based MCQ.
43. If the coefficient of variation of Dataset A is 20% and Dataset B is 35%, which dataset is more consistent?
A. Dataset A
B. Dataset B
C. Both are equally consistent
D. Cannot be determined
Explanation:
Key Idea: Lower coefficient of variation means higher consistency.
Mistake: Comparing absolute values instead of relative dispersion.
Application: Used in comparing performance of datasets.
Key Idea: Lower coefficient of variation means higher consistency.
Mistake: Comparing absolute values instead of relative dispersion.
Application: Used in comparing performance of datasets.
44. Which measure of dispersion is expressed in the same units as the data?
A. Standard deviation
B. Variance
C. Coefficient of variation
D. Relative range
Explanation:
Concept: Standard deviation is expressed in original units.
Trap: Confusing it with variance (squared units).
Exam Use: Helps in real-world interpretation of data spread.
Concept: Standard deviation is expressed in original units.
Trap: Confusing it with variance (squared units).
Exam Use: Helps in real-world interpretation of data spread.
45. If variance is 25, the standard deviation is:
A. 5
B. 10
C. 12.5
D. 625
Explanation:
Logic: Standard deviation is the square root of variance.
Why Correct: √25 = 5 restores unit consistency.
Exam Use: Basic but high-frequency numerical question.
Logic: Standard deviation is the square root of variance.
Why Correct: √25 = 5 restores unit consistency.
Exam Use: Basic but high-frequency numerical question.
💡 Concept Insight
Variance and standard deviation are closely linked measures of dispersion, but only standard deviation is directly interpretable because it is expressed in the same units as the data.
⚠️ Examiner Trap:
Students often report variance instead of standard deviation. Always check what the question asks—units matter!
46. In a normal distribution, approximately 68% of observations lie within:
A. ±1 standard deviation
B. ±2 standard deviations
C. ±3 standard deviations
D. Range only
Explanation:
Concept: Empirical rule defines spread in normal distribution.
Trap: Mixing 68%, 95%, and 99.7% ranges.
Exam Use: Very common in theoretical MCQs.
Concept: Empirical rule defines spread in normal distribution.
Trap: Mixing 68%, 95%, and 99.7% ranges.
Exam Use: Very common in theoretical MCQs.
47. The square of the coefficient of variation gives information about:
A. Relative variance
B. Absolute deviation
C. Range
D. Mean deviation
Explanation:
Key Idea: Squaring CV relates to relative variance.
Mistake: Confusing it with absolute variance.
Application: Useful in comparative statistical analysis.
Key Idea: Squaring CV relates to relative variance.
Mistake: Confusing it with absolute variance.
Application: Useful in comparative statistical analysis.
48. If two distributions have equal means but different standard deviations, the one with higher standard deviation is:
A. More dispersed
B. More consistent
C. Symmetrical
D. Uniform
Explanation:
Concept: Higher standard deviation means greater spread.
Trap: Assuming equal mean implies equal dispersion.
Exam Use: Common comparison-based question.
Concept: Higher standard deviation means greater spread.
Trap: Assuming equal mean implies equal dispersion.
Exam Use: Common comparison-based question.
49. Which measure of dispersion cannot be negative?
A. Variance
B. Mean deviation
C. Standard deviation
D. All of the above
Explanation:
Logic: Variance is based on squared values.
Why Correct: Squares cannot produce negative results.
Exam Use: Tests fundamental properties of dispersion.
Logic: Variance is based on squared values.
Why Correct: Squares cannot produce negative results.
Exam Use: Tests fundamental properties of dispersion.
50. The primary purpose of measures of dispersion is to:
A. Quantify variability around a central value
B. Determine correlation
C. Measure central tendency
D. Test hypotheses
Explanation:
Concept: Dispersion measures variability around central value.
Trap: Confusing it with central tendency.
Exam Use: Basic conceptual MCQ.
Concept: Dispersion measures variability around central value.
Trap: Confusing it with central tendency.
Exam Use: Basic conceptual MCQ.
💡 Concept Insight
Measures of dispersion complement central tendency by showing how data is spread or scattered around the average.
⚠️ Examiner Trap:
The mean alone is not enough—two datasets can have the same mean but very different variability.
PART-6 (MCQs 51–60)
51. If each observation of a dataset is transformed as X' = 3X − 2, the new variance will be:
A. 9σ²
B. 3σ²
C. σ² − 2
D. 9σ² − 2
Explanation:
Concept: In transformation X' = aX + b, variance becomes a²σ².
Trap: Students often ignore the square on the coefficient.
Exam Use: Very high-yield transformation rule in CSS/FPSC exams.
Concept: In transformation X' = aX + b, variance becomes a²σ².
Trap: Students often ignore the square on the coefficient.
Exam Use: Very high-yield transformation rule in CSS/FPSC exams.
52. Two datasets have identical means. Dataset A has variance 16, Dataset B has variance 25. Which statement is correct?
A. Dataset B is more dispersed
B. Dataset A is more dispersed
C. Both have equal dispersion
D. Cannot be compared
Explanation:
Logic: Variance directly reflects spread of observations.
Why Correct: 25 > 16 means Dataset B is more dispersed.
Exam Use: Common comparison-based MCQ in exams.
Logic: Variance directly reflects spread of observations.
Why Correct: 25 > 16 means Dataset B is more dispersed.
Exam Use: Common comparison-based MCQ in exams.
53. If the coefficient of variation is zero, this implies:
A. All observations are identical
B. Mean is zero
C. Variance is negative
D. Standard deviation equals mean
Explanation:
Concept: CV = (SD / Mean) × 100; if SD = 0, CV = 0.
Trap: Confusing zero CV with zero mean.
Exam Use: Indicates perfect consistency in dataset.
Concept: CV = (SD / Mean) × 100; if SD = 0, CV = 0.
Trap: Confusing zero CV with zero mean.
Exam Use: Indicates perfect consistency in dataset.
54. In a normal distribution, approximately 95% of observations lie within:
A. ±1 standard deviation
B. ±2 standard deviations
C. ±3 standard deviations
D. Interquartile range
Explanation:
Key Idea: Empirical rule: 95% data lies within ±2 SD.
Mistake: Mixing it with 68% or 99.7% ranges.
Application: Frequently tested in theoretical MCQs.
Key Idea: Empirical rule: 95% data lies within ±2 SD.
Mistake: Mixing it with 68% or 99.7% ranges.
Application: Frequently tested in theoretical MCQs.
55. The variance of a constant added to every observation is:
A. Unchanged
B. Increased by that constant
C. Multiplied by that constant
D. Reduced to zero
Explanation:
Logic: Adding a constant shifts data but does not affect spread.
Why Correct: Deviations from mean remain unchanged.
Exam Use: Classic transformation concept in exams.
Logic: Adding a constant shifts data but does not affect spread.
Why Correct: Deviations from mean remain unchanged.
Exam Use: Classic transformation concept in exams.
💡 Concept Insight
In linear transformations, adding a constant shifts the data (changes mean only), while multiplying by a constant scales the dispersion (changes variance and standard deviation).
⚠️ Examiner Trap:
Addition does NOT affect variance—only multiplication changes dispersion.
56. If standard deviation doubles while mean remains constant, the coefficient of variation will:
A. Double
B. Halve
C. Remain constant
D. Become zero
Explanation:
Concept: CV depends directly on standard deviation when mean is constant.
Trap: Ignoring proportional relationship.
Exam Use: Important in relative variability questions.
Concept: CV depends directly on standard deviation when mean is constant.
Trap: Ignoring proportional relationship.
Exam Use: Important in relative variability questions.
57. Which measure of dispersion is most affected by extreme values?
A. Range
B. Interquartile range
C. Quartile deviation
D. Coefficient of variation
Explanation:
Logic: Range uses only maximum and minimum values.
Why Correct: Extreme values strongly influence it.
Exam Use: Tests sensitivity of dispersion measures.
Logic: Range uses only maximum and minimum values.
Why Correct: Extreme values strongly influence it.
Exam Use: Tests sensitivity of dispersion measures.
58. If variance is minimized, which condition must hold?
A. Observations cluster closely around the mean
B. Mean equals zero
C. Range is maximum
D. Sample size is small
Explanation:
Concept: Lower variance means observations are tightly clustered.
Trap: Relating it incorrectly with mean value.
Exam Use: Used in interpretation-based MCQs.
Concept: Lower variance means observations are tightly clustered.
Trap: Relating it incorrectly with mean value.
Exam Use: Used in interpretation-based MCQs.
59. For grouped data, standard deviation is computed using:
A. Class midpoints and frequencies
B. Cumulative frequencies only
C. Highest class interval
D. Median class boundaries
Explanation:
Key Idea: Grouped data calculations rely on class midpoints.
Mistake: Ignoring frequencies in calculations.
Application: Essential for practical statistics problems.
Key Idea: Grouped data calculations rely on class midpoints.
Mistake: Ignoring frequencies in calculations.
Application: Essential for practical statistics problems.
60. If mean = 50 and standard deviation = 5, the coefficient of variation is:
A. 10%
B. 5%
C. 20%
D. 50%
Explanation:
Logic: CV = (SD / Mean) × 100 formula applies.
Why Correct: (5/50)×100 = 10%.
Exam Use: Frequently asked numerical MCQ.
Logic: CV = (SD / Mean) × 100 formula applies.
Why Correct: (5/50)×100 = 10%.
Exam Use: Frequently asked numerical MCQ.
💡 Concept Insight
Coefficient of Variation (CV) is a relative measure of dispersion that allows comparison between datasets even when their units or scales differ.
⚠️ Examiner Trap:
CV becomes meaningless or undefined when the mean is zero or extremely small.
PART-7 (MCQs 61–70)
61. Assertion (A): Standard deviation is always non-negative.
Reason (R): It is the square root of variance.
Reason (R): It is the square root of variance.
A. Both A and R are true, and R is the correct explanation of A
B. Both A and R are true, but R is not the correct explanation
C. A is true, R is false
D. A is false, R is true
Explanation:
Concept: Standard deviation is square root of variance.
Trap: Expecting negative values.
Exam Use: Assertion-reason MCQs frequently test this.
Concept: Standard deviation is square root of variance.
Trap: Expecting negative values.
Exam Use: Assertion-reason MCQs frequently test this.
62. If two distributions have equal standard deviations but different means, their absolute dispersion is:
A. Equal
B. Higher for the larger mean
C. Lower for the smaller mean
D. Cannot be determined
Explanation:
Key Idea: Standard deviation measures absolute dispersion.
Mistake: Linking it with mean difference.
Application: Equal SD means equal spread.
Key Idea: Standard deviation measures absolute dispersion.
Mistake: Linking it with mean difference.
Application: Equal SD means equal spread.
63. When comparing variability of two series with different units, the most appropriate measure is:
A. Coefficient of variation
B. Variance
C. Standard deviation
D. Range
Explanation:
Concept: CV standardizes dispersion relative to mean.
Trap: Using SD when units differ.
Exam Use: Essential for cross-dataset comparison.
Concept: CV standardizes dispersion relative to mean.
Trap: Using SD when units differ.
Exam Use: Essential for cross-dataset comparison.
64. If every value in a dataset is multiplied by −4, the standard deviation will:
A. Be multiplied by 4
B. Be multiplied by −4
C. Remain unchanged
D. Become zero
Explanation:
Logic: SD changes by |a| under multiplication.
Why Correct: −4 affects direction, not magnitude.
Exam Use: Important transformation MCQ.
Logic: SD changes by |a| under multiplication.
Why Correct: −4 affects direction, not magnitude.
Exam Use: Important transformation MCQ.
65. In a highly skewed dataset, the most robust measure of dispersion is:
A. Interquartile range
B. Variance
C. Standard deviation
D. Range
Explanation:
Concept: IQR focuses on central 50% of data.
Trap: Using SD for skewed datasets.
Exam Use: Preferred measure for outlier-resistant analysis.
Concept: IQR focuses on central 50% of data.
Trap: Using SD for skewed datasets.
Exam Use: Preferred measure for outlier-resistant analysis.
💡 Concept Insight
Robust measures like the Interquartile Range (IQR) ignore extreme values, making them ideal for analyzing skewed datasets.
⚠️ Examiner Trap:
Standard deviation is highly affected by outliers, so it can give misleading results in skewed distributions.
Figure: Interquartile range as a robust measure of dispersion.
66. If variance of a dataset is 0, the range must be:
A. 0
B. 1
C. Equal to mean
D. Undefined
Explanation:
Logic: Zero variance means all values are identical.
Why Correct: No difference → range becomes zero.
Exam Use: Fundamental property MCQ.
Logic: Zero variance means all values are identical.
Why Correct: No difference → range becomes zero.
Exam Use: Fundamental property MCQ.
67. Which measure of dispersion minimizes the sum of squared deviations?
A. Mean (as reference for variance)
B. Median
C. Mode
D. Range
Explanation:
Concept: Mean minimizes sum of squared deviations.
Trap: Confusing with median or mode.
Exam Use: Core theoretical concept in statistics.
Concept: Mean minimizes sum of squared deviations.
Trap: Confusing with median or mode.
Exam Use: Core theoretical concept in statistics.
68. If coefficient of variation of Series A is 15% and Series B is 10%, which series is more consistent?
A. Series B
B. Series A
C. Both equally consistent
D. Cannot be compared
Explanation:
Key Idea: Lower CV indicates higher consistency.
Mistake: Choosing higher CV incorrectly.
Application: Used in performance comparison.
Key Idea: Lower CV indicates higher consistency.
Mistake: Choosing higher CV incorrectly.
Application: Used in performance comparison.
69. If mean = 100 and coefficient of variation = 20%, the standard deviation equals:
A. 20
B. 5
C. 80
D. 120
Explanation:
Logic: SD = (CV × Mean) / 100.
Why Correct: (20 × 100)/100 = 20.
Exam Use: Common reverse-calculation MCQ.
Logic: SD = (CV × Mean) / 100.
Why Correct: (20 × 100)/100 = 20.
Exam Use: Common reverse-calculation MCQ.
70. In sampling theory, dividing by (n − 1) instead of n in variance formula provides:
A. Unbiased estimate of population variance
B. Larger dispersion intentionally
C. Smaller standard deviation
D. Correction for mean
Explanation:
Concept: Dividing by (n − 1) removes bias in estimation.
Trap: Using n instead of (n − 1).
Exam Use: Critical concept in inferential statistics.
Concept: Dividing by (n − 1) removes bias in estimation.
Trap: Using n instead of (n − 1).
Exam Use: Critical concept in inferential statistics.
🎯 How to Solve Dispersion MCQs in Exams
💡 Smart Strategy: In exams, speed comes from clarity. Train your mind to recognize the type of question before jumping into calculations.
- 🔍 Identify first: Is it conceptual or numerical? This saves time instantly
- 🔄 Check transformations: Addition does not change dispersion, multiplication does
- 🧠 Spot keywords: Words like "increase", "multiply", or "compare" reveal the method
- ❌ Use elimination: Remove wrong options quickly in tricky MCQs
⚡ Exam Hack: "Identify → Analyze → Eliminate → Answer 🎯"
⚠️ Common Mistakes Students Make
🚫 Avoid these frequent errors to boost your score:
- ⚠️ Confusing variance with standard deviation
- 📉 Forgetting to take square root when required
- 📊 Applying coefficient of variation when mean is zero (invalid case)
- 🔁 Ignoring transformation rules in conceptual MCQs
💡 Quick Fix: Revise concepts, not just formulas—this prevents repeated mistakes.
📊 FPSC / CSS Paper Trend
📈 In recent papers, statistics MCQs are shifting from direct calculation to conceptual interpretation. Examiners now focus more on how well you understand dispersion in statistics rather than how fast you apply formulas.
🎯 Questions commonly test data variability, transformation rules, and real-life application. For effective CSS statistics preparation, prioritize understanding patterns, relationships, and logic behind the formulas instead of rote learning.
🚀 Final Advice: "Concept clarity = High score in competitive exams"
⚡ 1-Minute Revision Table (Must Review Before Exam)
| Concept | Formula / Idea | Key Insight |
|---|---|---|
| Range | Max − Min | Uses only extreme values |
| Variance | σ² | Measured in squared units |
| Standard Deviation | √Variance | Most widely used measure |
| Coefficient of Variation | (SD / Mean) × 100 | Unit-free comparison |
🎯 Last-Minute Hack:
👉 Range → simplest but weakest
👉 SD → most reliable
👉 CV → best for comparison
💡 Learn these 3 → solve most MCQs instantly!
📊 Comparison of Measures of Dispersion
| Measure | Uses All Data? | Unit | Sensitivity to Outliers |
|---|---|---|---|
| Range | No | Same | Very High |
| Variance | Yes | Squared | High |
| Standard Deviation | Yes | Same | Moderate |
| IQR | Partial (middle 50%) | Same | Low |
🎯 Exam Insight:
👉 Range → most affected by outliers
👉 Variance → squared units (less interpretable)
👉 Standard Deviation → most practical measure
👉 IQR → best for skewed data
💡 Choose measure based on data type!
🔑 Key Differences
📊 Variance is measured in squared units, while
standard deviation is in original units.
📏 Range uses only extreme values and
ignores most data points.
📦 IQR is resistant to outliers, making it suitable for skewed data.
📐 Coefficient of Variation is a relative measure, while others are absolute.
🎯 Exam Shortcut:
👉 Squared units → Variance
👉 Same units → Standard Deviation
👉 Outliers present → IQR
👉 Comparison → CV
💡 Identify the question type → pick the right measure instantly!
🧠 Key Concepts Students Should Remember
Dispersion measures how data spreads around the mean. While range gives a quick estimate, standard deviation provides deeper insight by considering all observations.
- Range is simple but ignores most data points
- Standard deviation is the most reliable and widely used measure
- Variance is in squared units, making it less interpretable
- Coefficient of variation is best for comparing datasets
- IQR is ideal for skewed data and outliers
- Transformation rules are frequently tested in exams
🎯 Concept Tip:
👉 Focus on interpretation and relationships, not just formulas
💡 Most MCQs test understanding, not calculation
🔁 Concept Reminder
Always identify whether the question is asking for absolute or relative dispersion. This determines the correct measure to use.
Remember the key transformation rule: adding a constant does not affect dispersion, while multiplication changes it significantly.
🎯 Quick Recall:
👉 Same units → use SD
👉 Different units → use CV
👉 Outliers present → use IQR
💡 Identify question type first → avoid mistakes instantly!
⚡ 10-Second Revision Flashcards (Measures of Dispersion)
📏 What is Range?
Difference between maximum and minimum values.
📊 What is Variance?
Average of squared deviations from the mean.
📉 What is Standard Deviation?
Square root of variance; in original units.
📐 What is CV?
(SD / Mean) × 100; used for comparison.
📦 What is IQR?
Q3 − Q1; middle 50% of data.
⚠️ Range Limitation?
Affected heavily by outliers.
⚠️ SD Property?
Always non-negative and widely used.
🔄 Transformation Rule?
Add → no change, Multiply → variance × a².
📊 Best for Comparison?
Coefficient of Variation (CV).
📦 Best for Skewed Data?
Interquartile Range (IQR).
🎯 Key Takeaways
📉 Standard Deviation is the most reliable and widely used measure of dispersion.
📊 Variance is mathematically useful but less interpretable due to squared units.
📏 Range is the simplest but least reliable as it ignores most data.
📦 IQR is best for skewed distributions and resistant to outliers.
🎯 Final Exam Strategy:
👉 Use SD for general analysis
👉 Use CV for comparison
👉 Use IQR when outliers exist
💡 Choosing the right measure = correct answer!
Mastering measures of dispersion is essential for solving advanced exam questions. Concepts like data spread, variability, and comparison help students approach problems logically rather than relying on memorization.
Whether you are solving standard deviation or variance-based questions, the key lies in understanding their application. Strong conceptual clarity enables students to handle both theoretical and numerical problems effectively in competitive exams.
📌 Concluding Analytical Perspective
Understanding dispersion is essential for interpreting real-world data, as it reveals how values are distributed around the mean rather than relying solely on averages.
Two datasets may have the same mean but behave entirely differently depending on their spread and variability, making dispersion a critical tool for deeper analysis.
In competitive examinations, success depends on conceptual clarity and correct application rather than memorization. Knowing when to use range, standard deviation, IQR, or coefficient of variation provides a clear advantage.
🎯 Final Tip:
👉 Focus on interpretation + selecting the right measure
💡 Most MCQs test “which measure to use”, not just formulas
❓ Frequently Asked Questions
These frequently asked questions clarify key concepts of measures of dispersion and help reinforce exam preparation.
Which is the best measure of dispersion?
Standard deviation is considered the most reliable because it uses all data points and is expressed in the same units.
Why is variance squared?
Squaring removes negative values and gives more weight to larger deviations, making variability clearer.
When should I use IQR?
Use IQR for skewed distributions or when outliers are present, as it focuses on the middle 50% of data.
Can variance be negative?
No, variance can never be negative because it is based on squared deviations.
🔗 Related Topics for Better Understanding
To strengthen your concepts in statistics MCQs, you should also explore related topics that are frequently asked in CSS, FPSC, PMS, and GAT exams:
- 📊 Measures of Central Tendency MCQs (Mean, Median, Mode)
- 📈 Skewness and Kurtosis MCQs
- 📉 Frequency Distribution MCQs
💡 Tip: These topics are interconnected and often appear together in exams.
For theoretical understanding, you may also refer to the official explanation of Standard Deviation – Wikipedia , which explains the mathematical foundation and applications in inferential statistics.
Disclaimer: These MCQs are created for educational and practice purposes only. They are designed to support competitive exam preparation including CSS, FPSC, PMS, GAT, and university-level assessments.
Author Credentials: The author has extensive teaching experience in statistics and has guided students for CSS, FPSC, PMS, and university exams for several years.
Last Updated: 29 March 2026
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