Chapter Overview
Measures of Dispersion in One Minute
An average shows where the data sits. Dispersion shows whether values hug that centre or scatter far from it.
R
Range
Largest − Smallest
Q
Quartile Deviation
Middle 50% spread
M
Mean Deviation
Average absolute distance
σ
Standard Deviation
Most powerful measure
1. Meaning and Need for Dispersion
Measures of Central Tendency tell us where the observations are centred. However, they do not tell us how closely the observations are grouped around that centre. This is measured by Dispersion.
Statistics answers two important questions.
1. Where is the centre of the data?
This is answered by the Measures of Central Tendency such as Mean, Median and Mode.
2. How far are the observations spread around that centre?
This is answered by the Measures of Dispersion.
1. Where is the centre of the data?
This is answered by the Measures of Central Tendency such as Mean, Median and Mode.
2. How far are the observations spread around that centre?
This is answered by the Measures of Dispersion.
Why Average Alone is Not Enough?
Factory A
98, 99, 100, 101, 102
Mean = 100 units
Very little variation.
Production is highly consistent.
Factory B
70, 90, 100, 110, 130
Mean = 100 units
Large variation.
Production is unpredictable.
Both factories have the same average production (100 units), but Factory A is much more reliable because its production remains close to the average every day.
Average tells us where the data is centred.
Dispersion tells us how widely the observations are scattered around that centre.
Average tells us where the data is centred.
Dispersion tells us how widely the observations are scattered around that centre.
Definition of Dispersion
Dispersion is the degree to which observations are scattered or deviate from an appropriate measure of central tendency such as the Mean or Median.
Exam Point
Measures of Central Tendency are known as First Order Characteristics because they describe the centre of the data.
Measures of Dispersion are known as Second Order Characteristics because they measure how far the observations deviate from that centre.
Measures of Central Tendency are known as First Order Characteristics because they describe the centre of the data.
Measures of Dispersion are known as Second Order Characteristics because they measure how far the observations deviate from that centre.
Classification of Measures of Dispersion
Measures of Dispersion
Main classification
→
Absolute Measures
Actual amount of spread
→
Relative Measures
Spread relative to the size of the data
Absolute Measures
- Range
- Quartile Deviation
- Mean Deviation
- Standard Deviation
Relative Measures
- Coefficient of Range
- Coefficient of Quartile Deviation
- Coefficient of Mean Deviation
- Coefficient of Variation (CV)
Characteristics of an Ideal Measure of Dispersion
| Characteristic | Importance |
|---|---|
| Rigidly Defined | Everyone should obtain the same result for the same data. |
| Easy to Understand | The result should be simple to interpret. |
| Easy to Compute | Calculations should not be unnecessarily complicated. |
| Based on All Observations | Every observation should contribute to the measure. |
| Unaffected by Sampling Fluctuations | Small changes in the sample should not cause large changes in the result. |
| Amenable to Mathematical Treatment | It should be suitable for further statistical analysis. |
Remember
A good measure of dispersion should be easy to calculate, easy to understand, based on all observations, reasonably stable and suitable for further mathematical analysis.
A good measure of dispersion should be easy to calculate, easy to understand, based on all observations, reasonably stable and suitable for further mathematical analysis.
Business Application
- Measures the consistency of production and quality.
- Compares the stability of sales, profits and costs.
- In finance, greater dispersion generally indicates higher risk or volatility.
2. Absolute and Relative Measures of Dispersion
Measures of dispersion are broadly classified into two categories depending on what they measure.
Absolute Measures
Absolute measures show the actual amount of dispersion in the original units of measurement.
Examples:
- Range
- Quartile Deviation
- Mean Deviation
- Standard Deviation
Relative Measures
Relative measures express dispersion as a ratio or percentage, making comparisons between different series possible.
Examples:
- Coefficient of Range
- Coefficient of Quartile Deviation
- Coefficient of Mean Deviation
- Coefficient of Variation (CV)
Why Do We Need Relative Measures?
Student A
Mean Marks = 90
Standard Deviation = 10
Student B
Mean Marks = 40
Standard Deviation = 10
Both students have the same Standard Deviation (10 marks).
However, losing 10 marks from an average of 90 is very different from losing 10 marks from an average of 40.
Therefore, comparing only the absolute measure may give misleading conclusions.
Relative measures solve this problem by comparing the dispersion with the average.
Difference Between Absolute and Relative Measures
| Basis | Absolute Measures | Relative Measures |
|---|---|---|
| Meaning | Actual amount of dispersion. | Dispersion expressed relative to the average. |
| Units | Same units as the original data. | Unit-free (ratio or percentage). |
| Purpose | Measures spread within a single series. | Compares variability between different series. |
| Comparison | Suitable only when the units and averages are similar. | Can compare different units and different averages. |
| Ease of Computation | Generally easier to calculate and understand. | Slightly more difficult because they involve ratios or percentages. |
| Examples | Range, Q.D., M.D., S.D. | Coefficient of Range, Coefficient of Q.D., Coefficient of M.D., C.V. |
Memory Tip
Absolute Measure answers: "How much is the spread?"
Relative Measure answers: "Is this spread large or small compared to the average?"
Absolute Measure answers: "How much is the spread?"
Relative Measure answers: "Is this spread large or small compared to the average?"
Exam Point
Whenever two or more series have different means, different units or different scales, comparison should always be made using relative measures of dispersion, especially the Coefficient of Variation (CV).
Whenever two or more series have different means, different units or different scales, comparison should always be made using relative measures of dispersion, especially the Coefficient of Variation (CV).
3. Range
The Range is the simplest measure of dispersion. It measures the total spread of a series by using only its largest and smallest observations.
Definition: Range is the difference between the largest value and the smallest value in a series.
Range (R) = L − S
Coefficient of Range = (L − S)/(L + S)
where L = largest observation and S = smallest observation.
Coefficient of Range = (L − S)/(L + S)
where L = largest observation and S = smallest observation.
Illustration: Individual Observations
Data
12, 18, 7, 25, 16, 10
Extremes
L = 25 and S = 7
Range
R = 25 − 7 = 18
Coefficient
(25 − 7)/(25 + 7) = 18/32 = 0.5625
Range in Different Types of Series
| Series | How to Find L and S | Important Note |
|---|---|---|
| Individual Series | Take the largest and smallest observations. | Arrange the data if the extremes are not obvious. |
| Discrete Series | Take the largest and smallest values of x having non-zero frequency. | Frequency does not enter the formula directly. |
| Continuous Series | Use the upper boundary of the highest class and lower boundary of the lowest class. | Convert inclusive classes into class boundaries where necessary. |
Simple Solved Numerical: Continuous Series
Question: Find the Range and Coefficient of Range for the classes 10–20, 20–30, 30–40, 40–50.
Step 1: Lowest class boundary, S = 10.
Step 2: Highest class boundary, L = 50.
Step 3: Range = L − S = 50 − 10 = 40.
Step 4: Coefficient of Range = (L − S)/(L + S) = 40/60 = 0.667 approximately.
Properties of Range
| Situation | Effect on Range |
|---|---|
| All observations are equal | Range = 0 |
| A constant a is added to or subtracted from every value | No change in Range |
| Every value is multiplied or divided by b | Range is multiplied or divided by |b| |
| Extreme values change | Range may change sharply |
If y = a + bx, then Ry = |b|Rx.
Merits
- Very simple to understand and calculate.
- Provides a quick indication of total spread.
- Useful in preliminary comparisons and quality-control work.
- Useful when only the extreme values matter.
Limitations
- Uses only two observations and ignores the rest.
- Highly affected by extreme values.
- Unstable from sample to sample.
- Not suitable for open-ended distributions.
- Has almost no algebraic usefulness.
Best Use: Range is suitable when a quick estimate is required, such as daily temperature variation, stock-price high–low movement, or checking whether manufactured dimensions remain within limits.
Exam Traps: Adding a constant does not change Range. For a continuous distribution, use class boundaries—not merely the printed class limits. Range generally cannot be calculated for open-ended classes.
4. Quartile Deviation
Quartile Deviation measures the spread of the central 50% of observations. Since it ignores the lowest 25% and highest 25%, it is less affected by extreme values.
Interquartile Range (IQR) is the difference between the third quartile and the first quartile. Quartile Deviation is one-half of this range and is therefore also called the Semi-Interquartile Range.
Lowest 25%Ignored
→
Q₁ to Q₃Middle 50% studied
→
Highest 25%Ignored
IQR = Q₃ − Q₁
Q.D. = (Q₃ − Q₁)/2
Coefficient of Q.D. = (Q₃ − Q₁)/(Q₃ + Q₁)
Q.D. = (Q₃ − Q₁)/2
Coefficient of Q.D. = (Q₃ − Q₁)/(Q₃ + Q₁)
How to Locate Q₁ and Q₃
| Series | Q₁ | Q₃ |
|---|---|---|
| Individual Series | Size of (N + 1)/4th item | Size of 3(N + 1)/4th item |
| Discrete Series | Locate N/4th item using cumulative frequency | Locate 3N/4th item using cumulative frequency |
| Continuous Series | L + [(N/4 − c.f.)/f] × h | L + [(3N/4 − c.f.)/f] × h |
In the continuous-series formula: L = lower boundary of the quartile class, c.f. = cumulative frequency before that class, f = frequency of the quartile class and h = class width.
Illustration
Ordered Data
4, 7, 9, 12, 15, 18, 22, 25, 30
Quartiles
N = 9; Q₁ = size of 2.5th item = 8; Q₃ = size of 7.5th item = 23.5
Answer
Q.D. = (23.5 − 8)/2 = 7.75; Coefficient ≈ 0.492
Simple Solved Numerical: Discrete Series
| x | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| f | 2 | 3 | 4 | 3 | 2 |
Step 1: N = Σf = 14. The cumulative frequencies are 2, 5, 9, 12, 14.
Step 2: Q₁ is the size of N/4 = 14/4 = 3.5th item. The 3.5th item lies at x = 10. Therefore, Q₁ = 10.
Step 3: Q₃ is the size of 3N/4 = 10.5th item. The 10.5th item lies at x = 20. Therefore, Q₃ = 20.
Step 4: Q.D. = (20 − 10)/2 = 5.
Step 5: Coefficient of Q.D. = (20 − 10)/(20 + 10) = 10/30 = 0.333.
Properties of Quartile Deviation
| Situation | Effect |
|---|---|
| All observations are equal | Q.D. = 0 |
| Change of origin | No effect on Q.D. |
| Change of scale | Q.D. is multiplied by |b| |
| Extreme observations change | Usually little or no effect unless quartile positions change |
Merits
- Simple and easy to interpret.
- Less affected by extreme values.
- Suitable for skewed distributions.
- Can be used with certain open-ended distributions.
Limitations
- Ignores 50% of the observations.
- Not based on all values.
- Not suitable for algebraic treatment.
- Less stable than Standard Deviation.
Best Use: Quartile Deviation is preferred for skewed distributions, data containing extreme values, and certain open-ended frequency distributions.
Exam Traps: IQR and Q.D. are not the same. Q.D. is one-half of IQR. For comparison between two series, use the Coefficient of Q.D., not Q.D. alone.
5. Mean Deviation
Mean Deviation is the arithmetic mean of the absolute deviations of observations from a selected average—usually the Mean, Median or Mode.
Select a CentreMean, Median or Mode
Find Deviationsx − A
Ignore Signs|x − A|
Take the AverageMean Deviation
Individual Series: M.D.(A) = Σ|x − A|/N
Discrete or Continuous Series: M.D.(A) = Σf|x − A|/Σf
Coefficient of M.D. = M.D.(A)/A
Discrete or Continuous Series: M.D.(A) = Σf|x − A|/Σf
Coefficient of M.D. = M.D.(A)/A
For a continuous series, take class mid-points as x. The denominator of the coefficient must be the same average from which Mean Deviation was calculated.
Illustration
Data
2, 4, 6, 8, 10; Mean = 6
Absolute Deviations
4, 2, 0, 2, 4; Σ|x − x̄| = 12
Answer
M.D. about Mean = 12/5 = 2.4; Coefficient = 2.4/6 = 0.40
Why Are Absolute Signs Used?
Positive and negative deviations cancel each other. In fact, Σ(x − x̄) = 0. Absolute signs prevent cancellation and allow us to find the average distance from the selected centre.
Important Mathematical Result
The sum of absolute deviations, and therefore Mean Deviation, is minimum when deviations are taken from the Median. This is why the Median is often considered the most appropriate centre for Mean Deviation.
Properties of Mean Deviation
| Situation | Effect |
|---|---|
| All observations are equal | M.D. = 0 |
| Change of origin | No effect |
| Change of scale | M.D. is multiplied by |b| |
| Deviation taken from Median | M.D. is minimum |
Merits
- Uses all observations.
- Easy to understand as an average distance.
- Less affected by extremes than S.D. because deviations are not squared.
- Can be calculated about Mean, Median or Mode.
Limitations
- Absolute signs make algebraic treatment difficult.
- Not widely used in advanced statistical analysis.
- The result varies with the average selected.
- Calculation may be lengthy for grouped data.
Exam Traps: Never remove the modulus sign. Do not use Mean in the coefficient if M.D. was calculated about Median. For continuous data, use class mid-points.
6. Standard Deviation and Variance
Standard Deviation is the most important and widely used measure of dispersion. It is based on every observation and measures spread around the arithmetic mean.
How Standard Deviation Develops
The logic behind the formula
Find Meanx̄
→
Find Deviationsx − x̄
→
Square(x − x̄)²
→
AverageVariance
→
Square RootS.D.
Variance is the arithmetic mean of the squared deviations from the arithmetic mean. Standard Deviation is the positive square root of Variance.
Variance: σ² = Σ(x − x̄)²/N
Standard Deviation: σ = √[Σ(x − x̄)²/N]
Frequency Series: σ = √[Σf(x − x̄)²/Σf]
Standard Deviation: σ = √[Σ(x − x̄)²/N]
Frequency Series: σ = √[Σf(x − x̄)²/Σf]
Why Are Deviations Squared?
Squaring prevents positive and negative deviations from cancelling. It also gives greater weight to large deviations, making Standard Deviation sensitive to unusual or extreme observations.
Variance versus Standard Deviation
| Basis | Variance | Standard Deviation |
|---|---|---|
| Meaning | Average squared deviation | Square root of Variance |
| Notation | σ² | σ |
| Unit | Squared unit | Same unit as original data |
| Interpretation | Mainly mathematical | Easier to interpret practically |
If the data are measured in rupees, Variance is measured in rupees squared, whereas Standard Deviation is again measured in rupees.
Why Standard Deviation Is Considered the Best Measure
- Rigidly defined.
- Based on all observations.
- Relatively stable from sample to sample.
- Suitable for algebraic treatment.
- Widely used in correlation, regression and probability.
- Useful in finance, quality control and research.
- Forms the basis of Coefficient of Variation.
- Measures total variation around the Mean.
7. Methods of Calculating Standard Deviation
The method selected depends on the form and size of the data. All methods produce the same Standard Deviation when applied correctly.
Actual Mean Method
Use when the arithmetic mean and deviations are easy to calculate.
Assumed Mean Method
Use when values are large but deviations from a convenient assumed mean are manageable.
Step-Deviation Method
Use when deviations have a common factor, particularly in equal-width continuous classes.
A. Actual Mean or Direct Method
Individual: σ = √[Σ(x − x̄)²/N]
Frequency: σ = √[Σf(x − x̄)²/Σf]
Frequency: σ = √[Σf(x − x̄)²/Σf]
B. Assumed Mean or Short-Cut Method
Let d = x − A.
Individual: σ = √[(Σd²/N) − (Σd/N)²]
Frequency: σ = √[(Σfd²/Σf) − (Σfd/Σf)²]
Individual: σ = √[(Σd²/N) − (Σd/N)²]
Frequency: σ = √[(Σfd²/Σf) − (Σfd/Σf)²]
C. Step-Deviation Method
Let u = (x − A)/h.
σ = h × √[(Σfu²/Σf) − (Σfu/Σf)²]
σ = h × √[(Σfu²/Σf) − (Σfu/Σf)²]
Worked Illustration: Individual Series
Data
2, 4, 6, 8, 10
Mean
x̄ = 6
Squares
Σ(x − 6)² = 16 + 4 + 0 + 4 + 16 = 40
Answer
Variance = 40/5 = 8; S.D. = √8 = 2.83
Simple Solved Numerical: Discrete Frequency Series
| x | 2 | 4 | 6 |
|---|---|---|---|
| f | 1 | 2 | 1 |
Step 1: Σf = 4 and Σfx = (1×2) + (2×4) + (1×6) = 16.
Step 2: Mean = Σfx/Σf = 16/4 = 4.
Step 3: Σf(x−x̄)² = 1(2−4)² + 2(4−4)² + 1(6−4)² = 4 + 0 + 4 = 8.
Step 4: Variance = 8/4 = 2.
Step 5: S.D. = √2 = 1.414 approximately.
Simple Solved Numerical: Step-Deviation Method
| Class | 0–10 | 10–20 | 20–30 |
|---|---|---|---|
| f | 1 | 2 | 1 |
Mid-points x are 5, 15 and 25. Take A = 15 and h = 10.
u = (x−A)/h gives −1, 0 and 1.
Σfu = 1(−1) + 2(0) + 1(1) = 0; Σfu² = 1(1) + 2(0) + 1(1) = 2; Σf = 4.
σ = 10 × √[(2/4) − (0/4)²] = 10 × √0.5 = 7.071 approximately.
For Continuous Frequency Distribution
First find the mid-point of every class. Treat the mid-point as x. In the step-deviation method, h is the common class width or another convenient common factor.
Frequent Errors: Dividing by the number of classes instead of Σf; forgetting the second term in the short-cut formula; forgetting to multiply by h in step-deviation; and stopping at Variance when S.D. is required.
8. Properties of Standard Deviation
| Property | Explanation or Rule |
|---|---|
| Non-negative | σ ≥ 0 because it is the square root of a non-negative quantity. |
| Zero dispersion | σ = 0 only when all observations are equal. |
| Change of origin | Adding or subtracting a constant does not change S.D. |
| Change of scale | Multiplying all observations by b multiplies S.D. by |b|. |
| Variance and scale | Variance is multiplied by b². |
| Unit | S.D. has the same unit as the variable. |
| Extreme values | S.D. is sensitive to extremes because deviations are squared. |
| Based on all values | Every observation influences the result. |
If y = a + bx, then
x̄y = a + bx̄x
σy = |b|σx
σ²y = b²σ²x
x̄y = a + bx̄x
σy = |b|σx
σ²y = b²σ²x
Effect of Common Transformations
| Transformation | Mean | S.D. | Variance |
|---|---|---|---|
| Add a | Increases by a | No change | No change |
| Subtract a | Decreases by a | No change | No change |
| Multiply by b | Multiplied by b | Multiplied by |b| | Multiplied by b² |
| Divide by b | Divided by b | Divided by |b| | Divided by b² |
Simple Solved Numerical: Change of Origin and Scale
Question: The Mean and S.D. of x are 20 and 4. Find the Mean, S.D. and Variance of y = 10 + 3x.
Mean of y: 10 + 3(20) = 70.
S.D. of y: |3| × 4 = 12.
Variance of y: 3² × 4² = 9 × 16 = 144.
The added constant 10 changes the Mean but does not affect S.D.; multiplication by 3 multiplies S.D. by 3.
Typical ICAI Pattern: If the Mean and S.D. of x are given and y = a + bx, students may be asked to find the Mean, S.D., Variance or C.V. of y.
9. Combined Standard Deviation
Combined Standard Deviation is used when two or more groups are merged and the overall variability of the combined group is required.
Step 1: Combined Mean
x̄c = (N₁x̄₁ + N₂x̄₂)/(N₁ + N₂)
Step 2: Combined Variance and S.D.
σ²c = [N₁(σ₁² + d₁²) + N₂(σ₂² + d₂²)]/(N₁ + N₂)
where d₁ = x̄₁ − x̄c and d₂ = x̄₂ − x̄c.
σc = √σ²c
where d₁ = x̄₁ − x̄c and d₂ = x̄₂ − x̄c.
σc = √σ²c
Total combined variation has two parts: variation within each group (σ₁² and σ₂²) and variation caused by the difference between the group means (d₁² and d₂²).
Illustration
Groups
A: N₁ = 40, Mean = 50, σ₁ = 6; B: N₂ = 60, Mean = 60, σ₂ = 8
Combined Mean
x̄c = (40×50 + 60×60)/100 = 56
Distances
d₁ = −6 and d₂ = 4
Result
σ²c = 76.8; σc = 8.76 approximately
Simple Solved Numerical
Question: Group A has 2 observations with Mean 10 and S.D. 2. Group B has 2 observations with Mean 14 and S.D. 2. Find the combined S.D.
Step 1: Combined Mean = (2×10 + 2×14)/4 = 48/4 = 12.
Step 2: d₁ = 10−12 = −2 and d₂ = 14−12 = 2.
Step 3: Combined Variance = [2(2² + (−2)²) + 2(2² + 2²)]/4.
= [2(4+4) + 2(4+4)]/4 = 32/4 = 8.
Step 4: Combined S.D. = √8 = 2.828 approximately.
The combined S.D. can be greater than both group S.D.s when the group means are far apart, because between-group variation is added.
Frequent Errors: Using σ instead of σ² inside the formula; forgetting d²; calculating d from the wrong mean; and forgetting the final square root.
10. Coefficient of Variation
Coefficient of Variation is a relative measure that expresses Standard Deviation as a percentage of the Mean. It is used to compare consistency across series having different averages or units.
C.V. = (σ/x̄) × 100
Interpretation: Lower C.V. means lower variability relative to the Mean and therefore greater consistency. Higher C.V. means greater relative variability and lower consistency.
Illustration
Series
A: Mean = 40, σ = 4; B: Mean = 60, σ = 9
C.V.
A = (4/40)×100 = 10%; B = (9/60)×100 = 15%
Conclusion
Series A is more consistent because it has the lower C.V.
Simple Solved Numerical: Same S.D., Different Means
Question: Series P has Mean 50 and S.D. 5. Series Q has Mean 100 and S.D. 5. Which is more consistent?
C.V. of P = (5/50)×100 = 10%.
C.V. of Q = (5/100)×100 = 5%.
Conclusion: Series Q is more consistent because its C.V. is lower, even though both series have the same S.D.
Why S.D. Alone May Mislead
A larger S.D. does not automatically mean lower consistency. The size of S.D. must be judged relative to the Mean. That is precisely what C.V. does.
Applications
| Area | Use |
|---|---|
| Investment | Compare risk per unit of expected return. |
| Production | Compare process consistency across products with different dimensions. |
| Sales | Compare stability of sales across regions of different sizes. |
| Employee Performance | Compare consistency where average output differs. |
Exam Traps: Lower C.V.—not lower Mean—indicates greater consistency. Do not compare C.V. when the Mean is zero or extremely close to zero because the ratio becomes meaningless or unstable.
11. Comparison and Selection of Measures
Which Measure Should You Use?
Select the measure according to the purpose and nature of data
Quick total spread
Range
Skewed or open-ended data
Quartile Deviation
Average absolute distance
Mean Deviation
Detailed statistical analysis
Standard Deviation
Compare consistency
Coefficient of Variation
| Basis | Range | Q.D. | M.D. | S.D. |
|---|---|---|---|---|
| Observations used | Only two | Middle 50% | All | All |
| Effect of extremes | Very high | Low | Moderate | High |
| Ease of calculation | Very easy | Easy | Moderate | Comparatively detailed |
| Sampling stability | Low | Better than Range | Reasonable | Highest |
| Algebraic treatment | Not suitable | Not suitable | Limited | Excellent |
| Best use | Quick check | Skewed/open-ended data | Average distance | Scientific analysis |
Decision Table
| Requirement | Preferred Measure | Reason |
|---|---|---|
| Quickest measure | Range | Uses only largest and smallest values. |
| Extreme values or skewness | Quartile Deviation | Focuses on the middle 50%. |
| Open-ended distribution | Quartile Deviation | Quartiles can often be located without both extreme boundaries. |
| Average absolute distance | Mean Deviation | Uses absolute deviations. |
| Further mathematical analysis | Standard Deviation | Strong algebraic properties. |
| Comparison of consistency | Coefficient of Variation | Relative and unit-free. |
12. Formula Review, ICAI Patterns and Exam Traps
Master Formula Table
| Measure | Absolute Formula | Relative Formula |
|---|---|---|
| Range | L − S | (L − S)/(L + S) |
| Quartile Deviation | (Q₃ − Q₁)/2 | (Q₃ − Q₁)/(Q₃ + Q₁) |
| Mean Deviation | Σ|x − A|/N or Σf|x − A|/Σf | M.D.(A)/A |
| Variance | Σ(x − x̄)²/N | — |
| Standard Deviation | √Variance | C.V. = (σ/x̄)×100 |
Common ICAI Question Patterns
- Calculate Range, Q.D., M.D., Variance or S.D.
- Calculate coefficients and compare two series.
- Determine the more consistent series using C.V.
- Apply y = a + bx to Mean, S.D. and Variance.
- Calculate combined Mean and combined S.D.
- Select the most suitable measure for a situation.
- Identify theoretical properties, merits and limitations.
High-Frequency Traps
- IQR ≠ Q.D.; Q.D. = IQR/2.
- Modulus is compulsory in Mean Deviation.
- Use the same average in the coefficient of M.D.
- Variance is not Standard Deviation.
- Multiply by h in step-deviation.
- Use variances in the combined formula.
- Lower C.V. means greater consistency.
Effect of Origin and Scale: One-View Summary
| Measure | Change of Origin | Change of Scale |
|---|---|---|
| Range | No effect | Multiplied by |b| |
| Quartile Deviation | No effect | Multiplied by |b| |
| Mean Deviation | No effect | Multiplied by |b| |
| Standard Deviation | No effect | Multiplied by |b| |
| Variance | No effect | Multiplied by b² |
Final Calculation Check: Identify the type of series → choose the correct formula → verify the denominator → check whether the answer required is absolute or relative → interpret the result in words.
13. ICAI Question Patterns and Complete Exam Coverage
Questions from this unit are commonly framed as concept MCQs, formula recognition, direct calculations, transformations, consistency comparisons and combined-group problems.
A. Identify the Measure from the Wording
| Question Wording | Use |
|---|---|
| Difference between largest and smallest value | Range |
| Middle 50%, skewed data or open-end classification | Quartile Deviation |
| Average absolute distance from Mean, Median or Mode | Mean Deviation |
| Most useful, stable or algebraically suitable measure | Standard Deviation |
| Compare consistency, stability or risk | Coefficient of Variation |
| Two groups are merged | Combined Mean and Combined S.D. |
| Every item increased or decreased by a constant | Mean changes; Range, Q.D., M.D. and S.D. do not |
| Every item multiplied or divided by a constant | Absolute dispersion changes in the same proportion; Variance by its square |
B. High-Frequency Concept Answers
Classification
- Dispersion measures scatterness or variability.
- Central tendency gives first-order averages; dispersion gives second-order averages.
- Absolute measures show the amount of variation.
- Relative measures show the degree of variation and are used for comparison.
- Range, Q.D., M.D. and S.D. are absolute measures.
- Their coefficients and C.V. are relative measures.
Choice of Measure
- Range is the easiest measure and is used in quality control.
- Q.D. is least affected by extreme observations.
- Q.D. is suitable for open-ended classifications.
- M.D. is based on absolute deviations.
- M.D. and S.D. use all observations.
- S.D. is the most useful general measure.
C. Formula Recognition
| Measure | Formula | Key Reference |
|---|---|---|
| Coefficient of Range | (L − S)/(L + S) | Sum of extremes |
| Q.D. | (Q₃ − Q₁)/2 | Half of IQR |
| Coefficient of Q.D. | (Q₃ − Q₁)/(Q₃ + Q₁) | Relative middle spread |
| M.D. | Σ|x − A|/N or Σf|x − A|/Σf | A = Mean, Median or Mode |
| Coefficient of M.D. | M.D.(A)/A | Use the same average |
| Variance | Σ(x − x̄)²/N | Squared unit |
| S.D. | √Variance | Positive square root |
| C.V. | (σ/x̄) × 100 | Arithmetic Mean |
D. Transformation Rules
If y = a + bx:
Mean(y) = a + b Mean(x)
Range(y) = |b| Range(x)
Q.D.(y) = |b| Q.D.(x)
M.D.(y) = |b| M.D.(x)
S.D.(y) = |b| S.D.(x)
Variance(y) = b² Variance(x)
Mean(y) = a + b Mean(x)
Range(y) = |b| Range(x)
Q.D.(y) = |b| Q.D.(x)
M.D.(y) = |b| M.D.(x)
S.D.(y) = |b| S.D.(x)
Variance(y) = b² Variance(x)
Origin versus Scale: Addition or subtraction changes the Mean but not dispersion. Multiplication or division changes the Mean and absolute dispersion. C.V. remains unchanged when every value is multiplied by the same positive constant.
E. Standard Results Used in MCQs
| Result | Answer |
|---|---|
| Σ(x − x̄) | 0 |
| Minimum sum of squared deviations | About Arithmetic Mean |
| Minimum sum of absolute deviations | About Median |
| S.D. when all values are equal | 0 |
| Variance of first n natural numbers | (n² − 1)/12 |
| S.D. of first n natural numbers | √[(n² − 1)/12] |
| Greater consistency | Lower C.V. |
| Greater volatility or risk | Higher dispersion |
F. Numerical Question Types Covered
Direct
Range, Q.D., M.D., Variance, S.D. and their coefficients.
Frequency Series
Individual, discrete and continuous data through direct, assumed-mean and step-deviation methods.
Comparison
Consistency and stability through C.V.
Transformation
Addition, subtraction, multiplication, division and y = a + bx.
Combined Groups
Combined Mean, Variance and Standard Deviation.
Concept MCQs
Definitions, selection of measure, extremes, units and open-end classes.
G. Final Exam Checklist
1
Identify actual spread or relative comparison.
2
Identify individual, discrete or continuous series.
3
Select the correct formula and use boundaries or mid-points where required.
4
Check for origin, scale, transformation or combined groups.
5
Interpret correctly: lower dispersion or lower C.V. means greater consistency.
Revision Mind Map
14. Measures of Dispersion — One-Page Recall
DISPERSION
ABSOLUTE MEASURES
RANGE
L − S
Only extremes
Fast · Unstable
Q.D.
(Q₃ − Q₁)/2
Middle 50%
Skewed data
M.D.
Σ|x−A|/N
Min about Median
Absolute distance
S.D.
√Variance
Best algebra
Most powerful
RELATIVE MEASURES
Coeff. of Range
(L − S)/(L + S)
Relative extreme gap
Coeff. of Q.D.
(Q₃ − Q₁)/(Q₃ + Q₁)
Relative middle spread
Coeff. of M.D.
M.D. ÷ same average
Relative absolute distance
C.V.
(σ / Mean) × 100
Lower C.V. = more consistent
Final Memory:
Absolute → actual spread (Range · Q.D. · M.D. · S.D.) |
Relative → fair comparison (Coefficients · C.V.) |
Consistency test → lower C.V. wins