Measures of Dispersion: Range, Quartile Deviation, Mean Deviation and Standard Deviation Questions

Question 1

Define dispersion.

Accepted Answer

Dispersion refers to the extent of variation or spread of data values around a central value.

Question 2

Define range.

Accepted Answer

Range is the difference between the largest and the smallest value in the data set (R = L − S).

Question 3

What is quartile deviation (QD)?

Accepted Answer

Quartile deviation is half of the interquartile range: QD = (Q3 − Q1)/2.

Question 4

Define mean deviation.

Accepted Answer

Mean deviation is the average of absolute deviations of observations from a central value (mean/median).

Question 5

Define standard deviation.

Accepted Answer

Standard deviation is the square root of the average of squared deviations from the mean.

Question 6

What is coefficient of variation (CV)?

Accepted Answer

CV = (SD/Mean) × 100; it measures relative variation and helps compare consistency.

Question 7

Write any three uses/importance of dispersion in business.

Accepted Answer

Uses of dispersion in business include:

1) Measuring stability/consistency of sales, costs or profits.
2) Studying risk and uncertainty (higher variability = higher risk).
3) Quality control by analyzing variation in production/defects.

(Any three uses can be written.)

Question 8

Write any three merits and demerits of range.

Accepted Answer

Range merits/demerits:

Merits:
1) Very simple and quick to compute.
2) Useful for rough comparison.

Demerits:
3) Depends only on extreme values and ignores the rest.

(Any three points can be written.)

Question 9

Differentiate variance and standard deviation (any three points).

Accepted Answer

Variance and SD differ as follows:

| Point | Variance | Standard deviation |
|---|---|---|
| Definition | average of squared deviations | square root of variance |
| Unit | squared unit | original unit |
| Use | theoretical calculations | practical interpretation |

(Any three points can be written.)

Question 10

Write any three features/merits of standard deviation.

Accepted Answer

Merits of SD include:

1) Based on all observations.
2) Rigidly defined and widely used.
3) Useful for further analysis (correlation, regression, normal distribution).

(Any three merits can be written.)

Question 11

Explain quartile deviation and its coefficient (any three points).

Accepted Answer

Quartile deviation (QD) measures dispersion of the middle 50% data:

1) QD = (Q3 − Q1)/2.
2) It is less affected by extreme values.
3) Coefficient of QD = (Q3 − Q1)/(Q3 + Q1), used for comparison.

(Any three points can be written.)

Question 12

Explain coefficient of variation (CV) and its interpretation (any three points).

Accepted Answer

CV measures relative dispersion:

1) CV = (SD/Mean) × 100.
2) It allows comparison of variability between series with different units/means.
3) Lower CV indicates higher consistency and stability.

(Any three points can be written.)

Question 13

Explain range and quartile deviation with formulas and a short example.

Accepted Answer

Dispersion measures show how spread out the data values are.

**Range** is the simplest measure of dispersion. It is the difference between the largest value (L) and the smallest value (S):

R = L − S

Coefficient of range = (L − S)/(L + S)

**Quartile deviation (QD)** is based on quartiles and measures spread of the middle 50% of data:

QD = (Q3 − Q1)/2

Coefficient of QD = (Q3 − Q1)/(Q3 + Q1)

**Example:** Data: 5, 8, 10, 12, 15
Range = 15 − 5 = 10
Here Q1 = 8 and Q3 = 12 (middle of halves), so QD = (12 − 8)/2 = 2

Thus, range is quick but depends only on extremes, whereas QD is more stable because it focuses on the middle part of data.

Question 14

Explain standard deviation and coefficient of variation with formulas and interpretation.

Accepted Answer

Standard deviation (SD) is the most commonly used measure of dispersion. It is the square root of the average of squared deviations from the mean.

For individual series:

σ = √[ Σ(x − x̄)² / n ]

Variance is σ² = Σ(x − x̄)² / n

Because SD is in the same units as the original data, it is easy to interpret: higher SD means values are more spread out; lower SD means more consistency.

Coefficient of variation (CV) is used to compare variability between different series:

CV = (σ / x̄) × 100

Interpretation:
- Lower CV ⇒ more stability/consistency
- Higher CV ⇒ more relative variation/risk

Thus, SD measures absolute dispersion and CV measures relative dispersion, making them very useful for comparing business performance across products or time periods.

Question 15

Write a detailed note comparing measures of dispersion and selecting the best measure. Use a table.

Accepted Answer

Different measures of dispersion have different uses. The choice depends on the nature of data and purpose.

| Measure | Uses | Limitation |
|---|---|---|
| Range | quick rough comparison | depends only on extremes |
| QD | useful in skewed data; middle 50% spread | ignores other 50% data |
| MD | average of absolute deviations; more stable than range | not convenient for algebraic work |
| SD | best overall; used in many methods | calculation is longer; affected by extremes |
| CV | compares consistency across series | depends on mean and SD accuracy |

In practice, **standard deviation** is considered the best measure because it uses all observations and is widely applied in statistical analysis. If data sets have different units or different means, **CV** should be used for comparison.

Therefore, selection of dispersion measure should be based on data distribution, need for further analysis and comparison requirements.

Measures of Dispersion: Range, Quartile Deviation, Mean Deviation and Standard Deviation

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