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Coefficient of Variation

Introduction

The Coefficient of Variation (CV) is a key statistical measure that compares the degree of variation between two or more datasets, regardless of their units or scales. It is especially useful for determining which dataset is more consistent or stable.

This pattern is important because it allows comparison of variability even when the means are different - something that raw standard deviations cannot do directly.

Pattern: Coefficient of Variation

Pattern

The key concept: CV measures the relative spread of data as a percentage of the mean - smaller CV means more consistency.

Formula:
CV = (Standard Deviation ÷ Mean) × 100

Interpretation:
• Lower CV → more consistent data
• Higher CV → more variable data

Step-by-Step Example

Question

The average marks of two students in a test are as follows: Student A → Mean = 60, SD = 6; Student B → Mean = 80, SD = 10. Find whose performance is more consistent using Coefficient of Variation.

Solution

  1. Step 1: Identify the given data.

    Student A → Mean = 60, SD = 6.
    Student B → Mean = 80, SD = 10.

  2. Step 2: Apply the formula for CV.

    CV = (Standard Deviation ÷ Mean) × 100

  3. Step 3: Compute CV for each student.

    CV (A) = (6 ÷ 60) × 100 = 10%
    CV (B) = (10 ÷ 80) × 100 = 12.5%

  4. Step 4: Compare the CV values.

    Lower CV → Higher consistency. Here, 10% (A) is less than 12.5% (B).

  5. Final Answer:

    Student A is more consistent as his CV is smaller.

  6. Quick Check:

    Even though B’s mean is higher, his variability is proportionally greater → CV confirms A’s steadier performance ✅

Quick Variations

1. Comparing performance of two machines, factories, or investments.

2. Use CV when means differ but you want to check stability.

3. For grouped data, first find SD and Mean before applying the CV formula.

Trick to Always Use

  • Step 1: Always compute both Mean and SD before finding CV.
  • Step 2: The smaller the CV, the more consistent the data - always check the direction of comparison carefully.
  • Step 3: Use CV only for ratio-scale data (e.g., marks, profits, speeds).

Summary

Summary

In the Coefficient of Variation (CV) pattern:

  • Formula: CV = (Standard Deviation ÷ Mean) × 100
  • Lower CV → higher consistency, higher CV → more variability.
  • Used widely to compare performance, returns, or reliability across different scales.
  • Always calculate both mean and SD accurately before applying the CV formula.
  • CV is useful when comparing datasets that have different units or magnitudes.

Practice

(1/5)
1. Student A has Mean = 60 and SD = 6. Student B has Mean = 80 and SD = 10. Who is more consistent?
easy
A. Student A
B. Student B
C. Both equally consistent
D. Cannot decide

Solution

  1. Step 1: Identify data

    Student A → Mean = 60, SD = 6. Student B → Mean = 80, SD = 10.

  2. Step 2: Compute CV = (SD ÷ Mean) × 100

    CV(A) = (6 ÷ 60) × 100 = 10%
    CV(B) = (10 ÷ 80) × 100 = 12.5%

  3. Step 3: Compare CVs

    Smaller CV indicates greater consistency. 10% < 12.5% → Student A is more consistent.

  4. Final Answer:

    Student A → Option A.

  5. Quick Check:

    Though B has higher mean, B's variability is proportionally larger (12.5%) so A is steadier ✅

Hint: Compute CV for each set and pick the smaller percentage.
Common Mistakes: Comparing SDs directly without accounting for different means.
2. Company X: Mean return = 200, SD = 20. Company Y: Mean return = 150, SD = 15. Which company has the lower CV?
easy
A. Company X
B. Company Y
C. Both have equal CV
D. Cannot decide

Solution

  1. Step 1: Identify values

    X → Mean 200, SD 20. Y → Mean 150, SD 15.

  2. Step 2: Compute CVs

    CV(X) = (20 ÷ 200) × 100 = 10%
    CV(Y) = (15 ÷ 150) × 100 = 10%

  3. Step 3: Compare

    Both CVs are equal (10%), so both companies have the same relative variability.

  4. Final Answer:

    Both have equal CV → Option C.

  5. Quick Check:

    SD is 10% of mean in both cases, so consistency is identical ✅

Hint: If SD is the same fraction of mean, CVs are equal.
Common Mistakes: Assuming higher mean automatically means lower CV.
3. A dataset has Mean = 40 and SD = 4. What is the Coefficient of Variation (CV)?
easy
A. 8%
B. 10%
C. 12%
D. 15%

Solution

  1. Step 1: Identify values

    Mean = 40, SD = 4.

  2. Step 2: Apply CV formula

    CV = (SD ÷ Mean) × 100 = (4 ÷ 40) × 100 = 10%

  3. Final Answer:

    10% → Option B.

  4. Quick Check:

    SD is one-tenth of mean → CV 10% ✅

Hint: CV = (SD/Mean)×100 - if SD = 1/10 of mean, CV = 10%.
Common Mistakes: Forgetting to multiply by 100 to get percentage.
4. Dataset A: Mean = 30, SD = 6. Dataset B: Mean = 20, SD = 3. Which dataset is relatively more consistent?
medium
A. Dataset A
B. Dataset B
C. Both same
D. Dataset B is more consistent

Solution

  1. Step 1: Identify values

    A → Mean 30, SD 6. B → Mean 20, SD 3.

  2. Step 2: Compute CVs

    CV(A) = (6 ÷ 30) × 100 = 20%
    CV(B) = (3 ÷ 20) × 100 = 15%

  3. Step 3: Compare

    Lower CV = more consistency. 15% < 20% → Dataset B is more consistent.

  4. Final Answer:

    Dataset B is more consistent → Option D.

  5. Quick Check:

    Though A has higher mean and higher SD, B's relative variability is smaller ✅

Hint: Always convert to percentage and compare - smaller percentage wins.
Common Mistakes: Comparing absolute SDs instead of relative CVs.
5. Which of the following operations leaves the Coefficient of Variation (CV) unchanged for a dataset?
medium
A. Multiplying every value by a constant k
B. Adding a constant c to every value
C. Both multiplying and adding
D. Neither

Solution

  1. Step 1: Recall effect of transformations

    If values x become y = a + b×x, Mean scales as a + b×Mean, SD scales as |b|×SD.

  2. Step 2: Consider CV after multiplying by k (a = 0, b = k)

    New CV = (|k|×SD) ÷ (k×Mean) × 100 = (SD ÷ Mean) × 100 → unchanged.

  3. Step 3: Consider adding constant c (b = 1)

    Adding shifts mean but SD stays same, so CV changes.

  4. Final Answer:

    Multiplying every value by a constant k leaves CV unchanged → Option A.

  5. Quick Check:

    Multiplication scales both numerator and denominator by same factor, cancelling out ✅

Hint: CV unaffected by scaling (multiplication), but changed by shifting (addition).
Common Mistakes: Thinking addition keeps CV same - it does not.

Mock Test

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