Complete Magical Math: Coefficient of Variation

1 The standard deviation and mean of a data are 6.5 and 12.5 respectively. Find the coefficient of variation.

Step 1: Identify what we know

Standard Deviation (σ) = 6.5

Mean (μ) = 12.5

Step 2: Write the magic formula

CV = (σ ÷ μ) × 100%

Step 3: Plug in the numbers

CV = (6.5 ÷ 12.5) × 100%

Step 4: Perform the calculation

CV = 0.52 × 100% = 52%

2 The standard deviation and coefficient of variation of a data are 1.2 and 25.6 respectively. Find the value of mean.

Step 1: Identify what we know

Standard Deviation (σ) = 1.2

CV = 25.6%

Step 2: Rearrange the magic formula

CV = (σ ÷ μ) × 100% ⇒ μ = (σ ÷ CV) × 100

Step 3: Plug in the numbers

μ = (1.2 ÷ 25.6) × 100

Step 4: Perform the calculation

μ = 0.046875 × 100 = 4.6875

3 If the mean and coefficient of variation of a data are 15 and 48 respectively, then find the value of standard deviation.

Step 1: Identify what we know

Mean (μ) = 15

CV = 48%

Step 2: Rearrange the magic formula

CV = (σ ÷ μ) × 100% ⇒ σ = (CV × μ) ÷ 100

Step 3: Plug in the numbers

σ = (48 × 15) ÷ 100

Step 4: Perform the calculation

σ = 720 ÷ 100 = 7.2

4 If n = 5, x̄ = 6, ∑x² = 765, then calculate the coefficient of variation.

Step 1: Gather our magical ingredients

n = 5, Mean (x̄) = 6, ∑x² = 765

Step 2: Brew the standard deviation potion

σ = √[(∑x²/n) - (x̄)²]

Step 3: Mix in the numbers

σ = √[(765/5) - (6)²] = √[153 - 36] = √117 ≈ 10.8167

Step 4: Complete the CV enchantment

CV = (σ / x̄) × 100% = (10.8167 / 6) × 100% ≈ 180.28%

5 Find the coefficient of variation of 24, 26, 33, 37, 29, 31.

Step 1: Calculate the magical average (Mean)

μ = (24 + 26 + 33 + 37 + 29 + 31) / 6 = 180 / 6 = 30

Step 2: Prepare the variance ingredients

Variance = [(24-30)² + (26-30)² + (33-30)² + (37-30)² + (29-30)² + (31-30)²] / 6

= [36 + 16 + 9 + 49 + 1 + 1] / 6 = 112 / 6 ≈ 18.6667

Step 3: Extract standard deviation essence

σ = √18.6667 ≈ 4.3205

Step 4: Complete the CV potion

CV = (4.3205 / 30) × 100% ≈ 14.40%

6 The time taken (in minutes) to complete a homework by 8 students in a day are given by 38, 40, 47, 44, 46, 43, 49, 53. Find the coefficient of variation.

Step 1: Calculate the average time

μ = (38 + 40 + 47 + 44 + 46 + 43 + 49 + 53) / 8 = 360 / 8 = 45

Step 2: Prepare time variance ingredients

Variance = [(38-45)² + (40-45)² + (47-45)² + (44-45)² + (46-45)² + (43-45)² + (49-45)² + (53-45)²] / 8

= [49 + 25 + 4 + 1 + 1 + 4 + 16 + 64] / 8 = 164 / 8 = 20.5

Step 3: Extract standard deviation from time sands

σ = √20.5 ≈ 4.5277

Step 4: Complete the temporal CV potion

CV = (4.5277 / 45) × 100% ≈ 10.06%

7 The total marks scored by two students Sathya and Vidhya in 5 subjects are 460 and 480 with standard deviation 4.6 and 2.4 respectively. Who is more consistent in performance?

Step 1: Calculate means for both wizards

Sathya's Mean = 460 / 5 = 92

Vidhya's Mean = 480 / 5 = 96

Step 2: Brew their CV potions

Sathya's CV = (4.6 / 92) × 100% = 5%

Vidhya's CV = (2.4 / 96) × 100% = 2.5%

Step 3: Analyze the magical results

Since Vidhya's CV (2.5%) is less than Sathya's CV (5%),

Vidhya is more consistent! ✨

8 The mean and standard deviation of marks obtained by 40 students of a class in three subjects Mathematics, Science and Social Science are given below.

Subject	Mean	SD
Mathematics	56	12
Science	65	14
Social Science	60	10

Which of the three subjects shows more consistent and which shows less consistent in marks?

Step 1: Calculate CV for each subject

We'll use our magic formula: CV = (SD ÷ Mean) × 100%

Step 2: Mathematics

CV = (12 ÷ 56) × 100% ≈ 21.43%

Step 3: Science

CV = (14 ÷ 65) × 100% ≈ 21.54%

Step 4: Social Science

CV = (10 ÷ 60) × 100% ≈ 16.67%

Step 5: Analyze the results

- Most consistent: Social Science (lowest CV ~16.67%)

- Least consistent: Science (highest CV ~21.54%)

Magical Math Adventure

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