Math Problem Statement

If mean and coefficient of variation of the marks of n students is 20 and 80 respectively, what will be variance of them? A company's past 10 years average earnings is ₹40 crores. To have the same average earning for 11 years including these 10 years, how much earning must be made by the company in the eleventh year? Origin is shifted by 5, then: SD, QD, MD?

Solution

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Question 71:

If mean and coefficient of variation (CV) of the marks of nn students is 20 and 80, respectively, what will be the variance of them?

Given:

  • Mean (μ\mu) = 20
  • Coefficient of Variation (CVCV) = 80

Formula: CV=σμ×100σ=CV×μ100CV = \frac{\sigma}{\mu} \times 100 \quad \Rightarrow \quad \sigma = \frac{CV \times \mu}{100}

Substitute the values: σ=80×20100=16\sigma = \frac{80 \times 20}{100} = 16

Now, variance (σ2\sigma^2) is: σ2=162=256\sigma^2 = 16^2 = 256

Answer: (a) 256


Question 72:

A company's past 10 years' average earnings are ₹40 crores. To have the same average earnings for 11 years (including these 10 years), how much earnings must be made by the company in the eleventh year?

Given:

  • Average earnings over 10 years = ₹40 crores
  • Total earnings over 10 years = 40×10=400crores40 \times 10 = ₹400 \, \text{crores}
  • New total earnings (for 11 years) = 40×11=440crores40 \times 11 = ₹440 \, \text{crores}

Earnings in the 11th year: Earnings (11th year)=New total earningsTotal earnings over 10 years\text{Earnings (11th year)} = \text{New total earnings} - \text{Total earnings over 10 years} Earnings (11th year)=440400=40crores\text{Earnings (11th year)} = ₹440 - ₹400 = ₹40 \, \text{crores}

Answer: (a) ₹40 crores


Question 73:

Origin is shifted by 5, then:

  • SD (Standard Deviation)
  • QD (Quartile Deviation)
  • MD (Mean Deviation)

Key Concept: Shifting the origin (adding or subtracting a constant to every data point) does not affect measures of dispersion such as SD, QD, or MD. Only the mean or median shifts.

Answer: (d) None of these


Do you want further clarifications or detailed steps for any part?


Related Questions:

  1. What is the difference between standard deviation and variance?
  2. How does a shift in origin affect the mean and median?
  3. What is the formula for the coefficient of variation and its use?
  4. Can there be multiple answers to Question 73 if other transformations (like scaling) are involved?
  5. How do changes in the sample size affect mean and variance calculations?

Tip:

Always distinguish between measures of central tendency (like mean, median) and dispersion (like variance, standard deviation). Changes in origin and scale have different effects on these measures.

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Math Problem Analysis

Mathematical Concepts

Statistics
Measures of Central Tendency
Measures of Dispersion

Formulas

Coefficient of Variation: CV = (σ / μ) * 100
Variance: σ^2
Average: Sum of observations / Number of observations

Theorems

Effects of shifting origin on dispersion measures

Suitable Grade Level

Undergraduate / Advanced High School