If the two Quartiles $N(\mu, \sigma^2)$ are $14.6$ and $25.4$ respectively. What is the standard deviation of the distribution?

The correct answer is Option d: $8$. **Finding Standard Deviation from Quartiles**\n\nFor a normal distribution $N(\\mu, \\sigma^2)$, the first quartile ($Q_1$) and the third quartile ($Q_3$) are symmetric about the mean $\\mu$ and are given by:\n$$Q_1 = \\mu - 0.675\\sigma$$\n$$Q_3 = \\mu + 0.675\\sigma$$\n\nThe **Quartile Deviation (QD)** is defined as:\n$$\\text{QD} = \\frac{Q_3 - Q_1}{2}$$\n\nWe also know the relationship between QD and Standard Deviation ($\\sigma$):\n$$\\text{QD} = 0.675\\sigma$$\n\n**Given:**\n- Lower Quartile $Q_1 = 14.6$\n- Upper Quartile $Q_3 = 25.4$\n\n**Step 1: Calculate the Quartile Deviation (QD)**\n$$\\text{QD} = \\frac{25.4 - 14.6}{2}$$\n$$\\text{QD} = \\frac{10.8}{2} = 5.4$$\n\n**Step 2: Calculate the Standard Deviation ($\\sigma$)**\nUsing $\\text{QD} = 0.675\\sigma$:\n$$5.4 = 0.675\\sigma$$\n$$\\sigma = \\frac{5.4}{0.675}$$\n$$\\sigma = \\frac{5.4}{\\frac{27}{40}} = 5.4 \\times \\frac{40}{27} = 0.2 \\times 40 = 8$$\n\nThus, the standard deviation is $8$.\n\nHence, **Option D** is the correct answer.

Theoretical DistributionsMTP Dec 2023 Series IQuestion 3996 of 230

All Questions

If the two Quartiles $\displaystyle N(\mu, \sigma^2)$ are $\displaystyle 14.6$ and $\displaystyle 25.4$ respectively. What is the standard deviation of the distribution?

Options

\displaystyle 9

\displaystyle 6

\displaystyle 10

\displaystyle 8

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Correct Answer

✅ Option d — $\displaystyle 8$

All Options:

A $\displaystyle 9$
B $\displaystyle 6$
C $\displaystyle 10$
D $\displaystyle 8$

Detailed Solution & Explanation

**Finding Standard Deviation from Quartiles**\n\nFor a normal distribution

\displaystyle N(\\mu, \\sigma^2)

, the first quartile (

\displaystyle Q_1

) and the third quartile (

\displaystyle Q_3

) are symmetric about the mean

\displaystyle \\mu

and are given by:\n

Q_1 = \\mu - 0.675\\sigma

Q_3 = \\mu + 0.675\\sigma

\n\nThe **Quartile Deviation (QD)** is defined as:\n

\\text{QD} = \\frac{Q_3 - Q_1}{2}

\n\nWe also know the relationship between QD and Standard Deviation (

\displaystyle \\sigma

):\n

\\text{QD} = 0.675\\sigma

\n\n**Given:**\n- Lower Quartile

\displaystyle Q_1 = 14.6

\n- Upper Quartile

\displaystyle Q_3 = 25.4

\n\n**Step 1: Calculate the Quartile Deviation (QD)**\n

\\text{QD} = \\frac{25.4 - 14.6}{2}

\\text{QD} = \\frac{10.8}{2} = 5.4

\n\n**Step 2: Calculate the Standard Deviation (

\displaystyle \\sigma

)**\nUsing

\displaystyle \\text{QD} = 0.675\\sigma

:\n

5.4 = 0.675\\sigma

\\sigma = \\frac{5.4}{0.675}

\\sigma = \\frac{5.4}{\\frac{27}{40}} = 5.4 \\times \\frac{40}{27} = 0.2 \\times 40 = 8

\n\nThus, the standard deviation is

\displaystyle 8

.\n\nHence, **Option D** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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If the two Quartiles N(μ,σ2)\displaystyle N(\mu, \sigma^2)N(μ,σ2) are 14.6\displaystyle 14.614.6 and 25.4\displaystyle 25.425.4 respectively. What is the standard deviation of the distribution?