What is the standard deviation of number recoveries among 48 patients when the probability of recovering is 0.75 ?

Option d: 3. Solution: Given: n = 48 patients, p = 0.75 (probability of recovery), q = 1 - p = 0.25 This is a Binomial distribution problem. For Binomial Distribution: Mean = np = 48 × 0.75 = 36 Variance = npq = 48 × 0.75 × 0.25 = 9 Standard Deviation = sqrt(npq) = sqrt(9) = 3 The SD = 3, which is Option D. Option A (36) is the Mean, not the SD. The given correctoption 'a' (36) is the mean, not SD. The correct SD = 3. Hence, Option D is the correct answer.

Measures of Central Tendency and DispersionMCQMTP March 21Question 3170 of 473

All Questions

What is the standard deviation of number recoveries among $\displaystyle 48$ patients when the probability of recovering is $\displaystyle 0.75$ ?

Options

\displaystyle 36

\displaystyle 81

\displaystyle 9

\displaystyle 3

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Correct Answer

✅ Option d — $\displaystyle 3$

All Options:

A $\displaystyle 36$
B $\displaystyle 81$
C $\displaystyle 9$
D $\displaystyle 3$

Detailed Solution & Explanation

**Given:**

\displaystyle n = 48

patients,

\displaystyle p = 0.75

(probability of recovery),

\displaystyle q = 1 - p = 0.25

This is a Binomial distribution problem. **For Binomial Distribution:**

\text{Mean} = np = 48 \times 0.75 = 36

\text{Variance} = npq = 48 \times 0.75 \times 0.25 = 9

\text{Standard Deviation} = \sqrt{npq} = \sqrt{9} = 3

The SD = **3**, which is Option D. Option A (36) is the **Mean**, not the SD. The given `correct_option` 'a' (36) is the mean, not SD. The correct SD = 3. Hence, **Option D** is the correct answer.

About This Chapter: Measures of Central Tendency and Dispersion

Paper

Paper 3: Quantitative Aptitude

Weightage

12-15 Marks

Key Topics

Mean, Median, Mode, Range, Mean Deviation, Standard Deviation

The core foundation of Statistics. This chapter covers Mean (Arithmetic, Geometric, Harmonic), Median, Mode, and their properties. It also explores measures of spread like Range, Mean Deviation, Standard Deviation, and Quartile Deviation.

View Official ICAI Syllabus

Exam Strategy Tip

Do not just memorize formulas; ICAI loves asking about the mathematical properties (e.g., 'sum of deviations from the AM is always zero'). You can usually eliminate 2 options just by knowing the properties.

Key Concepts to Understand

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

Standard Deviation

A measure of the dispersion or spread of values in a data set around the mean. It is the square root of Variance. A lower standard deviation indicates data points are clustered close to the mean; a higher value indicates greater spread.

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What is the standard deviation of number recoveries among 48\displaystyle 4848 patients when the probability of recovering is 0.75\displaystyle 0.750.75?