The given question is a data sufficiency question in statistics and average. Concept covered. Understanding of the formula to compute standard deviation of a set of numbers if the data about sum of the numbers and sum of the squares of the numbers is known.

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- EACH statement ALONE is sufficient to answer the question asked.
- Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

All numbers used are real numbers.

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 12: What is the standard deviation (SD) of the four numbers p, q, r, and s?

- The sum of p, q, r, and s is 24.
- The sum of the squares of p, q, r, and s is 224.

**What kind of an answer will the question fetch?**

The question is an **"What is"** question. Answer to the question should be a number that is the standard deviation of p, q, r, and s.

**When is the data sufficient?**

The data is sufficient if we are able to get a UNIQUE value for the SD of the four numbers from the information given in the statements.

**How to find the standard deviation of a set of numbers?**

Standard deviation = \\sqrt {\text{Mean of squares of the numbers} - \text{square of mean of the numbers}})

From the information in statement 1 we can find the mean of the four numbers to be 6 and the square of the mean of the numbers to be 36.

We need additional information to find the SD.

This statement does not provide any information about the mean of the squares of the numbers.

**Statement 1 alone is NOT sufficient.**

Eliminate choices A and D. Choices narrow down to B, C, or E.

Hence, the mean of the squares of the numbers is 56.

However, this statement does not provide any information about the square of the mean of the numbers.

**Statement 2 alone is NOT sufficient.**

Eliminate choice B. Choices narrow down to C and E.

From statement 1 we know that the square of the means is 36.

From statement 2 we know that the mean of the squares is 56.

Using the formula,

Standard deviation = \\sqrt {\text{Mean of squares of the numbers} - \text{square of mean of the numbers}}),

we can find the SD of the 4 numbers.

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