[GMAT math practice question]
If the average (arithmetic mean) of 100 numbers is 50, what is the standard deviation of the numbers?
1) The smallest number is 50
2) The largest number is 50
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If the average (arithmetic mean) of 100 numbers is 50, what
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Max@Math Revolution
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Terry@ThePrincetonReview
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You will never be asked to calculate an actual standard deviation on the GMAT. The test wants you to recognize what's needed in order to calculate a standard deviation. (Data Sufficiency is about knowing whether or not you have what you need to perform a calculation, not about actually performing it!)
The formula for standard deviation is: $$\sqrt{\frac{sum\left(x-mean\left(x\right)\right)^2}{n-1}}$$
So you need the difference between each member of a set and the average, and you need a count of the members of the set. That is what we're looking for when evaluating the statements.
Statement 1: If the smallest number of 100 is 50 AND the average (central value) is also 50, then that means that all 100 numbers in the set equal 50. The difference between each value in the set and the average is 0, making the standard deviation 0. Statement 1 is sufficient.
Statement 2: If the largest number in the set is 50 AND the average is also 50, then (by the same logic used for statement 1), then all 100 numbers in the set equal 50 and the standard deviation is again 0. Statement 2 is sufficient.
Statements 1 and 2 are sufficient independently, so the answer is choice D.
The logic of these statements is that if one of the limits of a set of number (greatest or least value) is equal to the mean, then all values must equal the mean. Try it with a small data set: If there are three numbers with a mean of 4 and the greatest value is 4, what does that say about the values of the other numbers? Because mean = sum/count and mean * count = sum, three numbers with a mean of 4 must add to 3 * 4 = 12. If the greatest number is 4, then the two other numbers must add to 12 - 4 = 8. Many pairs of numbers can add to 8: 1 + 7, 2 + 6, 3 + 5, 4 + 4. But only 4 + 4 doesn't use a number that exceeds the maximum value of 4.
An actual GMAT question would probably not say "smallest" and "largest". More commonly, it would say "least" and "greatest".
Terry
The formula for standard deviation is: $$\sqrt{\frac{sum\left(x-mean\left(x\right)\right)^2}{n-1}}$$
So you need the difference between each member of a set and the average, and you need a count of the members of the set. That is what we're looking for when evaluating the statements.
Statement 1: If the smallest number of 100 is 50 AND the average (central value) is also 50, then that means that all 100 numbers in the set equal 50. The difference between each value in the set and the average is 0, making the standard deviation 0. Statement 1 is sufficient.
Statement 2: If the largest number in the set is 50 AND the average is also 50, then (by the same logic used for statement 1), then all 100 numbers in the set equal 50 and the standard deviation is again 0. Statement 2 is sufficient.
Statements 1 and 2 are sufficient independently, so the answer is choice D.
The logic of these statements is that if one of the limits of a set of number (greatest or least value) is equal to the mean, then all values must equal the mean. Try it with a small data set: If there are three numbers with a mean of 4 and the greatest value is 4, what does that say about the values of the other numbers? Because mean = sum/count and mean * count = sum, three numbers with a mean of 4 must add to 3 * 4 = 12. If the greatest number is 4, then the two other numbers must add to 12 - 4 = 8. Many pairs of numbers can add to 8: 1 + 7, 2 + 6, 3 + 5, 4 + 4. But only 4 + 4 doesn't use a number that exceeds the maximum value of 4.
An actual GMAT question would probably not say "smallest" and "largest". More commonly, it would say "least" and "greatest".
Terry
Terry Serres
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Max@Math Revolution
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of VA(Variable Approach) method is modifying the original condition and the question, and rechecking the number of variables and the number of equations. We often encounter this type of question in the GMAT quant exam these days. If the mean is equal to either the maximum or the minimum, or the range ( = Max - Min ) is zero, then the standard deviation is zero.
Condition 1)
All data items are greater than or equal to 50, and their average is 50.
It means that all data items are equal to 50.
Since all of the data items are equal, their standard deviation is 0.
This is sufficient.
Condition 2)
All data items are less than or equal to 50, and their average is 50.
It means that all data items are equal to 50.
Since all of the data items are equal, their standard deviation is 0.
This is sufficient too.
Therefore, the answer is D.
Answer: D
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of VA(Variable Approach) method is modifying the original condition and the question, and rechecking the number of variables and the number of equations. We often encounter this type of question in the GMAT quant exam these days. If the mean is equal to either the maximum or the minimum, or the range ( = Max - Min ) is zero, then the standard deviation is zero.
Condition 1)
All data items are greater than or equal to 50, and their average is 50.
It means that all data items are equal to 50.
Since all of the data items are equal, their standard deviation is 0.
This is sufficient.
Condition 2)
All data items are less than or equal to 50, and their average is 50.
It means that all data items are equal to 50.
Since all of the data items are equal, their standard deviation is 0.
This is sufficient too.
Therefore, the answer is D.
Answer: D
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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