The concept of standard deviation is often used in data sufficiency questions on the GMAT. You will not need to calculate standard deviation, but you must understand what it is and how it is used.

Below are several examples of how to answer data sufficiency questions that involve standard deviation.

##### Example

Set W consists of 6 values. What is the standard deviation of set W?

(1) The mean of the set is 10.
(2) The range of the set is 10.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

### Section

The first statement gives the mean of the set as 10. But this does NOT define how far apart the values are. Therefore the first statement is NOT sufficient by itself.

The second statement gives the range of the set as 10. The range gives the distance between the greatest and least values. But it does not give the distribution within the range. If all the values are near the mean, then the standard deviation will be less than if all the values are far from the mean. Therefore the second statement by itself is NOT sufficient.

Combining the statements, the same reasoning applies. Though the distance between the greatest and least values is known, the distance between each value and the mean is still missing. Therefore the statements combined are NOT sufficient to answer the question.

The correct answer is choice (E).

##### Example

Set V consists of 2 distinct positive values. What is the standard deviation of set V?

(1) The mean of the set is 10.
(2) The range of the set is 10.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

### Section

Notice that the set has only two elements. Use the Plug-In strategy. Let the elements be x and y, with 0 < x < y.

The first statement gives the mean of the set as 10. The mean of the two numbers will be (x + y)/2. But this information does NOT define how far apart x and y are. For example, they can be x = 10, y = 10 with distance 0, or x = 0, y = 20 with distance 20. Therefore the first statement by itself is NOT sufficient.

The second statement gives the range as 10, so yx = 10. The elements in the set are {x, x + 10}. The mean will be x + 5. The distance between the two elements will be same, 10, for any value of x. Using the mean and the distance, you can plug in any value for x and calculate the standard deviation. Therefore the second statement is sufficient by itself to answer the question.

The correct answer is choice (B).

Note the question does not ask for the actual value of the standard deviation, just whether it can be found and is unique. For reference, here are sample values and calculations:
Let x = 0. Then the set is {0, 10} and the mean is 5.

standard deviation =√(0 – 5)2 + (10 – 5)2/2=√25 + 25/2= √25 = 5

##### Example

George scored 85% on a test. Did George do better than 90% of his classmates on the test?

(1) The mean score on the test was 80%.
(2) The standard deviation was 2.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

### Section

The first statement gives the mean. Having the mean is not enough to compare data values. The comparison needs the standard deviation. Therefore the first statement is NOT sufficient by itself.
Note: Be careful to not assume that the standard deviation is 5 (85 – 80 = 5).

The second statement gives the standard deviation. Having the standard deviation without the mean is not enough. Therefore the second statement is NOT sufficient by itself.

Combining statements (1) and (2), the score of 85% is 2.5 standard deviations above the mean. This means that George scored higher than more than 95% of the other students.

The correct answer is choice (C).

##### Example

In a normal distribution, an interval contains 81.5% of the values in a set. What are the lower and upper endpoints of the interval?

(1) The standard deviation of the set is 2.
(2) The mean of the set is 10.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

### Section

Statement (1) by itself is not sufficient because it does not allow us to determine the start or end of the interval. Statement (2) by itself is not sufficient for the same reason.

Notice that the interval of 81.5% = 34% + 34% + 13.5%. Combining this information with both statements seems to imply the interval is one deviation on both sides of the mean plus a second deviation on one side. The interval values could be between -2 and +1 standard deviations from the mean, so with a mean of 10 the interval would be 6 ≤ x ≤ 12.

But the interval could also be between -1 and +2 standard deviations from the mean, so the interval would be 8 ≤ x ≤ 14.

The mean and standard deviation are not enough to define the interval.

The correct answer is choice (E).

Note: In addition, an interval does not need to start or end on a standard deviation. The 81.5% interval could start at about -1.8 standard deviations and end at about 1.1 standard deviations.