GRE Standard Deviation Questions Practice Questions with Answers

Standard deviation measures the spread of data points around the arithmetic mean in a given set of numbers. Succeeding on the Quantitative Reasoning section requires a firm grasp of how this statistical tool behaves when you add, subtract, or multiply values within a list. While the Educational Testing Service (ETS) rarely asks you to calculate the full formula from scratch, they frequently test your ability to compare the dispersion of two different datasets. Understanding the qualitative properties of GRE Standard Deviation Questions is often more valuable than memorizing the complex square-root formula itself.

To build a solid foundation for your studies, you might explore our comprehensive GRE Prep hub, which organizes these mathematical concepts into manageable lessons. Since the GRE is an adaptive exam, the difficulty of the data analysis questions you encounter will depend on your performance in earlier sections. Utilizing tools like an AI Question Generator can help you simulate this experience by providing problems that match your current skill level.

Concept Explanation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of numerical values. A low standard deviation indicates that the data points tend to be very close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.

On the GRE, you should focus on these key properties:

• The Distance Rule: Standard deviation depends on the distance of each data point from the mean. The further the points are from the average, the higher the standard deviation.
• Constant Addition/Subtraction: If you add or subtract the same constant $k$ to every number in a set, the standard deviation remains unchanged. The entire distribution shifts, but the spacing between the numbers stays the same.
• Constant Multiplication: If you multiply every number in a set by a constant $k$, the new standard deviation is the original standard deviation multiplied by $|k|$.
• The Zero Deviation: The standard deviation is zero if and only if all numbers in the set are identical.

The formula for sample standard deviation $s$ is given by:

$$s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}$$

However, for the GRE, you usually only need to understand that the process involves squaring the differences from the mean, which ensures that the result is always non-negative.

Solved Examples

1. Example: Comparing Spread
Set A: {10, 20, 30, 40, 50}
Set B: {10, 25, 30, 35, 50}
Which set has a higher standard deviation?

1. Calculate the mean for both sets. Both sets sum to 150, so the mean for both is $\frac{150}{5} = 30$.
2. Examine the distance of individual points from 30. In Set A, the values 20 and 40 are 10 units away from the mean. In Set B, the values 25 and 35 are only 5 units away from the mean.
3. Since Set A has points that are generally further from the mean than Set B, Set A has a higher standard deviation.

2. Example: The Constant Addition Rule
The standard deviation of set $\{x, y, z\}$ is $s$. What is the standard deviation of the set $\{x+10, y+10, z+10\}$?

1. Recall that adding a constant to every element in a set does not change the spread.
2. The mean of the new set will be the old mean plus 10.
3. The distance of each new point from the new mean remains exactly the same as the original distances.
4. Therefore, the standard deviation is still $s$.

3. Example: Determining the Smallest Deviation
Which of the following sets has the smallest standard deviation?
A: {5, 5, 5, 5}
B: {1, 2, 3, 4}
C: {-10, 0, 10}

1. Check for the set where all numbers are closest to the mean.
2. In Set A, every number is identical. The mean is 5, and every deviation is $5 - 5 = 0$.
3. A standard deviation of 0 is the smallest possible value. Therefore, Set A has the smallest standard deviation.

Practice Questions

1. Set $X$ contains the elements {2, 4, 6, 8, 10}. Set $Y$ is created by multiplying each element of Set $X$ by 3 and then adding 5 to each resulting product. If the standard deviation of Set $X$ is $s$, what is the standard deviation of Set $Y$ in terms of $s$?

2. Quantity A: The standard deviation of {12, 15, 18, 21, 24}
Quantity B: The standard deviation of {112, 115, 118, 121, 124}

3. A set of 5 distinct integers has a mean of 20 and a standard deviation of $\sigma$. If the largest integer is increased by 10 and the smallest integer is decreased by 10, what happens to the standard deviation?

4. Which of the following sets of numbers has the greatest standard deviation?
A: {2, 2, 8, 8}
B: {4, 4, 6, 6}
C: {5, 5, 5, 5}
D: {2, 3, 7, 8}

5. If the standard deviation of the set {a, b, c} is 0, what must be true about the values of a, b, and c?

6. Set S consists of the values {10, 20, 30, 40, 50}. If the value 30 is removed from the set, does the standard deviation increase, decrease, or stay the same?

7. Quantity A: The standard deviation of {1, 2, 3}
Quantity B: The standard deviation of {1, 1, 5}

8. A data set has a mean of 50 and a standard deviation of 5. If every value in the set is multiplied by 0.5, what is the new standard deviation?

Answers & Explanations

1. Answer: $3s$. Multiplying each element by 3 scales the standard deviation by 3. Adding 5 to each element does not change the standard deviation. Thus, the new standard deviation is $3 \times s$.
2. Answer: The two quantities are equal. Set B is simply Set A with 100 added to each element. Since adding a constant does not change the spread, the standard deviations are identical.
3. Answer: The standard deviation increases. By moving the extremes further away from the mean, the average distance from the mean increases, which results in a larger standard deviation.
4. Answer: A. The mean for all sets is 5. In Set A, every point is 3 units away from the mean. In Set B, every point is 1 unit away. In Set C, the deviation is 0. In Set D, two points are 3 units away and two are 2 units away. Set A has the most total "distance" from the mean.
5. Answer: a = b = c. A standard deviation of 0 occurs if and only if there is no variation in the data, meaning all elements are identical.
6. Answer: Increase. The value 30 is the mean of the set. Removing the value that is exactly at the mean increases the relative weight of the values that are far from the mean, thereby increasing the standard deviation.
7. Answer: Quantity B is greater. The mean of {1, 2, 3} is 2, with distances {1, 0, 1}. The mean of {1, 1, 5} is 2.33, with much larger distances (especially the 5, which is 2.67 away from the mean). More dispersion equals higher standard deviation.
8. Answer: 2.5. Multiplying every value by a constant $k$ multiplies the standard deviation by $|k|$. Here, $5 \times 0.5 = 2.5$.

1. If the standard deviation of a set is 4 and every number in the set is increased by 10, what is the new standard deviation?

• A4
• B14
• C40
• D0

Frequently Asked Questions

Do I need to use the standard deviation formula on the GRE?

You rarely need to perform the full calculation using the formula. Most GRE questions test your conceptual understanding of how changes to a data set affect the spread or ask you to compare two sets visually.

Can standard deviation be negative?

No, standard deviation can never be negative because it is calculated by taking the square root of a sum of squared differences. The smallest possible value is zero, which occurs when all numbers in a set are identical.

How does adding a new number equal to the mean affect standard deviation?

Adding a number that is exactly equal to the mean will decrease the standard deviation. This is because you are increasing the count of numbers with zero deviation, which pulls the average deviation downward.

What is the difference between sample and population standard deviation for the GRE?

For the purposes of the GRE, the distinction between dividing by $n$ or $n-1$ is almost never tested. You should focus on the general concept of dispersion rather than the specific denominator used in the formula.

How is standard deviation related to the normal distribution?

In a normal distribution, standard deviation defines the percentage of data within certain ranges: approximately 68% of data falls within one standard deviation of the mean, and 95% falls within two. You can practice these probability concepts with our AI Exam Simulator.