GRE Statistics Practice Questions Practice Questions with Answers

Concept Explanation

GRE statistics involves the analysis, interpretation, and presentation of data sets using measures of central tendency, dispersion, and probability distributions. To succeed on the Quantitative Reasoning section, you must understand how to calculate the arithmetic mean, median, mode, and range, while also interpreting more complex concepts like standard deviation and the properties of the normal distribution. According to Educational Testing Service (ETS), data analysis accounts for a significant portion of the exam, requiring students to move beyond basic arithmetic into data interpretation. For those building a comprehensive study plan, utilizing a GRE Prep hub can help organize these mathematical concepts into a structured timeline. Statistics on the GRE often tests your ability to determine how adding or removing a data point affects the mean and standard deviation, rather than just asking for simple calculations.

Solved Examples

1. Problem: A set of five integers has a mean of 12, a median of 10, and a mode of 8. What is the maximum possible value for the largest integer in the set?
   1. Determine the sum of the integers: Since the mean of 5 integers is 12, the sum is $5 \ \times 12 = 60$.
   2. Use the mode: The mode is 8, so at least two integers must be 8. Since the median is 10, the set in increasing order looks like: $8, 8, 10, x, y$.
   3. Minimize the unknown values: To maximize $y$, we must minimize $x$. Since the median is 10 and the integers are in increasing order, the smallest possible value for $x$ is 10.
   4. Solve for the maximum: $8 + 8 + 10 + 10 + y = 60$. Simplified: $36 + y = 60$, so $y = 24$. The maximum possible value is 24.
2. Problem: If the standard deviation of the set $\{x, y, z\}$ is $s$, what is the standard deviation of the set $\{x+5, y+5, z+5\}$?
   1. Understand the property: Standard deviation measures the spread of data points from the mean.
   2. Analyze the shift: Adding a constant to every term in a set shifts the mean by that constant but does not change the distance between the points.
   3. Conclusion: Because the relative spacing remains identical, the standard deviation remains exactly $s$.
3. Problem: In a normal distribution, approximately what percentage of the data falls within two standard deviations of the mean?
   1. Recall the Empirical Rule: The 68-95-99.7 rule describes data distribution in a bell curve.
   2. Apply the rule: Approximately 68% of data falls within 1 standard deviation, and 95% falls within 2 standard deviations.
   3. Answer: 95%.

Practice Questions

1. A list of 7 numbers has a mean of 20. If a number $x$ is added to the list, the new mean is 21. What is the value of $x$?

2. Set A consists of the elements $\{2, 4, 6, 8, 10\}$. Set B consists of the elements $\{102, 104, 106, 108, 110\}$. Which set has a greater standard deviation?

3. The weights of a population of 2,000 birds are normally distributed with a mean of 1.5 kg and a standard deviation of 0.2 kg. Approximately how many birds weigh more than 1.9 kg?

4. In a set of 15 different integers, the median is 30. If the largest 7 integers are each increased by 10, what happens to the median of the set?

5. A data set consists of the values $\{10, 15, 20, 25, 30\}$. If a new value, 20, is added to the set, will the standard deviation increase, decrease, or stay the same?

6. The average (arithmetic mean) of five numbers is 10. If the sum of three of the numbers is 24, what is the average of the other two numbers?

7. A set of numbers has a range of 14 and a smallest value of -5. If every number in the set is multiplied by -2, what is the range of the new set?

8. For a set of 100 values, the 75th percentile is 82. If 10 more values, all greater than 90, are added to the set, how does the 75th percentile change?

Answers & Explanations

1. 28. The original sum is $7 \ \times 20 = 140$. The new sum for 8 numbers is $8 \ \times 21 = 168$. The value added is $168 - 140 = 28$.

2. They are equal. Standard deviation measures spread. Set B is simply Set A with 100 added to each element. Shifting a set does not change its spread or standard deviation.

3. 50 birds. A weight of 1.9 kg is 2 standard deviations above the mean $(1.5 + 2 \ \times 0.2 = 1.9)$. In a normal distribution, 95% of data is within 2 standard deviations, meaning 5% is outside that range (2.5% in each tail). 2.5% of 2,000 is 50.

4. It remains 30. The median is the middle value (the 8th value in a set of 15). Increasing the values above the median does not change the position or the value of the 8th term.

5. Decrease. The mean of the original set is 20. Adding a value equal to the mean reduces the average squared distance from the mean, thereby decreasing the standard deviation. You can verify this using the AI Question Generator for more variance-related drills.

6. 13. The total sum of the five numbers is $5 \ \times 10 = 50$. The sum of the remaining two numbers is $50 - 24 = 26$. The average of these two is $\ \frac{26}{2} = 13$.

7. 28. Range is the difference between the maximum and minimum. Range is always non-negative. If you multiply every number by $k$, the new range is $|k| \ \times \ \text{old range}$. Here, $|-2| \ \times 14 = 28$.

8. It increases. The 75th percentile is the value below which 75% of the data falls. Adding values to the upper end of the distribution shifts the rank of existing values, pushing the 75th percentile threshold higher. For more high-level practice, check out the general pathology questions which, while medical, use similar statistical logic for lab values.

1. If the mean of a set of 6 numbers is 35 and one number is removed, the new mean is 32. What number was removed?

• A45
• B50
• C38
• D53

Frequently Asked Questions

What is the difference between mean and median on the GRE?

The mean is the average calculated by dividing the sum of all values by the count, while the median is the middle value when the data is ordered. The mean is more sensitive to outliers, whereas the median provides a better measure of center for skewed distributions.

How does standard deviation appear on the GRE?

The GRE rarely requires you to calculate standard deviation using the complex formula involving square roots. Instead, it tests your conceptual understanding of how data spread affects the value and how transformations (like adding or multiplying constants) change it.

What is the 68-95-99.7 rule?

This rule, also known as the Empirical Rule, states that in a normal distribution, 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. You will need this to solve probability questions involving bell curves.

Can the standard deviation ever be negative?

No, standard deviation is always zero or positive because it is calculated using the square root of squared differences. A standard deviation of zero indicates that all values in the data set are exactly the same.

Does the GRE test interquartile range (IQR)?

Yes, the GRE occasionally tests IQR, which is the difference between the 75th percentile (Q3) and the 25th percentile (Q1). It represents the range of the middle 50% of the data and is a robust measure of dispersion against outliers.