GRE Statistics Exam Questions Practice Questions with Answers

Statistics accounts for approximately 15% to 25% of the GRE Quantitative Reasoning section, requiring a firm grasp of data interpretation and descriptive measures. Preparing for GRE Statistics Exam Questions involves more than just memorizing formulas; you must understand how to analyze distributions, calculate probabilities, and interpret standard deviation within the context of varied data sets. This guide provides the conceptual foundation and rigorous practice necessary to excel in this high-weight area of the exam.

1. Concept Explanation

GRE Statistics Exam Questions focus on the collection, analysis, interpretation, and presentation of data through measures of central tendency, dispersion, and frequency distributions. At its core, the GRE tests your ability to summarize data using the arithmetic mean, median, mode, and range. Beyond these basics, you must understand weighted averages, quartiles, percentiles, and the properties of the normal distribution. Standard deviation is a particularly crucial concept, representing how much the data points in a set vary from the mean. A higher standard deviation indicates a wider spread, while a lower one suggests data points are clustered closely around the average. For a deeper dive into overall test structures, you can explore our GRE Prep hub. Additionally, the exam frequently uses graphical representations like boxplots, histograms, and scatterplots to test your ability to extract numerical values from visual data.

2. Solved Examples

Review these worked examples to understand the logical steps required for common statistics problems.

1. Example 1: Weighted Average
   A student scores 80 on two tests and 95 on three tests. What is the average score for all five tests?
   
   1. Calculate the sum of the first two scores: $2 \times 80 = 160$.
   2. Calculate the sum of the next three scores: $3 \times 95 = 285$.
   3. Add the sums together: $160 + 285 = 445$.
   4. Divide by the total number of tests: $\frac{445}{5} = 89$.
   5. The average score is 89.
2. Example 2: Standard Deviation Logic
   Set A is $\{10, 20, 30\}$ and Set B is $\{100, 110, 120\}$. Which set has a higher standard deviation?
   
   1. Find the mean of Set A: $\frac{10+20+30}{3} = 20$. The distances from the mean are $10, 0, 10$.
   2. Find the mean of Set B: $\frac{100+110+120}{3} = 110$. The distances from the mean are $10, 0, 10$.
   3. Since the distances from the mean are identical for both sets, the standard deviations are equal.
3. Example 3: Median of a Frequency Distribution
   In a group of 10 students, 3 students are 15 years old, 4 students are 16 years old, and 3 students are 17 years old. Find the median age.
   
   1. List the ages in order: 15, 15, 15, 16, 16, 16, 16, 17, 17, 17.
   2. Identify the middle positions for an even number of data points ($n=10$): the 5th and 6th values.
   3. The 5th value is 16 and the 6th value is 16.
   4. The average of 16 and 16 is 16. The median age is 16.

3. Practice Questions

Test your knowledge with these GRE Statistics Exam Questions ranging from basic arithmetic to complex data analysis.

1. The mean of a set of 7 numbers is 12. If one number is removed, the mean of the remaining 6 numbers is 11. What was the value of the number that was removed?
2. Set X consists of the integers $\{2, 4, 6, 8, 10\}$. If a new integer $k$ is added to the set, the mean increases by 1. What is the value of $k$?
3. A distribution of scores has a mean of 70 and a standard deviation of 5. If every score in the distribution is increased by 10, what is the new mean and the new standard deviation?

4. In a set of 25 measurements, the 20th percentile is 40. How many measurements are less than or equal to 40?
5. The range of a set of 5 positive integers is 15. If the smallest integer is 4, what is the largest possible value for the mean of the set?
6. A normal distribution has a mean of 50 and a standard deviation of 8. Approximately what percent of the data falls between 42 and 58?
7. Quantity A: The standard deviation of $\{5, 5, 5, 5\}$.
   Quantity B: The standard deviation of $\{4, 5, 5, 6\}$.
8. If the average of $x, y, \text{ and } z$ is 12, and the average of $x \text{ and } y$ is 9, what is the value of $z$?
9. A set contains the values $\{12, 15, 18, 21, x\}$. If the median of the set is 18, what is the minimum possible value for $x$?
10. In a group of 100 people, the 75th percentile for height is 180 cm. If 5 more people, all taller than 190 cm, are added to the group, what happens to the 75th percentile?

4. Answers & Explanations

1. Answer: 18. The sum of the original 7 numbers is $7 \times 12 = 84$. The sum of the remaining 6 numbers is $6 \times 11 = 66$. The removed number is $84 - 66 = 18$.
2. Answer: 12. The original mean of $\{2, 4, 6, 8, 10\}$ is 6. The sum is 30. With $k$ added, the new mean is $6 + 1 = 7$. The new sum for 6 numbers must be $6 \times 7 = 42$. Thus, $30 + k = 42$, so $k = 12$.
3. Answer: Mean = 80, SD = 5. Adding a constant to every term in a set increases the mean by that constant but does not change the standard deviation, as the relative spacing between numbers remains the same.
4. Answer: 5. The 20th percentile means 20% of the data points are at or below that value. $0.20 \times 25 = 5$.
5. Answer: 16. Range is $Max - Min$. $Max - 4 = 15$, so $Max = 19$. To maximize the mean, make the other three integers as large as possible (19). The set is $\{4, 19, 19, 19, 19\}$. Mean = $\frac{4 + 19 \times 4}{5} = \frac{80}{5} = 16$.
6. Answer: 68%. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Since $50 - 8 = 42$ and $50 + 8 = 58$, the range 42-58 represents $\pm 1$ standard deviation.
7. Answer: Quantity B is greater. The standard deviation of a set where all numbers are identical (Quantity A) is 0. Since Quantity B has variation, its standard deviation must be greater than 0.
8. Answer: 18. Sum of $x, y, z = 3 \times 12 = 36$. Sum of $x, y = 2 \times 9 = 18$. Therefore, $z = 36 - 18 = 18$.
9. Answer: 18. For 18 to be the median of 5 numbers, at least two numbers must be less than or equal to 18 and at least two must be greater than or equal to 18. The current set is $\{12, 15, 18, 21, x\}$. If $x < 18$, the median shifts. If $x = 18$, the median remains 18. Thus, 18 is the minimum.
10. Answer: It increases or stays the same. Adding values at the top end of a distribution shifts the rank of existing values. The new 75th percentile will now be the value at the 79th or 80th position of the original sorted list, which must be greater than or equal to the original 75th percentile.

1. If a set of data has a standard deviation of 0, which of the following must be true?

• AThe mean is 0
• BThe median is 0
• CAll values in the set are identical
• DThe range is equal to the mean

Frequently Asked Questions

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

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

How does standard deviation change if I multiply every number in a set by 2?

If you multiply every value in a set by a constant $k$, the standard deviation is also multiplied by the absolute value of $k$. In this case, the standard deviation would double. This differs from adding a constant, which does not change the standard deviation at all.

What is a "weighted average" and when should I use it?

A weighted average is used when different data points contribute differently to the final average, often seen in grade calculations or mixture problems. You calculate it by multiplying each value by its corresponding weight (frequency or percentage), summing those products, and dividing by the total weight. For similar quantitative logic, you might find our USMLE Physiology practice questions helpful in seeing how variables interact.

Can the standard deviation ever be a negative number?

No, the standard deviation can never be negative because it is calculated as the square root of the variance, which is a sum of squared differences. The minimum possible value for standard deviation is zero, occurring only when all data points are identical.

How do quartiles relate to percentiles on the GRE?

Quartiles divide a data set into four equal parts: the first quartile (Q1) is the 25th percentile, the second quartile (Q2) is the 50th percentile (the median), and the third quartile (Q3) is the 75th percentile. These are frequently tested using boxplots, which visually represent the interquartile range. To further sharpen your skills, try using an AI Question Generator for customized statistics drills.