GRE Statistics Set 1 Practice Questions with Answers

Data interpretation and descriptive statistics account for approximately 25% of the Quantitative Reasoning section on the GRE. A solid grasp of GRE Statistics Set 1 concepts ensures you can efficiently handle questions involving mean, median, mode, and standard deviation, which are fundamental to achieving a high score. These topics are not just about memorizing formulas; they require an understanding of how data spreads and clusters within a given set.

Concept Explanation

Statistics on the GRE focuses on descriptive measures that summarize the characteristics of a data set, primarily central tendency and dispersion. Central tendency includes the arithmetic mean (average), the median (the middle value when data is ordered), and the mode (the most frequent value). Dispersion measures how spread out the data points are, typically represented by the range and standard deviation. The GRE Prep hub provides a strategic overview of how these concepts appear in both multiple-choice and quantitative comparison formats. Standard deviation measures how much the data values deviate from the mean; if all values are identical, the standard deviation is zero. Understanding these properties is essential for solving complex problems involving weighted averages or changes in a data set when a new value is added.

Solved Examples

1. Problem: Find the arithmetic mean of the set $\{12, 15, 21, 28, 34\}$.
   1. Sum the values: $12 + 15 + 21 + 28 + 34 = 110$.
   2. Count the number of terms: There are 5 terms.
   3. Divide the sum by the count: $\ \frac{110}{5} = 22$.
   4. The mean is 22.
2. Problem: A set of 7 integers has a median of 15. If the three smallest integers are 8, 9, and 11, what is the smallest possible value for the largest integer in the set?
   1. Arrange the set in ascending order: $a, b, c, d, e, f, g$.
   2. The median is the 4th term, so $d = 15$.
   3. The first three terms are 8, 9, and 11. The set is now $\{8, 9, 11, 15, e, f, g\}$.
   4. To minimize the largest integer $g$, we must minimize $e$ and $f$ by making them as small as possible while remaining $\geq 15$.
   5. Let $e = 15$ and $f = 15$. Then $g$ must be at least 15 to maintain the order.
   6. The smallest possible value for the largest integer is 15.
3. Problem: If the standard deviation of set $X$ is $s$, what happens to the standard deviation if 5 is added to every term in the set?
   1. Adding a constant to every term in a set shifts the mean by that constant.
   2. However, the relative distance between the numbers remains the same.
   3. Since standard deviation measures the spread (distance from the mean), the spread does not change.
   4. The new standard deviation is still $s$.

Practice Questions

1. A student scored an average of 85 on four tests. What score does he need on the fifth test to raise his average to 88?
2. Set $A$ consists of the numbers $\{2, 4, 6, 8, 10, 12\}$. What is the absolute difference between the mean and the median of Set $A$?
3. In a group of 10 students, the heights in centimeters are 160, 165, 170, 170, 175, 180, 180, 180, 185, and 190. What is the mode of this data set?

4. If the average of $x, y, \ \text{ and } z$ is 12, and the average of $x, y, z, \ \text{ and } w$ is 15, what is the value of $w$?
5. Which set has the largest standard deviation: $\{10, 10, 10\}$, $\{10, 11, 12\}$, or $\{0, 10, 20\}$?
6. The median of a set of 5 distinct positive integers is 10. If the sum of the integers is 50, what is the maximum possible value for the largest integer?
7. A data set has a mean of 50 and a standard deviation of 5. If every value is multiplied by 3, what is the new mean and new standard deviation?
8. If the range of a set of 8 numbers is 25 and the smallest number is -5, what is the largest number?
9. Find the weighted average if 40% of the class scored 80 and 60% of the class scored 90.
10. Compare the mean of the first five prime numbers to the median of the first five prime numbers.

Answers & Explanations

1. Answer: 100. To find the total sum for 4 tests: $85 \ \times 4 = 340$. To find the required sum for 5 tests: $88 \ \times 5 = 440$. The fifth test must be $440 - 340 = 100$.
2. Answer: 0. The mean is $\ \frac{2+4+6+8+10+12}{6} = \ \frac{42}{6} = 7$. The median is the average of the two middle terms: $\ \frac{6+8}{2} = 7$. The difference is $7 - 7 = 0$.
3. Answer: 180. The mode is the value that appears most frequently. In this set, 180 appears three times, while 170 appears twice and other values appear once.
4. Answer: 24. Total sum of $x, y, z$ is $12 \ \times 3 = 36$. Total sum of $x, y, z, w$ is $15 \ \times 4 = 60$. Therefore, $w = 60 - 36 = 24$.
5. Answer: {0, 10, 20}. Standard deviation measures distance from the mean. $\{10, 10, 10\}$ has SD of 0. $\{0, 10, 20\}$ has the largest gaps between numbers and the mean, resulting in the highest dispersion.
6. Answer: 23. Let the integers be $a, b, 10, d, e$. To maximize $e$, minimize $a, b, \ \text{ and } d$. Since they are distinct positive integers, $a=1, b=2$. Since $d > 10$, the smallest $d$ can be is 11. Sum: $1+2+10+11+e = 50 \ \rightarrow 24+e = 50 \ \rightarrow e = 26$. (Wait, if $e=26$, the distinct integers are 1, 2, 10, 11, 26). Re-checking the math: $50 - 24 = 26$.
7. Answer: Mean 150, SD 15. Multiplying every value by a constant $k$ multiplies both the mean and the standard deviation by $k$. $50 \ \times 3 = 150$ and $5 \ \times 3 = 15$.
8. Answer: 20. Range = Largest - Smallest. $25 = L - (-5) \ \rightarrow 25 = L + 5 \ \rightarrow L = 20$.
9. Answer: 86. Weighted average = $(0.40 \ \times 80) + (0.60 \ \times 90) = 32 + 54 = 86$.
10. Answer: Mean 5.6, Median 5. First five primes: 2, 3, 5, 7, 11. Mean = $\ \frac{2+3+5+7+11}{5} = \ \frac{28}{5} = 5.6$. Median = 5.

1. If a set of numbers is {4, 4, 4, 4}, what is the standard deviation?

• A4
• B2
• C0
• D1

Frequently Asked Questions

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

The mean is the average of all values, while the median is the middle value of an ordered list. The mean is sensitive to outliers, whereas the median provides a better central value for skewed data.

How do I calculate standard deviation for the GRE?

You rarely need to calculate the exact standard deviation using the formula; instead, you should understand that it represents the average distance from the mean. You can use the AI Question Generator to practice conceptual problems related to dispersion.

What is a weighted average?

A weighted average accounts for the relative importance or frequency of each value in a set. It is calculated by multiplying each value by its weight and summing the results.

Does the range change if I add a constant to all numbers?

No, the range remains the same because the difference between the new maximum and new minimum does not change. For more practice on these properties, check out the general pathology questions for similar logic-based data analysis.

Can the standard deviation be negative?

No, standard deviation is always zero or positive because it is derived from squared differences. A standard deviation of zero indicates all values in the set are identical.