GRE Statistical Analysis Questions Practice Questions with Answers

Concept Explanation

GRE Statistical Analysis Questions assess your ability to interpret, analyze, and manipulate data sets using measures of central tendency, dispersion, and probability. At its core, this topic requires a firm grasp of the arithmetic mean (average), median, mode, range, and standard deviation. You will often encounter these concepts in the context of GRE Prep, where data interpretation is a significant component of the Quantitative Reasoning section. Understanding the relationship between these values—such as how adding a constant to every number in a set affects the mean but leaves the standard deviation unchanged—is vital for efficiency during the exam.

Beyond basic averages, statistical analysis on the GRE involves understanding frequency distributions and the properties of the normal distribution curve. For instance, in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Familiarity with standard deviation and how it measures the spread of data relative to the mean allows you to compare different data sets quickly. You should also be comfortable with quartiles, percentiles, and boxplots, which provide snapshots of data distribution. Utilizing tools like an AI Question Generator can help you practice these specific data interpretation formats through varied repetition.

Solved Examples

1. Example 1: Mean and Median
   A set of five integers has a mean of 12 and a median of 10. If the four smallest integers are 6, 8, 10, and 14, what is the fifth integer?
   1. First, use the mean formula: $\text{Mean} = \frac{ \text{Sum of terms}}{ \text{Number of terms}}$.
   2. Multiply the mean by the number of terms to find the total sum: $12 \times 5 = 60$.
   3. Sum the known integers: $6 + 8 + 10 + 14 = 38$.
   4. Subtract the sum of the known integers from the total sum: $60 - 38 = 22$.
   5. Check if the median remains 10 with the set {6, 8, 10, 14, 22}. The middle value is 10. The fifth integer is 22.
2. Example 2: Standard Deviation Concept
   Set A consists of {10, 20, 30, 40, 50} and Set B consists of {15, 25, 35, 45, 55}. Compare the standard deviations of the two sets.
   1. Observe the relationship between the sets. Each element in Set B is exactly the element in Set A plus 5.
   2. Recall the rule: Adding or subtracting a constant to every value in a set does not change the distances between the values.
   3. Since the spread or "dispersion" of the numbers remains identical, the standard deviations must be equal.
3. Example 3: Weighted Averages
   In a class of 20 students, the average score on a test was 80. In another class of 30 students, the average score was 90. What is the combined average for all 50 students?
   1. Calculate the total points for the first class: $20 \times 80 = 1,600$.
   2. Calculate the total points for the second class: $30 \times 90 = 2,700$.
   3. Add the total points: $1,600 + 2,700 = 4,300$.
   4. Divide by the total number of students: $\frac{4,300}{50} = 86$.
   5. The combined average is 86.

Practice Questions

1. If the average (arithmetic mean) of $x, x+3,$ and $x+6$ is 15, what is the value of $x$?

2. A list of numbers consists of {4, 7, 12, 12, 15, 18, 22}. What is the difference between the range and the median of this list?

3. Set S contains 7 distinct positive integers. If the median of Set S is 15, what is the smallest possible value for the largest integer in the set?

4. The standard deviation of a set of numbers $\{a, b, c, d\}$ is $s$. What is the standard deviation of the set $\{3a, 3b, 3c, 3d\}$?

5. In a group of 10 measurements, the mean is 50. If one measurement of 95 is removed, what is the new mean of the remaining 9 measurements?

6. A normal distribution has a mean of 70 and a standard deviation of 10. Approximately what percent of the data falls between 50 and 90?

7. The arithmetic mean of 4 numbers is 25. If three of the numbers are 18, 22, and 30, what is the fourth number?

8. List L: {10, 12, 14, 16, 18, 20}. If the number 30 is added to the list, which will increase more: the mean or the median?

9. A set of 5 numbers has a mean of 20. If each number is multiplied by 2 and then increased by 5, what is the new mean?

10. If the 25th percentile of a data set of 400 scores is 65, how many scores are less than 65?

Answers & Explanations

1. Answer: 12. Setup the equation: $\frac{x + (x+3) + (x+6)}{3} = 15$. This simplifies to $\frac{3x + 9}{3} = 15$, which reduces to $x + 3 = 15$. Therefore, $x = 12$.
2. Answer: 6. The range is $22 - 4 = 18$. The median (middle value) of the 7 numbers is 12. The difference is $18 - 12 = 6$.
3. Answer: 18. To minimize the largest integer, we need the integers above the median to be as small as possible. Since they are distinct, the three integers above 15 must be at least 16, 17, and 18. Thus, 18 is the smallest possible value for the largest integer.
4. Answer: 3s. When every value in a data set is multiplied by a constant $k$, the standard deviation is also multiplied by $|k|$. Here, $k = 3$, so the new standard deviation is $3s$.
5. Answer: 45. The original total sum was $10 \times 50 = 500$. After removing 95, the new total sum is $500 - 95 = 405$. The new mean is $405 \div 9 = 45$.
6. Answer: 95%. The range 50 to 90 represents two standard deviations from the mean ($70 \pm 2(10)$). According to the 68-95-99.7 rule, approximately 95% of data in a normal distribution falls within two standard deviations.
7. Answer: 30. Total sum = $4 \times 25 = 100$. Sum of known three = $18 + 22 + 30 = 70$. Fourth number = $100 - 70 = 30$.
8. Answer: The Mean. Original mean: 15. Original median: 15. New list: {10, 12, 14, 16, 18, 20, 30}. New mean: $120 \div 7 \approx 17.14$. New median: 16. The mean increased by 2.14, while the median increased by 1.
9. Answer: 45. Linear transformations apply to the mean directly. If the original mean is $M$, the new mean is $2M + 5$. So, $2(20) + 5 = 45$.
10. Answer: 100. The 25th percentile means 25% of the data points are below that value. $0.25 \times 400 = 100$.

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

• 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 calculated average of all values, while the median is the middle value when the data is ordered from least to greatest. On the GRE, the mean is more sensitive to outliers, whereas the median provides a better sense of the "center" for skewed data.

How does standard deviation appear on the GRE?

GRE questions typically focus on the conceptual understanding of standard deviation rather than complex manual calculations. You will often be asked to compare the spread of two sets or apply the properties of the normal distribution curve.

What is a weighted average?

A weighted average occurs when different groups contribute differently to the final mean based on their size or "weight." To solve these, you must find the total sum of all values across all groups and divide by the total number of items.

Can the standard deviation ever be negative?

No, standard deviation is always zero or positive because it is derived from the square root of the variance. A value of zero indicates all data points are identical, while higher values indicate more spread.

What is the 68-95-99.7 rule?

This rule describes the percentage of data within one, two, and three standard deviations of the mean in a normal distribution. It is a frequent shortcut used in GRE Quantitative Reasoning to estimate proportions of a population.