Hard GRE Statistics Practice Test Practice Questions

Concept Explanation

Statistics on the GRE Quantitative Reasoning section involves the analysis, interpretation, and manipulation of data sets using measures of central tendency, dispersion, and probability distributions. To succeed on a hard GRE Statistics practice test, you must go beyond simple averages and understand how adding or removing data points shifts the standard deviation, how normal distributions function in terms of percentiles, and how to calculate combinations and permutations in complex scenarios. The GRE often tests your ability to reason about data conceptually rather than just performing arithmetic calculations. For instance, knowing that the standard deviation measures the spread of data around the mean allows you to compare two sets without calculating the actual value of $\sigma$. These concepts are integral to a comprehensive GRE Prep strategy, as they frequently appear in Data Interpretation and Quantitative Comparison questions.

Key statistical concepts you will encounter include:

• Arithmetic Mean and Median: The mean is the average, while the median is the middle value. In skewed distributions, these two values diverge significantly.

• Standard Deviation ($\sigma$): A measure of how far data points are from the mean. A larger $\sigma$ indicates more spread out data.

• Normal Distribution: A symmetric, bell-shaped curve where approximately 68% of data falls within one standard deviation of the mean, and 95% falls within two.

• Interquartile Range (IQR): The difference between the 75th percentile (Q3) and the 25th percentile (Q1).

Harder problems typically combine these elements, requiring you to use the GRE practice questions with explanations to identify patterns in how the test makers disguise simple statistical properties within wordy problems.

Solved Examples

Review these worked examples to understand the logic required for high-difficulty statistics questions.

1. Example: Standard Deviation Shift
   A set of 10 distinct integers has a mean of 50 and a standard deviation of 12. If each integer in the set is multiplied by 3 and then increased by 5, what is the new mean and the new standard deviation?

   1. The mean is affected by both multiplication and addition. New Mean = $(50 \times 3) + 5 = 155$.

   2. The standard deviation is affected by multiplication but not by addition/subtraction.

   3. New Standard Deviation = $12 \times 3 = 36$. Adding 5 shifts the entire distribution but does not change the spread.

2. Example: Normal Distribution Percentiles
   The heights of a population are normally distributed with a mean of 170 cm and a standard deviation of 10 cm. What percentage of the population is between 150 cm and 180 cm?

   1. Identify the z-scores. 150 cm is 2 standard deviations below the mean $(z = -2)$. 180 cm is 1 standard deviation above the mean $(z = +1)$.

   2. Recall the 68-95-99.7 rule. The area from $z = -2$ to $z = 0$ is $95\% / 2 = 47.5\%$.

   3. The area from $z = 0$ to $z = 1$ is $68\% / 2 = 34\%$.

   4. Total percentage = $47.5\% + 34\% = 81.5\%$.

3. Example: Combined Mean
   Class A has 20 students with an average score of 80. Class B has $n$ students with an average score of 90. If the combined average of both classes is 84, what is the value of $n$?

   1. Use the weighted average formula: $\text{Total Sum} / \text{Total Students} = \text{Combined Mean}$.

   2. $\frac{(20 \times 80) + (n \times 90)}{20 + n} = 84$.

   3. $1600 + 90n = 84(20 + n) \rightarrow 1600 + 90n = 1680 + 84n$.

   4. $6n = 80 \rightarrow n = 13.33$. Since students must be integers, this setup usually implies a specific ratio in GRE problems (e.g., if the average was 82, $n$ would be 5).

Practice Questions

Test your knowledge with these hard GRE Statistics practice test practice questions. These are designed to mimic the difficulty of the 160+ score level.

1. Set X contains 7 consecutive even integers. If the greatest integer is removed and replaced with the next consecutive even integer, by how much does the mean of the set increase?

2. A distribution of 500 test scores has a mean of 72 and a standard deviation of 8. If the scores are normally distributed, approximately how many students scored between 64 and 88?

3. Quantity A: The standard deviation of {10, 20, 30, 40, 50}.
   Quantity B: The standard deviation of {110, 120, 130, 140, 150}.

4. A data set consists of 15 positive integers. If the median is 40 and the range is 50, what is the maximum possible value for the arithmetic mean of the set?

5. In a group of 100 people, the 40th percentile of their ages is 32. If 10 people aged 50 are added to the group, which of the following must be true about the new 40th percentile?

6. The probability of event A occurring is 0.6, and the probability of event B occurring is 0.5. If A and B are independent, what is the probability that exactly one of the events occurs?

7. Set S consists of the integers {2, 4, 6, 8, 10, 12}. If one number is removed from Set S, for which number's removal will the standard deviation decrease the most?

8. A box contains 5 red balls and 5 blue balls. If 3 balls are selected at random without replacement, what is the probability that at least 2 balls are red?

Answers & Explanations

1. Answer: 2/7. Let the set be $\{n, n+2, n+4, n+6, n+8, n+10, n+12\}$. The sum is $7n + 42$, and the mean is $n + 6$. If we replace $n+12$ with $n+14$, the new sum is $7n + 44$. The new mean is $(7n + 44)/7 = n + 6 + 2/7$. The increase is $2/7$.

2. Answer: 407. 64 is 1 SD below the mean, and 88 is 2 SDs above the mean. The area from -1 to 0 is 34.1%. The area from 0 to +2 is 47.7%. Total percentage = $34.1 + 47.7 = 81.8\%$. $0.818 \times 500 = 409$. (Standard approximations like 81.5% - 82% are acceptable).

3. Answer: The two quantities are equal. Standard deviation measures the distance of points from the mean. Set B is simply Set A with 100 added to every term. Shifting a set by a constant does not change its spread or standard deviation.

4. Answer: 64. To maximize the mean, maximize the sum. The 15 integers are $a_1, \dots, a_{15}$. The median $a_8 = 40$. To maximize the rest: $a_1$ to $a_7$ can be 40 (they must be $\leq$ median). $a_9$ to $a_{14}$ can be the maximum possible value. Since range is 50, and $a_1$ could be 1, $a_{15}$ could be 51. But if we make $a_1 = 40$, then $a_{15} = 90$. Sum = $8 \times 40 + 6 \times 90 + 90 = 320 + 540 + 90 = 950$. Mean = $950/15 \approx 63.3$. (Exact values depend on the constraint of distinctness; if not distinct, 64 is the ceiling).

5. Answer: The new 40th percentile will be greater than or equal to 32. Adding values above the current 40th percentile (ages 50) pulls the distribution upward or keeps it the same, shifting the rank of the original 32.

6. Answer: 0.5. $P( \text{exactly one}) = P(A \text{ and not } B) + P(B \text{ and not } A)$. $P(A)P( \text{not } B) = 0.6 \times 0.5 = 0.3$. $P(B)P( \text{not } A) = 0.5 \times 0.4 = 0.2$. Total = $0.3 + 0.2 = 0.5$.

7. Answer: 2 or 12. Standard deviation decreases the most when you remove an "outlier" or the value furthest from the mean. The mean of the set is 7. Both 2 and 12 are 5 units away from the mean.

8. Answer: 1/2. Total ways to pick 3: $\binom{10}{3} = 120$. Ways to get 2 red: $\binom{5}{2} \times \binom{5}{1} = 10 \times 5 = 50$. Ways to get 3 red: $\binom{5}{3} = 10$. Total favorable = 60. Probability = $60/120 = 1/2$.

1. If a constant \( k \) is added to every number in a list, how does the standard deviation change?

• AIt increases by \( k \)
• BIt decreases by \( k \)
• CIt is multiplied by \( k \)
• DIt remains the same

Frequently Asked Questions

What is the difference between standard deviation and variance?

Variance is the average of the squared differences from the mean, while standard deviation is the square root of the variance. Standard deviation is expressed in the same units as the data, making it more intuitive for interpreting spread.

Does the GRE provide the formula for standard deviation?

No, the GRE does not provide the formula, but you rarely need to calculate it manually. You are expected to understand its properties, such as how it reacts to transformations and how to compare the spread of two different sets visually or conceptually.

How do I handle normal distribution questions without a Z-table?

GRE normal distribution questions rely on the 68-95-99.7 rule. You should memorize that 34% of data is between the mean and 1 SD, and roughly 13.5% is between 1 SD and 2 SDs, which allows you to solve almost any related problem on the test.

What is the most efficient way to find the median of a large data set?

Organize the data in increasing order and use the formula $(n+1)/2$ to find the position of the median. If the result is an integer, that position is the median; if it ends in .5, average the two middle numbers.

Can the standard deviation ever be negative?

No, the standard deviation is always greater than or equal to zero. It is zero only if all numbers in the set are identical, indicating there is no spread or variation in the data whatsoever.

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