Hard GRE Statistical Analysis Questions Practice Questions

Statistical analysis on the GRE involves interpreting data sets, understanding distribution properties, and applying probability theory to solve complex quantitative problems. These problems often require more than just calculating a mean; they demand an understanding of how changes in data affect standard deviation, the nuances of normal distributions, and the mechanics of combinations and permutations. For students aiming for a high score, mastering Hard GRE Statistical Analysis Questions Practice Questions is essential for navigating the Quantitative Reasoning section with confidence.

Concept Explanation

Statistical analysis is the mathematical science of collecting, exploring, and presenting large amounts of data to discover underlying patterns and trends. In the context of the GRE, this involves three primary pillars: Descriptive Statistics, Probability, and Data Interpretation. Descriptive statistics focus on measures of central tendency (mean, median, mode) and measures of dispersion, such as range and standard deviation. A critical concept at the harder level is how adding or multiplying constants to a data set affects these measures. For instance, adding a constant to every value in a set increases the mean by that constant but leaves the standard deviation unchanged. Conversely, multiplying every value by a constant scales both the mean and the standard deviation by that factor.

Moving beyond basic averages, the GRE tests your ability to handle normal distributions and the 68-95-99.7 rule. You must be able to determine what percentage of data falls within a certain number of standard deviations from the mean. Additionally, hard questions frequently blend statistics with counting methods. You will need to distinguish between permutations (where order matters) and combinations (where order does not matter) to calculate probabilities in multi-step scenarios. Utilizing an Adaptive GRE Practice Test can help you identify which of these specific sub-topics requires the most focus during your study sessions.

Solved Examples

Review these worked examples to understand the logic required for high-difficulty statistical problems.

1. Example 1: Standard Deviation Shifts
   A set of 50 test scores has a mean of 75 and a standard deviation of 8. If every score is increased by 5 points and then multiplied by 1.2, what is the new mean and the new standard deviation?
   Solution:
   1. First, address the addition: Adding 5 to every score increases the mean to $75 + 5 = 80$. The standard deviation remains 8.
   2. Second, address the multiplication: Multiplying every score by 1.2 multiplies both the mean and the standard deviation.
   3. New Mean: $80 \times 1.2 = 96$.
   4. New Standard Deviation: $8 \times 1.2 = 9.6$.
2. Example 2: Probability and Combinations
   A bag contains 6 red marbles and 4 blue marbles. If 3 marbles are drawn at random without replacement, what is the probability that at least 2 are red?
   Solution:
   1. Total ways to choose 3 marbles from 10: $\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$.
   2. Case 1: Exactly 2 red (and 1 blue). Ways: $\binom{6}{2} \times \binom{4}{1} = 15 \times 4 = 60$.
   3. Case 2: Exactly 3 red (and 0 blue). Ways: $\binom{6}{3} \times \binom{4}{0} = 20 \times 1 = 20$.
   4. Total favorable outcomes: $60 + 20 = 80$.
   5. Probability: $\frac{80}{120} = \frac{2}{3}$.
3. Example 3: Normal Distribution Percentiles
   In a normally distributed set of 2,000 observations, the mean is 100 and the standard deviation is 10. Approximately how many observations are between 80 and 110?
   Solution:
   1. Identify the standard deviations from the mean: 80 is 2 standard deviations below the mean ($100 - 2(10)$). 110 is 1 standard deviation above the mean ($100 + 1(10)$).
   2. Use the 68-95-99.7 rule: The area between -2 and 0 standard deviations is $95\% / 2 = 47.5\%$.
   3. The area between 0 and +1 standard deviation is $68\% / 2 = 34\%$.
   4. Total percentage: $47.5\% + 34\% = 81.5\%$.
   5. Number of observations: $0.815 \times 2,000 = 1,630$.

Practice Questions

Test your skills with these Hard GRE Statistical Analysis Questions Practice Questions. Ensure you show your work for each step.

1. Set $S$ consists of 7 distinct integers. The median of $S$ is 15 and the range is 20. If the smallest possible integer is added to the set to create Set $T$, what is the maximum possible value of the new median?
2. A box contains 5 distinct light bulbs, 2 of which are defective. If a technician selects 3 bulbs at random, what is the probability that both defective bulbs are included in the selection?
3. The average (arithmetic mean) of a set of $n$ numbers is 40. If the number 60 is added to the set, the average increases to 44. Find the value of $n$.

4. In a group of 100 students, the heights are normally distributed with a mean of 170 cm and a standard deviation of 5 cm. If a student is selected at random, what is the probability that their height is greater than 175 cm?
5. A list of 10 numbers has a mean of 20 and a median of 18. If the largest number in the list is increased by 50, what is the new median?
6. If the standard deviation of set $\{a, b, c\}$ is $s$, what is the standard deviation of set $\{3a+4, 3b+4, 3c+4\}$?
7. How many different 4-letter codes can be formed using the letters in the word "STATISTICS"? (Note: This requires advanced combinatorial reasoning).
8. A data set has 11 values. If each value is doubled, by what percentage does the variance increase?
9. The interquartile range (IQR) of a set of data is 15. If every value in the set is multiplied by 3 and then 10 is subtracted, what is the new IQR?
10. If a fair coin is flipped 6 times, what is the probability of getting exactly 3 heads?

Answers & Explanations

1. Answer: 15.
   In a set of 7 integers, the median is the 4th term (15). When an 8th term is added (the smallest possible integer), the new median is the average of the 4th and 5th terms of the sorted set. To maximize this, the 5th term should be as small as possible. Since the integers are distinct, the 5th term must be at least 16, but the question asks for the median. Wait, if we add a value smaller than 15, the 4th term (15) remains in the middle. The new median of 8 terms is the average of the 4th and 5th terms. If the original 4th was 15, and we added a number smaller than 15, the new 4th is 15. The 5th was already $\geq 16$. However, if we add the smallest possible integer, the values shift. The original median 15 stays at position 4 or 5. If we add a very small number, the old 4th becomes the new 5th. The new median is $\frac{( \text{new 4th} + 15)}{2}$. Since the integers are distinct, the max value is 14.5 or 15 depending on the configuration. Actually, the median of 15 is fixed; adding a smaller number cannot increase the median.
2. Answer: 3/10 or 0.3.
   Total ways to choose 3 from 5: $\binom{5}{3} = 10$. To have both defective bulbs, you must choose both (1 way) and then 1 non-defective bulb from the remaining 3 (3 ways). Total favorable: $1 \times 3 = 3$. Probability: $3/10$.
3. Answer: 4.
   Original sum: $40n$. New sum: $40n + 60$. New average: $\frac{40n + 60}{n+1} = 44$. Solving gives $40n + 60 = 44n + 44$ $\rightarrow 16 = 4n$ $\rightarrow n = 4$.
4. Answer: 0.16 (or 16%).
   175 cm is exactly 1 standard deviation above the mean. In a normal distribution, approximately 68% of data is within 1 SD of the mean (34% above, 34% below). Since 50% is above the mean, the portion above 1 SD is $50\% - 34\% = 16\%$.
5. Answer: 18.
   The median is the middle value. Increasing the largest value does not change the order of the middle values, so the median remains 18.
6. Answer: 3s.
   Adding 4 to each term does not change the standard deviation. Multiplying each term by 3 multiplies the standard deviation by 3.
7. Answer: 234.
   This involves treating the repetitions (S:3, T:3, I:2, A:1, C:1). You must calculate cases for 4 distinct letters, 1 pair, 2 pairs, and 3-of-a-kind. For more practice on these types of logic puzzles, see GRE Sentence Equivalence Practice Questions to balance your verbal and quant study.
8. Answer: 300%.
   If values are doubled, the standard deviation doubles. Since variance is the square of standard deviation, the variance becomes $2^2 = 4$ times the original. A 4x increase is a 300% increase.
9. Answer: 45.
   Subtraction does not affect range-based measures like IQR. Multiplying by 3 scales the IQR by 3. $15 \times 3 = 45$.
10. Answer: 5/16.
    Total outcomes: $2^6 = 64$. Favorable: $\binom{6}{3} = 20$. Probability: $20/64 = 5/16$.

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

• AThe mean is 0
• BThe median is 0
• CAll numbers in the set are identical
• DThe range is greater than 0

Frequently Asked Questions

What is the difference between a permutation and a combination?

The primary difference lies in whether the order of selection matters. In a permutation, the sequence is important (like a passcode), whereas in a combination, the grouping is all that matters (like a hand of cards).

How does the GRE test standard deviation?

The GRE rarely asks you to calculate standard deviation manually from a large set. Instead, it tests your conceptual understanding of how the spread changes when data is added, removed, or transformed.

What is the 68-95-99.7 rule?

This rule describes the percentage of data within 1, 2, and 3 standard deviations of the mean in a normal distribution. It is a fundamental tool for solving GRE problems involving bell curves.

Can the median be higher than the mean?

Yes, the median can be higher than the mean in a negatively skewed distribution where there are several low-value outliers pulling the average down. The GRE often uses this property to test your understanding of data distribution.

What is the interquartile range?

The interquartile range (IQR) is the difference between the third quartile (75th percentile) and the first quartile (25th percentile). It measures the spread of the middle 50% of the data and is resistant to outliers.