Hard GRE Statistics Exam Questions Practice Questions

Standard deviation, normal distribution, and combinatorics frequently represent the most challenging hurdles in the Quantitative Reasoning section. To succeed, students must move beyond simple mean and median calculations to understand how data spreads and how variables interact under specific constraints. This collection of hard GRE statistics exam questions practice questions focuses on these high-level concepts to ensure you are prepared for the most rigorous problems the exam offers.

Concept Explanation

GRE statistics involves the collection, analysis, interpretation, and presentation of data, focusing heavily on measures of central tendency, dispersion, and probability distributions. While basic statistics covers the arithmetic mean, median, mode, and range, advanced GRE Prep requires a deep understanding of standard deviation and the properties of the normal distribution. Standard deviation measures how much the values in a data set vary from the mean; a low standard deviation indicates that the data points tend to be very close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values. The normal distribution, often called the "bell curve," is defined by its mean $\mu$ and standard deviation $\sigma$, with approximately 68% of data falling within one standard deviation, 95% within two, and 99.7% within three. Proficiency in these areas, along with counting methods like permutations and combinations, is essential for high-scoring students. You can explore more through GRE practice questions with explanations to see these theories in action.

Solved Examples

1. Example: Standard Deviation Shift
   A set of 10 numbers has a mean of 50 and a standard deviation of 8. If every number in the set is multiplied by 3 and then 10 is added to each result, what is the new mean and new standard deviation?
   1. Identify the effect of multiplication on the mean: $50 \times 3 = 150$.
   2. Identify the effect of addition on the mean: $150 + 10 = 160$. The new mean is 160.
   3. Identify the effect of multiplication on standard deviation: $8 \times 3 = 24$.
   4. Identify the effect of addition on standard deviation: Adding a constant does not change the spread. The standard deviation remains 24.
   5. Final Answer: Mean = 160, Standard Deviation = 24.
2. Example: Normal Distribution Percentiles
   In a normally distributed set of test scores with a mean of 75 and a standard deviation of 5, what percentage of scores fall between 70 and 85?
   1. Determine the z-scores for the boundaries. For 70: $\frac{70 - 75}{5} = -1$. For 85: $\frac{85 - 75}{5} = +2$.
   2. Recall the empirical rule (68-95-99.7). The area from -1 to 0 is $\frac{68\%}{2} = 34\%$.
   3. The area from 0 to +2 is $\frac{95\%}{2} = 47.5\%$.
   4. Add the areas: $34\% + 47.5\% = 81.5\%$.
   5. Final Answer: 81.5%.
3. Example: Combinatorial Probability
   A bag contains 5 red marbles and 5 blue marbles. If 3 marbles are drawn at random without replacement, what is the probability that exactly 2 are red?
   1. Calculate the total ways to choose 3 marbles: $\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$.
   2. Calculate the ways to choose 2 red marbles: $\binom{5}{2} = 10$.
   3. Calculate the ways to choose 1 blue marble: $\binom{5}{1} = 5$.
   4. Multiply favorable outcomes: $10 \times 5 = 50$.
   5. Calculate probability: $\frac{50}{120} = \frac{5}{12}$.

Practice Questions

1. Set A consists of 7 distinct integers with a median of 20 and a range of 15. If the smallest integer is increased by 5, what is the maximum possible new median?
2. A distribution of 500 values is approximately normal with a mean of 120 and a standard deviation of 10. How many values are expected to be greater than 140?
3. If the standard deviation of the set $\{x, y, z\}$ is $s$, what is the standard deviation of the set $\{x-4, y-4, z-4\}$?

4. A researcher has two sets of data. Set X has 10 items with a mean of 20. Set Y has 20 items with a mean of 35. What is the combined mean of the 30 items?
5. In a group of 100 students, 60 study French, 40 study Spanish, and 20 study both. If a student is chosen at random, what is the probability they study neither French nor Spanish?
6. Quantity A: The standard deviation of $\{10, 20, 30, 40, 50\}$. Quantity B: The standard deviation of $\{110, 120, 130, 140, 150\}$. Which is greater?
7. A set of numbers $S$ has a mean $m$. If a new number equal to $m$ is added to the set, how does the standard deviation change?
8. A committee of 3 people is to be chosen from a group of 5 men and 4 women. What is the probability that the committee contains at least one woman?
9. The probability of event A occurring is 0.6, and the probability of event B occurring is 0.4. If A and B are independent, what is the probability that neither A nor B occurs?
10. If the variance of a data set is 25, and every value in the set is doubled, what is the new variance?

Answers & Explanations

1. 20: Increasing the smallest value in a set of 7 distinct integers does not necessarily change the middle (4th) value. Since the integers must remain distinct and the original median was 20, the 4th value remains 20 unless the original 1st, 2nd, or 3rd values are pushed past 20. However, the question asks for the maximum possible median; since the original median was 20, shifting the bottom value doesn't force the 4th value to increase.
2. 12.5 (or approx 13): A value of 140 is exactly 2 standard deviations above the mean ($120 + 2 \times 10$). In a normal distribution, 95% of data is within 2 standard deviations. This leaves 5% in the tails (2.5% in each tail). $2.5\% \text{ of } 500 = 0.025 \times 500 = 12.5$.
3. s: Subtracting a constant from every member of a data set shifts the entire distribution but does not change the distance between the points. Therefore, the spread (standard deviation) remains exactly the same.
4. 30: Use a weighted average. Sum of X = $10 \times 20 = 200$. Sum of Y = $20 \times 35 = 700$. Total Sum = 900. Total Items = 30. Mean = $\frac{900}{30} = 30$.
5. 0.20: Use the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. $60 + 40 - 20 = 80$. 80 students study at least one language. Therefore, $100 - 80 = 20$ students study neither. Probability = $\frac{20}{100} = 0.20$.
6. They are equal: Standard deviation measures the dispersion from the mean. Both sets have the same spacing between elements (increments of 10). Shifting a set by 100 units does not change its internal dispersion.
7. It decreases: Standard deviation is the average distance from the mean. By adding a value exactly equal to the mean (distance = 0), you are adding a data point that has no deviation, which pulls the "average deviation" down.
8. $\frac{37}{42}$: It is easier to find the probability of the complement (no women). Total ways: $\binom{9}{3} = 84$. Ways with only men: $\binom{5}{3} = 10$. Probability of no women = $\frac{10}{84} = \frac{5}{42}$. Probability of at least one woman = $1 - \frac{5}{42} = \frac{37}{42}$.
9. 0.24: If $P(A) = 0.6$, then $P( \text{not } A) = 0.4$. If $P(B) = 0.4$, then $P( \text{not } B) = 0.6$. For independent events, multiply the probabilities: $0.4 \times 0.6 = 0.24$.
10. 100: Variance is the square of the standard deviation. If values are doubled, the standard deviation doubles. If the standard deviation doubles, the variance quadruples ($2^2 = 4$). $25 \times 4 = 100$.

Frequently Asked Questions

How does the GRE test standard deviation?

The GRE rarely asks you to calculate standard deviation using the complex formula; instead, it tests your understanding of how adding, multiplying, or shifting data points affects the spread. You should know that adding a constant doesn't change it, but multiplying every value by a constant scales the standard deviation by that same constant.

What is the difference between a permutation and a combination?

The primary difference lies in whether the order of selection matters. Use permutations when order is important (like seating arrangements or passwords) and combinations when the order does not matter (like choosing a committee or a hand of cards). For more practice on these logic-heavy problems, check out GRE text completion practice questions which often require similar analytical thinking.

What is the empirical rule for normal distributions?

The empirical rule, or the 68-95-99.7 rule, states that for a normal distribution, almost all data falls within three standard deviations of the mean. Specifically, 68% is within one SD, 95% is within two SDs, and 99.7% is within three SDs. This is a fundamental concept for solving unlimited GRE practice questions effectively.

Can the standard deviation ever be negative?

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

How do I calculate the weighted mean?

To calculate a weighted mean, multiply each value by its corresponding weight (or the number of items in its group), sum those products, and then divide by the total sum of the weights. This is common when combining two different groups with different averages into one large group.