Hard GRE Normal Distribution Questions Practice Questions

Concept Explanation

Normal distribution is a continuous probability distribution characterized by a symmetrical, bell-shaped curve where the mean, median, and mode are all equal. On the GRE, these distributions are defined by two parameters: the mean $\mu$ and the standard deviation $\sigma$. The GRE specifically tests your ability to apply the 68-95-99.7 rule, often called the Empirical Rule. This rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. Success on Hard GRE Normal Distribution Questions requires more than just memorizing these percentages; you must be able to calculate specific percentiles, compare different distributions using z-scores, and handle "greater than/less than" logic in complex word problems. For more broad practice, you can explore Free GRE Practice Questions to build your foundational skills.

To solve advanced problems, it is helpful to visualize the curve in segments. Since the curve is perfectly symmetrical, 50% of the data lies above the mean and 50% below. Breaking down the 68-95-99.7 rule further reveals that the area between the mean and one standard deviation is 34%. The area between one and two standard deviations is approximately 13.5%, and the area between two and three standard deviations is about 2.35%. Understanding these segments allows you to solve for any range of values provided in a question. If you are looking for a comprehensive study resource, the GRE Prep hub offers a centralized location for all quantitative and verbal topics.

Solved Examples

1. Example 1: The heights of a population are normally distributed with a mean of 170 cm and a standard deviation of 10 cm. What is the probability that a randomly selected individual is taller than 190 cm?
   1. Identify the mean $\mu = 170$ and standard deviation $\sigma = 10$.
   2. Determine how many standard deviations 190 is from the mean. Since $190 - 170 = 20$, and $20 / 10 = 2$, the value 190 is exactly $2\sigma$ above the mean.
   3. Recall the Empirical Rule: 95% of data is within $2\sigma$ of the mean ($170 \pm 20$). This means 5% of the data lies outside the range [150, 190].
   4. Because the distribution is symmetric, half of that 5% is in the upper tail. Therefore, the probability of being taller than 190 cm is $5\% / 2 = 2.5\%$.
2. Example 2: In a normally distributed dataset of 2,000 test scores with a mean of 75 and a standard deviation of 5, how many students scored between 70 and 85?
   1. The range 70 to 85 covers from $-1\sigma$ ($75 - 5$) to $+2\sigma$ ($75 + 10$).
   2. Calculate the area from $-1\sigma$ to the mean: 34%.
   3. Calculate the area from the mean to $+2\sigma$: $95\% / 2 = 47.5\%$.
   4. Total percentage = $34\% + 47.5\% = 81.5\%$.
   5. Calculate the number of students: $2,000 \times 0.815 = 1,630$.
3. Example 3: Two distributions, X and Y, are normal. X has a mean of 50 and standard deviation of 4. Y has a mean of 60 and standard deviation of 8. Which value is relatively higher: a score of 56 in X or a score of 72 in Y?
   1. Calculate the z-score for X: $z = \frac{56 - 50}{4} = \frac{6}{4} = 1.5$.
   2. Calculate the z-score for Y: $z = \frac{72 - 60}{8} = \frac{12}{8} = 1.5$.
   3. Since both z-scores are equal, both scores represent the same relative position (percentile) within their respective distributions.

Practice Questions

1. A set of measurements is normally distributed with mean $m$ and standard deviation $d$. What percent of the measurements are greater than $m + 2d$?
2. The annual rainfall in a city is normally distributed with a mean of 40 inches. If 16% of the years have rainfall greater than 45 inches, what is the standard deviation of the distribution?
3. A large batch of light bulbs has lifetimes that are normally distributed with a mean of 1,000 hours and a standard deviation of 50 hours. What is the approximate probability that a bulb chosen at random will last between 900 and 1,050 hours?

4. Quantity A: The fraction of data in a normal distribution within 0.5 standard deviations of the mean. Quantity B: 0.40.
5. The scores on a standardized test are normally distributed. If a score of 450 is 1 standard deviation below the mean and a score of 550 is 1 standard deviation above the mean, what percentage of scores are between 450 and 600?
6. A distribution of 500 values is normal with a mean of 100. If 340 values fall between 100 and 115, what is the standard deviation?
7. In a normal distribution, the 16th percentile is 40 and the 84th percentile is 60. What is the mean and standard deviation of this distribution?
8. A manufacturing process produces bolts with diameters that are normally distributed. If the mean diameter is 10.00 mm and 2.5% of bolts are rejected for being larger than 10.10 mm, what is the standard deviation?
9. If a value $x$ in a normal distribution has a z-score of -1.5, what percentage of the data is less than $x$?
10. A data set is normally distributed with mean $\mu$. If the probability of a value being between $\mu - \sigma$ and $\mu + 2\sigma$ is approximately $P$, what is the value of $P$?

Answers & Explanations

1. 2.5%: According to the Empirical Rule, 95% of the data falls within 2 standard deviations ($\pm 2d$). The remaining 5% is split equally between the two tails. Thus, 2.5% is greater than $m + 2d$.
2. 5 inches: In a normal distribution, the area above $+1\sigma$ is approximately 16% (since 68% is within $1\sigma$, 32% is outside, and 16% is in the upper tail). Since 45 inches is the 84th percentile (16% above it), 45 is $1\sigma$ above the mean. $45 - 40 = 5$, so $\sigma = 5$.
3. 81.5%: 900 is $2\sigma$ below the mean, and 1,050 is $1\sigma$ above the mean. Area from $-2\sigma$ to mean is 47.5%. Area from mean to $+1\sigma$ is 34%. $47.5\% + 34\% = 81.5\%$.
4. Quantity B is greater: Within $1\sigma$, there is 68%. Because the curve is tallest at the mean, the middle half-standard deviation ($\pm 0.5\sigma$) contains more than half of that 68%, but specifically, it is roughly 38%. Since 0.38 < 0.40, Quantity B is greater.
5. 81.5%: The mean is 500 and $\sigma = 50$. The range 450 to 600 is from $-1\sigma$ to $+2\sigma$. Area = $34\% ( \text{mean to } -1\sigma) + 47.5\% ( \text{mean to } +2\sigma) = 81.5\%$.
6. 15: 340 out of 500 is 68%. Since the distribution is symmetric, 340 values between the mean and 115 implies that 115 is $1\sigma$ above the mean (as 34% is the area from mean to $+1\sigma$). Therefore, $115 - 100 = 15$.
7. Mean = 50, SD = 10: The 16th percentile is $-1\sigma$ and the 84th percentile is $+1\sigma$. The distance between them is $2\sigma = 60 - 40 = 20$, so $\sigma = 10$. The mean is the midpoint: $(40+60)/2 = 50$.
8. 0.05 mm: 2.5% in the upper tail corresponds to the point $+2\sigma$ above the mean. So, $10.00 + 2\sigma = 10.10$. Solving for $\sigma$: $2\sigma = 0.10$, so $\sigma = 0.05$.
9. 6.7%: The area between $-1\sigma$ and $-2\sigma$ is 13.5%. A z-score of -1.5 is the midpoint. Approximately 6.7% of data lies below -1.5 standard deviations.
10. 81.5%: Area from $\mu - \sigma$ to $\mu$ is 34%. Area from $\mu$ to $\mu + 2\sigma$ is 47.5%. Total $P = 34 + 47.5 = 81.5\%$. For more complex scenarios, using an AI Exam Simulator can help you practice these multi-step probability calculations.

1. In a normal distribution, what percentage of the data falls between the mean and two standard deviations above the mean?

• A34%
• B47.5%
• C68%
• D95%

Frequently Asked Questions

What is the 68-95-99.7 rule on the GRE?

This rule describes the percentage of data that falls within one, two, and three standard deviations of the mean in a normal distribution. It is the primary tool used for solving GRE area-under-the-curve problems without a calculator.

Can I use a z-table during the GRE?

No, the GRE does not provide a z-table. You are expected to know the standard percentages associated with the 68-95-99.7 rule and apply them to solve for different ranges.

How do I identify a normal distribution question?

Look for keywords like "normally distributed," "bell curve," or mentions of specific percentiles like the 16th, 84th, or 97.5th. These values are mathematical "flags" indicating the problem relies on standard deviation properties.

What is a z-score and why is it used?

A z-score measures how many standard deviations a data point is from the mean. It is used to standardize different normal distributions so they can be compared directly, regardless of their original scales.

Is the normal distribution always symmetrical?

Yes, by definition, a true normal distribution is perfectly symmetrical. This symmetry is essential for calculating tail probabilities, as the area above the mean is always exactly 0.50.

What is the difference between standard deviation and variance?

Standard deviation is the square root of the variance. On the GRE, normal distribution problems almost exclusively use standard deviation to define the width of the bell curve.