Easy GRE Normal Distribution Questions Practice Questions

Concept Explanation

A normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve where the mean, median, and mode are all equal and located at the center. In the context of Easy GRE Normal Distribution Questions, the most critical concept to internalize is the 68-95-99.7 Rule, often referred to as the Empirical Rule. This rule dictates how data is dispersed around the mean in terms of standard deviations. This fundamental statistical concept is a staple of the GRE Prep curriculum because it allows test-takers to calculate proportions of a population without complex calculus.

On the GRE, you will typically encounter questions that assume a distribution is normal. You should be familiar with the following percentage breakdowns for a normal distribution with mean $\mu$ and standard deviation $\sigma$:

• Approximately 68% of the data falls within 1 standard deviation of the mean: $(\mu - \sigma, \mu + \sigma)$. This means 34% lies between the mean and one standard deviation above it.
• Approximately 95% of the data falls within 2 standard deviations of the mean: $(\mu - 2\sigma, \mu + 2\sigma)$.
• Approximately 99.7% of the data falls within 3 standard deviations of the mean: $(\mu - 3\sigma, \mu + 3\sigma)$.

Because the curve is perfectly symmetrical, exactly 50% of the data lies above the mean and 50% lies below the mean. Many easy-level questions involve simply adding or subtracting these known percentages to find the area under the curve for a specific range. For more interactive practice, you can use the AI Question Generator to produce variations of these distribution problems.

Solved Examples

Review these worked examples to understand the logic required for standard normal distribution problems on the GRE.

1. Example 1: A set of test scores is normally distributed with a mean of 75 and a standard deviation of 5. What percentage of students scored between 70 and 80?
   1. Identify the mean $\mu = 75$ and standard deviation $\sigma = 5$.
   2. Determine how many standard deviations the boundaries are from the mean. 70 is $75 - 5$ (1 SD below) and 80 is $75 + 5$ (1 SD above).
   3. Apply the 68-95-99.7 rule. The range $\mu \pm 1\sigma$ contains approximately 68% of the data.
   4. Answer: 68%.
2. Example 2: In a normal distribution with mean 100 and standard deviation 10, what is the probability that a randomly selected value is greater than 120?
   1. Identify that 120 is 2 standard deviations above the mean ($100 + 2 \times 10$).
   2. Recall that 95% of data is within 2 standard deviations. This leaves 5% in the "tails" (the areas outside 2 SDs).
   3. Since the curve is symmetric, half of that 5% is in the upper tail and half is in the lower tail.
   4. Calculate $5\% \div 2 = 2.5\%$.
   5. Answer: 2.5%.
3. Example 3: A population of heights is normally distributed with a mean of 66 inches. If 16% of the population is taller than 70 inches, what is the standard deviation?
   1. If 16% are above 70 inches, then 34% must be between the mean (50th percentile) and 70 inches ($50\% - 16\% = 34\%$).
   2. In a normal distribution, the area between the mean and 1 standard deviation above the mean is approximately 34%.
   3. Therefore, 70 inches must be exactly 1 standard deviation above the mean.
   4. Set up the equation: $66 + 1\sigma = 70$. Solve for $\sigma = 4$.
   5. Answer: 4 inches.

Practice Questions

Test your knowledge with these Easy GRE Normal Distribution Questions. Remember to use the properties of the bell curve for every calculation.

1. A distribution of weights is normal with a mean of 150 lbs and a standard deviation of 15 lbs. What percent of the weights are between 135 lbs and 165 lbs?

2. In a normal distribution, approximately what percent of the data is greater than the mean?

3. If a set of measurements is normally distributed with mean $\mu$ and standard deviation $\sigma$, what percentage of the data falls between $\mu - 2\sigma$ and $\mu$?

4. A factory produces lightbulbs with a life expectancy that is normally distributed. The mean life is 1,000 hours with a standard deviation of 50 hours. What fraction of bulbs lasts longer than 1,100 hours?

5. Quantity A: The percentage of data within 1 standard deviation of the mean in a normal distribution. Quantity B: 70%. Which is greater?

6. In a normal distribution with mean 50 and standard deviation 4, what value corresponds to the 16th percentile?

7. If 97.5% of a normally distributed population is less than 80, and the standard deviation is 5, what is the mean?

8. What is the approximate percentage of data that falls between 1 and 2 standard deviations above the mean?

Answers & Explanations

1. 68%: 135 is one SD below the mean and 165 is one SD above the mean. According to the Empirical Rule, 68% of data falls within $\pm 1$ SD.
2. 50%: By definition, the normal distribution is symmetric around the mean, meaning exactly half of the data is above and half is below.
3. 47.5%: We know 95% of data is between $\mu - 2\sigma$ and $\mu + 2\sigma$. Since the curve is symmetric, the area from the mean to 2 SDs is half of 95%, which is 47.5%.
4. 1/40 (or 2.5%): 1,100 hours is 2 standard deviations above the mean ($1000 + 2 \times 50$). Since 95% is within 2 SDs, 5% is outside. Only the top half (2.5%) is greater than 1,100.
5. Quantity B: Quantity A is approximately 68% (the area within 1 SD). Since 70% is greater than 68%, Quantity B is larger.
6. 46: The 16th percentile is 34% below the mean ($50\% - 16\% = 34\%$). This corresponds to exactly 1 standard deviation below the mean. $50 - 4 = 46$.
7. 70: 97.5% of the data is less than 2 standard deviations above the mean (50% below mean + 47.5% between mean and +2 SD). Thus, 80 is 2 SDs above the mean. $\mu + 2(5) = 80 \rightarrow \mu = 70$.
8. 13.5%: Approximately 34% is between the mean and 1 SD. Approximately 47.5% is between the mean and 2 SDs. The difference is $47.5\% - 34\% = 13.5\%$.

Frequently Asked Questions

What is the 68-95-99.7 rule?

This rule, also known as the Empirical Rule, describes the percentage of data that falls within one, two, and three standard deviations of the mean in a normal distribution. It is a fundamental tool for solving GRE statistics problems quickly without a calculator.

Can the standard deviation be negative in a normal distribution?

No, the standard deviation represents a measure of distance or spread and is always a non-negative value. If a GRE question mentions a normal distribution, the standard deviation will always be a positive number.

How does the GRE test normal distributions?

The GRE typically tests your ability to apply the Empirical Rule to find percentages, percentiles, or specific data values. You might also see Quantitative Comparison questions asking you to compare areas under different parts of the curve.

What is a z-score?

A z-score is a numerical measurement that describes a value's relationship to the mean in terms of standard deviations. A z-score of 0 is at the mean, while a z-score of 1.5 is one and a half standard deviations above the mean.

Is every bell-shaped curve a normal distribution?

While all normal distributions are bell-shaped, not all bell-shaped curves are normal distributions. However, for the purposes of the GRE, if the test describes a distribution as "normal," you should apply the specific properties of the Gaussian distribution, as detailed by mathematical theory.