GRE Normal Distribution Questions Practice Questions with Answers

Concept Explanation

A normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve where the mean, median, and mode all coincide at the center of the data. In the context of the GRE, this concept is pivotal for Quantitative Reasoning because it allows test-takers to predict the percentage of data points that fall within specific ranges using the 68-95-99.7 rule. This rule, often referred to as the Empirical Rule, states that approximately 68% of the data falls within one standard deviation $(\sigma)$ of the mean $(\mu)$, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

When you encounter GRE Normal Distribution Questions, you are usually expected to visualize the bell curve. Because the curve is perfectly symmetrical, exactly 50% of the values lie above the mean and 50% lie below the mean. For more complex calculations, understanding the mathematical properties of the normal distribution is helpful, but the GRE typically sticks to these standard increments. If you are looking to build a comprehensive study schedule, you might use an AI MasterPlan to allocate time for statistics and data analysis topics.

Range from Mean	Approximate Percentage of Data
Mean to $\pm 1\sigma$	68% (34% on each side)
Mean to $\pm 2\sigma$	95% (47.5% on each side)
Mean to $\pm 3\sigma$	99.7% (49.85% on each side)

To succeed on these questions during your GRE Prep, remember that the standard deviation measures the spread. A smaller standard deviation results in a taller, narrower bell, while a larger one creates a flatter curve. If you find these concepts challenging, using a AI Question Generator can provide endless variations of these problems to sharpen your skills.

Solved Examples

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. $70 = 75 - 5$, which is $-1\sigma$. $80 = 75 + 5$, which is $+1\sigma$.
3. Apply the Empirical Rule: The range from $-1\sigma$ to $+1\sigma$ contains approximately 68% of the data.
4. The answer is 68%.

Example 2: In a normally distributed population of 1,000 light bulbs, the mean life is 1,200 hours with a standard deviation of 100 hours. How many bulbs are expected to last more than 1,400 hours?

1. Calculate the distance from the mean in standard deviations: $\frac{1400 - 1200}{100} = 2\sigma$.
2. Recall that 95% of data falls between $-2\sigma$ and $+2\sigma$. This means 5% falls outside this range (2.5% in each tail).
3. The area "more than 1,400 hours" represents the upper tail beyond $+2\sigma$, which is 2.5%.
4. Calculate the number of bulbs: $0.025 \times 1000 = 25$.
5. The answer is 25 bulbs.

Example 3: A normal distribution has a mean of 50. If the value 62 is at the 97.5th percentile, what is the standard deviation?

1. The 97.5th percentile means 97.5% of the data is below this value. Since 50% is below the mean, 47.5% is between the mean and 62.
2. A range of 47.5% from the mean corresponds to exactly $2\sigma$ (half of 95%).
3. Set up the equation: $\mu + 2\sigma = 62$.
4. Substitute the mean: $50 + 2\sigma = 62 \rightarrow 2\sigma = 12 \rightarrow \sigma = 6$.
5. The answer is 6.

Practice Questions

1. A distribution of heights is normal with a mean of 170 cm and a standard deviation of 10 cm. What is the probability that a randomly selected individual is shorter than 160 cm?

2. In a normal distribution, approximately what percentage of the values are greater than the mean plus one standard deviation?

3. A manufacturer produces bolts with a mean diameter of 10 mm and a standard deviation of 0.2 mm. If the diameters are normally distributed, what range of diameters contains the middle 95% of the bolts?

4. Quantity A: The percentage of data between the mean and 1 standard deviation above the mean. Quantity B: 35%.

5. A set of 2,000 values is normally distributed. If 320 values are greater than 84 and the mean is 70, what is the approximate standard deviation?

6. If a value $x$ in a normal distribution has a z-score of -3, what percentage of the data is less than $x$?

7. A normal distribution has a mean of $m$ and a standard deviation of $d$. What percentage of values fall in the interval $[m - 2d, m + d]$?

8. If the 16th percentile of a normal distribution is 45 and the mean is 50, what is the 84th percentile?

Answers & Explanations

1. 16%. The value 160 is exactly one standard deviation below the mean $(170 - 10 = 160)$. We know 68% falls within $\pm 1\sigma$, leaving 32% in the tails. Since the curve is symmetric, 16% is in the lower tail below 160.

2. 16%. Half of the data (50%) is above the mean. We know 34% of the data is between the mean and $+1\sigma$. Therefore, the portion greater than $+1\sigma$ is $50\% - 34\% = 16\%$.

3. 9.6 mm to 10.4 mm. The middle 95% corresponds to $\pm 2\sigma$. Since $\sigma = 0.2$, $2\sigma = 0.4$. The range is $10 - 0.4$ to $10 + 0.4$.

4. Quantity B is greater. In a normal distribution, the area between the mean and $+1\sigma$ is approximately 34.1%. Since 34.1% is less than 35%, Quantity B is larger.

5. 14. 320 out of 2,000 is 16%. If 16% of the data is greater than 84, then 84 must be exactly $+1\sigma$ from the mean because 16% of data lies above $+1\sigma$. If $70 + \sigma = 84$, then $\sigma = 14$.

6. 0.15%. A z-score of -3 is 3 standard deviations below the mean. The Empirical Rule states 99.7% of data is within $\pm 3\sigma$. The remaining 0.3% is split between the two tails, so the lower tail is $0.3\% / 2 = 0.15\%$.

7. 81.5%. The interval from $m - 2d$ to the mean $m$ covers 47.5% of the data. The interval from the mean $m$ to $m + d$ covers 34%. Adding these together: $47.5\% + 34\% = 81.5\%$.

8. 55. The 16th percentile is $1\sigma$ below the mean (since 50% - 34% = 16%). This means $\sigma = 50 - 45 = 5$. The 84th percentile is $1\sigma$ above the mean (50% + 34% = 84%), so $50 + 5 = 55$.

1. In a normal distribution, what percentage of data points fall between the mean and two standard deviations below the mean?

• A34%
• B47.5%
• C68%
• D95%

Frequently Asked Questions

What is the 68-95-99.7 rule?

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 normal distribution problems without a calculator.

Can the standard deviation be negative?

No, the standard deviation is always a non-negative value because it is the square root of the variance. It represents a distance or spread, which cannot be less than zero.

How do I calculate a z-score on the GRE?

While the GRE rarely requires the formal z-score formula, it is calculated as $z = \frac{x - \text{mean}}{ \text{standard deviation}}$. This value tells you exactly how many standard deviations an observation is from the average.

Is every bell-shaped curve a normal distribution?

Not necessarily, as a distribution must meet specific mathematical criteria regarding its density function to be truly "normal." However, for the purposes of the GRE, if a question mentions a bell-shaped or symmetric distribution, you should apply normal distribution properties.

What is the difference between mean and median in a normal distribution?

In a perfectly normal distribution, there is no difference between the mean and the median. They are equal and located at the exact center of the distribution curve.