Easy GRE Z-Score Questions Practice Questions

A z-score represents the number of standard deviations a specific data point is from the mean of its distribution. Understanding how to calculate and interpret these values is a foundational skill for the quantitative reasoning portion of the GRE, especially when dealing with normal distributions. By converting raw scores into standard units, test-takers can compare disparate datasets, such as comparing a student's performance on a math test versus a verbal test. This article provides a comprehensive set of easy GRE z-score questions practice questions to help you build confidence in this essential statistical concept.

For more general practice, you might also find our guide to free GRE practice questions helpful as you build your study schedule. If you are targeting specific sections, check out GRE reading passage questions to round out your preparation.

Concept Explanation

A z-score, also known as a standard score, is a numerical measurement that describes a value's relationship to the mean of a group of values. It is calculated by taking the difference between a data point and the mean, then dividing that difference by the standard deviation. The formula for the z-score is written as:

$$z = \frac{x - \mu}{\sigma}$$

In this formula, $x$ represents the raw score, $\mu$ (mu) is the population mean, and $\sigma$ (sigma) is the population standard deviation. A z-score of 0 indicates that the data point is exactly equal to the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean. For example, a z-score of +1.5 means the value is 1.5 standard deviations above the average.

On the GRE, you will often apply this to the Empirical Rule (or the 68-95-99.7 rule). According to Wikipedia's entry on the 68-95-99.7 rule, approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. Grasping these percentages allows you to quickly solve many GRE-style problems without complex calculators. You can use AI-powered flashcards to memorize these common percentage distributions for faster recall during the exam.

Solved Examples

1. Basic Calculation: A set of test scores has a mean of 70 and a standard deviation of 5. What is the z-score for a student who scored 80?
   1. Identify the variables: $x = 80$, $\mu = 70$, $\sigma = 5$.
   2. Apply the formula: $z = \frac{80 - 70}{5}$.
   3. Calculate the numerator: $80 - 70 = 10$.
   4. Divide by the standard deviation: $\frac{10}{5} = 2$.
   5. The z-score is +2.
2. Finding the Raw Score: In a distribution with a mean of 100 and a standard deviation of 15, what value corresponds to a z-score of -1?
   1. Identify the variables: $z = -1$, $\mu = 100$, $\sigma = 15$.
   2. Set up the equation: $-1 = \frac{x - 100}{15}$.
   3. Multiply both sides by 15: $-15 = x - 100$.
   4. Add 100 to both sides: $x = 85$.
   5. The raw score is 85.
3. Comparing Two Scores: Alice scored 130 on a test where the mean was 100 and the standard deviation was 20. Bob scored 440 on a test where the mean was 400 and the standard deviation was 50. Relative to their respective groups, who performed better?
   1. Calculate Alice's z-score: $z_A = \frac{130 - 100}{20} = \frac{30}{20} = 1.5$.
   2. Calculate Bob's z-score: $z_B = \frac{440 - 400}{50} = \frac{40}{50} = 0.8$.
   3. Compare the results: Since 1.5 is greater than 0.8, Alice performed better relative to her group.

Practice Questions

1. A distribution has a mean of 50 and a standard deviation of 8. What is the z-score for the value 42?
2. A researcher finds that a specific data point has a z-score of 3.0. If the mean is 200 and the standard deviation is 10, what is the value of this data point?
3. If a data point is exactly 2.5 standard deviations below the mean, what is its z-score?

4. A group of runners has a mean finishing time of 240 minutes with a standard deviation of 20 minutes. What is the z-score for a runner who finishes in 210 minutes?
5. In a normal distribution, what percentage of data falls between a z-score of -1 and a z-score of +1?
6. A standardized test has a mean of 500 and a standard deviation of 100. If a student has a z-score of 0, what was their actual score?
7. If the z-score of a value is -2.0, the mean is 60, and the value is 50, what is the standard deviation?
8. Which z-score indicates a value further away from the mean: -2.5 or +2.1?

To master these types of quantitative problems, you can use an AI exam simulator to practice under timed conditions similar to the actual GRE. You might also want to explore GRE practice questions with explanations for more detailed walkthroughs.

Answers & Explanations

1. Answer: -1.0. Using the formula $z = \frac{42 - 50}{8}$, we get $\frac{-8}{8} = -1$. This means the value is one standard deviation below the mean.
2. Answer: 230. Using the formula $3 = \frac{x - 200}{10}$. Multiply 3 by 10 to get 30, then add 200. $x = 230$.
3. Answer: -2.5. By definition, the z-score is the number of standard deviations from the mean. Since it is "below" the mean, the sign is negative.
4. Answer: -1.5. Calculate $z = \frac{210 - 240}{20} = \frac{-30}{20} = -1.5$.
5. Answer: Approximately 68%. According to the Empirical Rule for normal distributions, roughly 68% of data points lie within one standard deviation ($z = \pm 1$) of the mean.
6. Answer: 500. A z-score of 0 always indicates that the raw score is exactly equal to the mean.
7. Answer: 5. Use the formula $-2 = \frac{50 - 60}{\sigma}$. This simplifies to $-2 = \frac{-10}{\sigma}$. Solving for $\sigma$, we get $\sigma = \frac{-10}{-2} = 5$.
8. Answer: -2.5. Distance from the mean is measured by the absolute value of the z-score. Since $|-2.5| = 2.5$ and $|2.1| = 2.1$, the score of -2.5 is further away.

1. What does a z-score of +1.0 represent in a normal distribution?

• AThe value is equal to the mean
• BThe value is 1 unit above the mean
• CThe value is 1 standard deviation above the mean
• DThe value is 10% higher than the mean

Frequently Asked Questions

What is the difference between a positive and negative z-score?

A positive z-score indicates that the data point is higher than the mean, while a negative z-score indicates that the data point is lower than the mean. The sign tells you the direction of the deviation from the center of the dataset.

Can a z-score be greater than 3?

Yes, a z-score can be any real number, including those greater than 3 or less than -3. However, in a normal distribution, values with z-scores beyond 3 are very rare, occurring less than 0.3% of the time.

Is a higher z-score always better?

Not necessarily, as it depends on the context of the data. While a high z-score is good for test results, a high z-score for something like "minutes to complete a task" might indicate slower-than-average performance.

How does the GRE typically test z-scores?

The GRE usually tests z-scores through Quantitative Comparison questions or Data Interpretation sets. You are often expected to know the percentages associated with 1, 2, and 3 standard deviations in a normal distribution.

What happens to the z-score if the standard deviation increases?

If the distance between the raw score and the mean stays the same but the standard deviation increases, the absolute value of the z-score will decrease. This is because the raw difference is being divided by a larger number.