GRE Z-Score Questions Practice Questions with Answers

Concept Explanation

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. In the context of the GRE, z-scores are fundamental to understanding the normal distribution, often referred to as the bell curve. When you calculate a z-score, you are essentially "standardizing" a raw score to see how many standard deviations it falls above or below the average. This allows for a direct comparison between different sets of data that may have different units or scales.

The formula for calculating a z-score is:

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

In this formula, $x$ represents the raw score, $\mu$ (mu) is the population mean, and $\sigma$ (sigma) is the standard deviation. 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 +1.0 means the value is exactly one standard deviation higher than the average. According to the empirical rule (or the 68-95-99.7 rule), approximately 68% of data in a normal distribution falls within one standard deviation of the mean (z-scores between -1 and 1). You can find more in-depth statistical breakdowns on Wikipedia's standard score page.

On the GRE, you will likely encounter GRE Z-Score Questions within the Data Interpretation or Quantitative Comparison sections. You may be asked to find the percentage of a population that falls above or below a certain score, or to compare two different scores from two different distributions. Efficiently managing these concepts is a core part of GRE Prep. Understanding that the area under the normal curve represents probability is the key to solving these problems quickly without a calculator.

Solved Examples

Review these step-by-step solutions to understand how to apply the z-score formula and normal distribution properties in a testing environment.

1. Example 1: Basic Calculation
   A set of test scores is normally distributed with a mean of 75 and a standard deviation of 5. What is the z-score for a student who scored an 85?
   

   1. Identify the variables: $x = 85$, $\mu = 75$, $\sigma = 5$.

   2. Apply the formula: $z = \ \frac{85 - 75}{5}$.

   3. Simplify: $z = \ \frac{10}{5} = 2$.

   4. The z-score is 2.0, meaning the score is 2 standard deviations above the mean.

2. Example 2: Finding a Raw Score
   In a distribution where the mean is 100 and the standard deviation is 15, what value corresponds to a z-score of -1.2?
   

   1. Identify the variables: $z = -1.2$, $\mu = 100$, $\sigma = 15$.

   2. Set up the equation: $-1.2 = \ \frac{x - 100}{15}$.

   3. Multiply both sides by 15: $-18 = x - 100$.

   4. Add 100 to both sides: $x = 82$.

   5. The raw score is 82.

3. Example 3: Percentile Comparison
   Heights of adult males are normally distributed. If a height of 70 inches corresponds to a z-score of 0, what percentage of males are taller than 70 inches?
   

   1. Recall that a z-score of 0 represents the mean.

   2. In a normal distribution, the mean is equal to the median.

   3. By definition, 50% of the data lies above the median.

   4. Therefore, 50% of males are taller than 70 inches.

Practice Questions

Test your knowledge with these GRE-style problems ranging from simple calculations to complex data interpretations.

1. A population of light bulbs has a mean life of 1,000 hours with a standard deviation of 50 hours. What is the z-score for a bulb that lasts 925 hours?

2. If a data point $x = 45$ in a distribution with $\mu = 50$ results in a z-score of -2.5, what is the standard deviation $\sigma$?

3. In a normal distribution, approximately what percentage of the data falls between a z-score of -1 and a z-score of +1?

4. A student takes two different exams. On the Math exam (mean 70, SD 10), the student scores 85. On the English exam (mean 60, SD 5), the student scores 72. On which exam did the student perform better relative to the other test-takers?

5. The weight of bags of flour is normally distributed with a mean of 5 lbs and a standard deviation of 0.1 lbs. What is the z-score for a bag weighing 5.15 lbs?

6. Quantity A: The z-score of a value 1.5 standard deviations below the mean.
   Quantity B: -1.5

7. A distribution has a mean of 200. If the value 230 has a z-score of 3, what is the value that would have a z-score of -2?

8. In a normal distribution, what is the probability (expressed as a percentage) that a randomly selected score is greater than the mean?

9. If the standard deviation of a set is 0, and the mean is 50, what is the z-score for the value 50?

10. A company's salaries are normally distributed with a mean of $40,000. If a salary of $55,000 has a z-score of 1.5, what is the standard deviation?

Answers & Explanations

1. -1.5: Using the formula $z = \ \frac{925 - 1000}{50} = \ \frac{-75}{50} = -1.5$.

2. 2: Use the formula $-2.5 = \ \frac{45 - 50}{\sigma}$. This becomes $-2.5 = \ \frac{-5}{\sigma}$, so $\sigma = \ \frac{-5}{-2.5} = 2$.

3. 68%: This is a standard property of the normal distribution (the empirical rule). For more on this, visit Khan Academy's stats section.

4. English Exam: Math z-score = $\ \frac{85 - 70}{10} = 1.5$. English z-score = $\ \frac{72 - 60}{5} = 2.4$. Since 2.4 > 1.5, the student performed better in English relative to the group.

5. 1.5: $z = \ \frac{5.15 - 5.0}{0.1} = \ \frac{0.15}{0.1} = 1.5$.

6. The two quantities are equal: By definition, a value "1.5 standard deviations below the mean" has a z-score of -1.5.

7. 180: First find $\sigma$: $3 = \ \frac{230 - 200}{\sigma} \ \rightarrow 3 = \ \frac{30}{\sigma} \ \rightarrow \sigma = 10$. Then find the value for z = -2: $-2 = \ \frac{x - 200}{10} \ \rightarrow -20 = x - 200 \ \rightarrow x = 180$.

8. 50%: In a perfectly normal distribution, the mean is the 50th percentile.

9. Undefined/Indeterminate: If the standard deviation is 0, all values in the set must be equal to the mean. The formula would involve division by zero, though conceptually, the value is exactly the mean.

10. $10,000: $1.5 = \ \frac{55,000 - 40,000}{\sigma} \ \rightarrow 1.5 = \ \frac{15,000}{\sigma} \ \rightarrow \sigma = 10,000$.

To further refine your skills with complex data sets, you might try the AI Question Generator to create custom practice sets based on these specific salary or height scenarios.

1. If a z-score is exactly 0, what does this indicate about the raw score?

• AThe score is the highest in the set
• BThe score is equal to the mean
• CThe score is equal to the standard deviation
• DThe score is invalid

Frequently Asked Questions

Can a z-score be used for non-normal distributions?

While you can calculate a z-score for any distribution to see how many standard deviations a point is from the mean, the percentile interpretations (like the 68-95-99.7 rule) only apply to normal distributions. For non-normal data, z-scores are less useful for determining specific probabilities.

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

A positive z-score indicates that the data point is above the mean, whereas a negative z-score indicates that the data point is below the mean. The magnitude of the number tells you how far away from the mean the point sits in units of standard deviation.

Does the GRE provide a z-table?

No, the GRE does not provide a z-table during the exam. Instead, you are expected to know the common percentages associated with the normal distribution, such as 68% for 1 SD and 95% for 2 SDs, to solve GRE Z-Score Questions.

Are z-scores and standard deviations the same thing?

No, they are different concepts; the standard deviation is a measure of spread for an entire dataset, while a z-score is a measure of position for a single data point. The z-score uses the standard deviation as a unit of measurement to describe that position.

Why is standardizing data with z-scores helpful?

Standardizing data allows researchers and students to compare scores from different scales or metrics on a single, common scale. For example, it allows you to compare a student's performance on a 100-point math test with their performance on a 1600-point SAT exam.