Hard GRE Z-Score Questions Practice Questions

A z-score represents the number of standard deviations a specific data point sits above or below the mean of a distribution. When tackling Hard GRE Z-Score Questions, you must navigate beyond simple formula application to interpret percentiles, compare disparate datasets, and solve for unknown population parameters within a normal distribution.

Concept Explanation

A z-score, also known as a standard score, is a dimensionless quantity calculated by subtracting the population mean from an individual raw score and dividing the result by the population standard deviation. This statistical tool allows for the comparison of scores from different normal distributions by "standardizing" them onto a common scale where the mean is 0 and the standard deviation is 1. On the GRE, these questions often appear in the Data Interpretation or Quantitative Comparison sections, frequently requiring knowledge of the 68-95-99.7 rule (the Empirical Rule).

The fundamental formula for calculating a z-score is:

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

Where:

• $x$ is the raw score.
• $\mu$ is the population mean.
• $\sigma$ $( \text{sigma})$ is the population standard deviation.

In high-difficulty scenarios, the GRE may ask you to find the raw score from a given percentile. For example, a score at the 84th percentile of a normal distribution corresponds roughly to a z-score of $+1$, because 50% of the data is below the mean and 34% (half of 68%) is between the mean and one standard deviation above. Understanding these standard score properties is essential for success in the GRE Prep journey. For more specialized practice, you might explore GRE Reading Passage Questions to balance your study sessions.

Solved Examples

Review these worked examples to understand the logic required for multi-step z-score problems.

1. Example: Solving for the Mean
   In a normally distributed set of test scores, a score of 82 is 1.5 standard deviations above the mean. If the standard deviation is 8, what is the mean score of the distribution?
   1. Identify the given variables: $x = 82$, $z = 1.5$, and $\sigma = 8$.
   2. Plug the values into the z-score formula: $1.5 = \frac{82 - \mu}{8}$.
   3. Multiply both sides by 8: $12 = 82 - \mu$.
   4. Solve for $\mu$: $\mu = 82 - 12 = 70$.
   5. The mean is 70.
2. Example: Comparing Relative Performance
   John scored 1400 on the SAT (mean 1000, SD 200) and 30 on the ACT (mean 21, SD 5). On which test did he perform better relative to the population?
   1. Calculate SAT z-score: $z = \frac{1400 - 1000}{200} = \frac{400}{200} = 2.0$.
   2. Calculate ACT z-score: $z = \frac{30 - 21}{5} = \frac{9}{5} = 1.8$.
   3. Compare the results: Since $2.0 > 1.8$, John performed better on the SAT.
3. Example: Percentile to Raw Score
   The heights of a population are normally distributed with a mean of 170 cm and a standard deviation of 10 cm. Approximately what height represents the 16th percentile?
   1. Recall the Empirical Rule: 68% of data falls within 1 SD of the mean ($\pm 1$).
   2. Since the distribution is symmetric, 16% of the data lies below $-1$ SD (calculated as $\frac{100 - 68}{2} = 16$).
   3. Find the value at $z = -1$: $-1 = \frac{x - 170}{10}$.
   4. Solve for $x$: $-10 = x - 170 \rightarrow x = 160$.
   5. The 16th percentile height is 160 cm.

Practice Questions

Test your knowledge with these Hard GRE Z-Score Questions. Use a calculator only if necessary, as the GRE often uses numbers that simplify logically.

1. A set of data is normally distributed with mean $m$ and standard deviation $d$. If a value $x$ has a z-score of 2.4, what is the z-score of $x + 2d$?
2. In a normal distribution, approximately what percentage of the data falls between $z = -1$ and $z = +2$?
3. The weight of organic apples is normally distributed with a mean of 150 grams. If 2.5% of the apples weigh more than 190 grams, what is the standard deviation of the distribution?

4. Quantity A: The z-score of a value 1.2 standard deviations below the mean.
   Quantity B: -1.3.
5. A distribution of scores has a mean of 50 and a standard deviation of 5. If every score in the distribution is increased by 10, what happens to the z-score of the original maximum value?
6. A machine produces metal rods with a mean length of 12 cm. It is known that 16% of the rods are shorter than 11.7 cm. What is the z-score of a rod that is 12.6 cm long?
7. Two students, Amy and Ben, are in different classes. Amy's score of 85 is 1.2 standard deviations above her class mean of 79. Ben's score of 85 is 1.5 standard deviations above his class mean. What is the mean of Ben's class if both classes have the same standard deviation?
8. In a normal distribution, if the 97.5th percentile is 120 and the 50th percentile is 100, find the value of the standard deviation.
9. If a value $y$ in a normal distribution is such that $P(X > y) = 0.84$, what is the z-score of $y$?

Answers & Explanations

1. 4.4: The z-score measures how many standard deviations a value is from the mean. If $x$ is 2.4 SDs above the mean, adding $2d$ (two more standard deviations) makes it $2.4 + 2 = 4.4$ SDs above the mean.
2. 81.5%: Using the Empirical Rule, the area between $z = -1$ and $z = 0$ is 34%. The area between $z = 0$ and $z = 1$ is 34%. The area between $z = 1$ and $z = 2$ is 13.5%. Total: $34 + 34 + 13.5 = 81.5\%$.
3. 20 grams: In a normal distribution, 2.5% of data is more than 2 SDs above the mean (since 95% is within 2 SDs, 5% is outside, and 2.5% is in the upper tail). Thus, $190 = 150 + 2\sigma$. $40 = 2\sigma$, so $\sigma = 20$.
4. Quantity A is greater: Quantity A is -1.2. Quantity B is -1.3. Since -1.2 is to the right of -1.3 on the number line, -1.2 is greater.
5. It remains the same: If every value and the mean increase by the same constant, the difference $(x - \mu)$ remains the same. Since the standard deviation also remains unchanged by addition, the z-score $\frac{x - \mu}{\sigma}$ does not change.
6. 2.0: 16% being shorter than 11.7 cm implies 11.7 is at $z = -1$. Thus, $12 - 11.7 = 0.3$, so $\sigma = 0.3$. For a rod of 12.6 cm: $z = \frac{12.6 - 12}{0.3} = \frac{0.6}{0.3} = 2.0$.
7. 77.5: First find the SD from Amy: $1.2 = \frac{85 - 79}{\sigma} \rightarrow 1.2 = \frac{6}{\sigma} \rightarrow \sigma = 5$. Use this for Ben: $1.5 = \frac{85 - \mu}{5} \rightarrow 7.5 = 85 - \mu \rightarrow \mu = 77.5$.
8. 10: The 50th percentile is the mean (100). The 97.5th percentile is 2 SDs above the mean. So, $100 + 2\sigma = 120$. $2\sigma = 20$, meaning $\sigma = 10$.
9. -1.0: If 84% of the area is to the right, then 16% is to the left. As established by the Empirical Rule, the 16th percentile corresponds to $z = -1$.

Studying for the GRE requires more than just memorizing formulas; you need to practice with GRE Practice Questions with Explanations to internalize the logic. Using tools like an AI Exam Simulator can help simulate the pressure of the actual test environment.

1. If a data point has a z-score of 0, which of the following must be true?

• AThe data point is equal to the mean
• BThe standard deviation is 0
• CThe data point is the maximum value
• DThe distribution is skewed

Frequently Asked Questions

What is the difference between a z-score and a standard deviation?

Standard deviation is a measure of the total spread of a dataset, whereas a z-score is a measure of a specific data point's position relative to the mean in units of standard deviation. Standard deviation applies to the whole group, while a z-score applies to a single value.

Can a z-score be negative?

Yes, a negative z-score indicates that the raw data point is below the mean of the distribution. For example, a z-score of -2.0 means the value is two standard deviations lower than the average.

How do I find the z-score if the GRE does not provide a table?

The GRE typically expects you to know the Empirical Rule (68-95-99.7) to estimate z-scores for percentiles. You will not need a full z-table; instead, focus on the values for 1, 2, and 3 standard deviations.

Does the z-score formula change for samples versus populations?

The logic remains the same, but the notation changes: for a sample, the mean is $\bar{x}$ and the standard deviation is $s$. On the GRE, the context usually involves population parameters unless otherwise specified.

Why is the z-score useful for comparing different tests?

Because different tests have different scales (like a 1600-point SAT vs. a 36-point ACT), the z-score provides a common language. It tells you how "rare" a score is relative to its own group, allowing for an apples-to-apples comparison.