Z-Score Practice Questions with Answers

Concept Explanation

A z-score, also known as a standard score, is a numerical value that describes a data point's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. If a z-score is 0, it indicates that the data point's score is identical to the mean score. A z-score of 1.0 indicates a value that is one standard deviation above the mean, while a z-score of -1.0 indicates a value that is one standard deviation below the mean.

The mathematical formula for calculating a z-score is:

z = (x – μ) / σ

Where:

• x is the raw score or individual data point.

• μ (mu) is the population mean.

• σ (sigma) is the population standard deviation.

Understanding z-scores is fundamental in statistics because it allows researchers to compare data from different distributions. For instance, if you want to compare a student's performance on a chemistry exam with their performance on a biology exam, you cannot simply look at the raw scores if the exams had different difficulty levels or grading scales. By converting both scores to z-scores, you can see which performance was higher relative to the rest of the class. This process is often called "standardizing" the data.

Z-scores are deeply connected to the Normal Distribution (or Bell Curve). In a standard normal distribution, approximately 68% of data points fall between z-scores of -1 and +1, 95% fall between -2 and +2, and 99.7% fall between -3 and +3. This is known as the Empirical Rule. You can learn more about how data distribution impacts scientific analysis in our guide on Mass Spectrometry Practice Questions, where data interpretation is key.

Solved Examples

The following examples demonstrate how to apply the z-score formula in various scenarios, from basic calculations to finding raw scores.

1. Basic Z-Score Calculation: A student scores 85 on a test where the mean (μ) is 75 and the standard deviation (σ) is 5. What is the student's z-score?

   • Identify the variables: x = 85, μ = 75, σ = 5.

   • Apply the formula: z = (85 - 75) / 5.

   • Subtract the mean from the raw score: 10.

   • Divide by the standard deviation: 10 / 5 = 2.0.

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

2. Negative Z-Score Calculation: A specific brand of lightbulb has a mean life of 1200 hours with a standard deviation of 100 hours. If a bulb lasts 1050 hours, what is its z-score?

   • Identify the variables: x = 1050, μ = 1200, σ = 100.

   • Apply the formula: z = (1050 - 1200) / 100.

   • Subtract the mean: -150.

   • Divide by the standard deviation: -150 / 100 = -1.5.

   • The z-score is -1.5, indicating it lasted less than the average bulb.

3. Finding the Raw Score (x): A distribution has a mean of 50 and a standard deviation of 8. If a value has a z-score of 1.25, what is the raw score?

   • Identify the variables: z = 1.25, μ = 50, σ = 8.

   • Rearrange the formula: x = μ + (z * σ).

   • Calculate: x = 50 + (1.25 * 8).

   • Multiply: 1.25 * 8 = 10.

   • Add to the mean: 50 + 10 = 60.

   • The raw score is 60.

4. Comparing Two Different Distributions: Sarah scored 1300 on the SAT (Mean = 1000, SD = 200) and 30 on the ACT (Mean = 21, SD = 5). On which test did she perform better?

   • Calculate SAT z-score: (1300 - 1000) / 200 = 300 / 200 = 1.5.

   • Calculate ACT z-score: (30 - 21) / 5 = 9 / 5 = 1.8.

   • Compare: Since 1.8 > 1.5, Sarah performed better on the ACT relative to other test-takers.

Practice Questions

1. A dataset has a mean of 100 and a standard deviation of 15. Calculate the z-score for a value of 130.

2. If a population mean is 40 and the standard deviation is 4, what is the z-score for a value of 34?

3. A researcher finds that a value of 55 corresponds to a z-score of 0.5 in a distribution with a standard deviation of 10. What is the mean of this distribution?

4. In a normal distribution with μ = 200 and σ = 25, find the raw score that corresponds to a z-score of -2.4.

5. Two students, Alex and Jordan, are in different math classes. Alex scored 88 (Class Mean = 80, SD = 4). Jordan scored 92 (Class Mean = 86, SD = 6). Who had the better score relative to their class? (Hint: Use z-scores).

6. A manufacturer produces steel rods with a mean length of 5 meters and a standard deviation of 0.02 meters. A rod is measured at 5.05 meters. Calculate the z-score and state if this is an outlier (usually defined as |z| > 3).

7. A standardized test has a mean of 500 and a standard deviation of 100. What raw score is required to be in the top 2.5% of test-takers? (Note: Top 2.5% corresponds to a z-score of approximately +1.96).

8. If the z-score for a value of 12 is -1.5 and the mean is 18, what is the standard deviation?

9. A data point x = 45 has a z-score of 0. If the standard deviation is 5, what is the mean?

10. A distribution of heights has a mean of 170 cm and a standard deviation of 10 cm. What is the z-score for a height of 165 cm?

Answers & Explanations

1. Answer: 2.0. Calculation: (130 - 100) / 15 = 30 / 15 = 2.

2. Answer: -1.5. Calculation: (34 - 40) / 4 = -6 / 4 = -1.5.

3. Answer: 50. Use the formula x = μ + zσ. 55 = μ + (0.5 * 10). 55 = μ + 5. μ = 50.

4. Answer: 140. Calculation: x = 200 + (-2.4 * 25) = 200 - 60 = 140.

5. Answer: Alex. Alex z-score: (88-80)/4 = 2. Jordan z-score: (92-86)/6 = 1. Alex's z-score is higher.

6. Answer: 2.5; Not an outlier. Calculation: (5.05 - 5.00) / 0.02 = 0.05 / 0.02 = 2.5. Since 2.5 is not greater than 3, it is not considered a standard outlier.

7. Answer: 696. Calculation: x = 500 + (1.96 * 100) = 500 + 196 = 696.

8. Answer: 4. Formula: z = (x - μ) / σ. -1.5 = (12 - 18) / σ. -1.5 = -6 / σ. σ = -6 / -1.5 = 4.

9. Answer: 45. A z-score of 0 always indicates that the raw score is exactly equal to the mean.

10. Answer: -0.5. Calculation: (165 - 170) / 10 = -5 / 10 = -0.5.

Quick Quiz

Frequently Asked Questions

What is a "good" z-score?

A "good" z-score depends entirely on the context of the data; in testing, a positive z-score is usually better, whereas in a measure of error or risk, a negative z-score is preferred. Generally, a z-score above 0 means you are above average.

Can a z-score be greater than 3?

Yes, z-scores can be greater than 3, though they are statistically rare in a normal distribution, representing less than 0.3% of the data. Such values are often categorized as outliers or extreme observations.

Why do we use z-scores instead of raw percentages?

Z-scores allow for comparison between different datasets that have different means and variations, which raw percentages cannot do. For example, a 90% on an easy test might be less impressive than a 70% on an extremely difficult test where the mean was 50%.

Does a z-score work for non-normal distributions?

While you can calculate a z-score for any distribution, the interpretation regarding percentiles (like the 68-95-99.7 rule) only applies strictly to normal distributions. For skewed data, z-scores are less informative about the exact proportion of data points.

What is the difference between a z-score and a t-score?

Standard z-scores are used when the population standard deviation is known and the sample size is large, whereas t-scores are used when the population standard deviation is unknown and must be estimated from a small sample. In chemistry, similar statistical distinctions are made when analyzing molecular properties, as seen in Medium Polarity Determination Practice Questions.

What does a z-score of 0 mean?

A z-score of 0 indicates that the specific data point is exactly equal to the mean of the distribution. It represents the center point of a standard normal distribution curve.

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