A large group of test scores is normally distributed with mean 65.2 and standard deviation 8.3.You scored 78.5 what is the z score? What percent do you think you are in?

To calculate the **z-score** for your score of 78.5, we use the formula for the z-score in a normal distribution:

\[
z = \frac{x - \mu}{\sigma}
\]

where:
- \( x \) is your score (78.5),
- \( \mu \) is the mean (65.2),
- \( \sigma \) is the standard deviation (8.3).

Substitute the values into the formula:

\[
z = \frac{78.5 - 65.2}{8.3} = \frac{13.3}{8.3} \approx 1.60
\]

### Z-score: **1.60**

#### What percent are you in?

A z-score of 1.60 corresponds to a value above the mean, so we look up the z-score in a standard normal distribution table or use a calculator. The cumulative probability for a z-score of 1.60 is approximately **0.9452**, meaning that:

- You scored better than **94.52%** of the test takers.

This places you in the **top 5.48%** of the distribution.

Would you like more details on this, or do you have any other questions?

Here are 5 related questions for further exploration:
1. How do you calculate the probability for any z-score?
2. What does it mean to be in the top 5% of a distribution?
3. How can you use z-scores to compare different distributions?
4. What is the significance of a z-score of 0 in a normal distribution?
5. How do you interpret negative z-scores?

**Tip:** Z-scores are a powerful tool for comparing individual scores to the overall population.

A large group of test scores is normally distributed with mean 65.2 and standard deviation 8.3. You scored 78.5. What is the z-score? What percent do you think you are in?

Learn how to calculate the z-score for a test score of 78.5 from a normally distributed set of scores with a mean of 65.2 and a standard deviation of 8.3. This solution uses the z-score formula and discusses percentile ranking in a normal distribution. Suitable for high school students (Grades 9-12).

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