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.