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Understanding how to calculate the z-score is crucial when dealing with the normal distribution. The z-score tells us how many standard deviations a data point is from the mean. It is calculated with the formula:
\[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is the value of the data point, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.
- **Example:** In our context, let's calculate for a GPA of 3.71 with a mean GPA of 3.56 and a standard deviation of 0.34, the z-score would be \( Z = \frac{3.71 - 3.56}{0.34} \).
- This results in a z-score of 0.44, meaning the GPA is 0.44 standard deviations above the mean.
This z-score helps us understand where this GPA falls in relation to the entire distribution of GPAs.

A percentile indicates the position or rank of a score in a distribution. If a score is at the 85th percentile, it means it is higher than 85% of the other scores. To find the percentile, one can use a z-table or statistical software.
- **Example of Use:** When we calculated a z-score of 0.44 for the GPA, we needed to convert this into a percentile.
- Using a z-table, the z-score of 0.44 converts approximately to the 67th percentile.
This tells us that the GPA of 3.71 is higher than approximately 67% of all GPAs in the applicant pool.
Understanding percentiles helps in making informed decisions, especially when a cut-off point is needed as seen in the evaluation of GPA requirements for medical school applications.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In a normal distribution:
- **The Role of Standard Deviation:** A small standard deviation means the data points are close to the mean. A large standard deviation indicates that the data are spread out over a wider range of values.
In our problem, the standard deviation is 0.34, which reflects how much students' GPAs vary around the mean GPA of 3.56.
Understanding standard deviation is fundamental since it impacts our calculation of the z-score, which in turn helps us determine the percentile rank of a specific GPA.