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Understanding the percentile of a normal distribution is crucial for interpreting various data sets, such as test scores or measurement data. In a normally distributed data set, percentiles are used to compare individual scores to the rest of the data.

Simply put, if you score in the 90th percentile on a test, it means you did better than 90% of the people who took the same test. Calculating this value requires a normal distribution curve, which is symmetrical and has a bell shape. Most scores cluster around the mean, decreasing in frequency as they move further away.

The calculation is straightforward when you have the Z-score for the desired percentile, which you can get from statistical tables or calculators. With the Z-score in hand and knowledge of the distribution's mean and standard deviation, you can pinpoint exactly where on the x-axis your percentile lies.

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is expressed in terms of standard deviations from the mean. A Z-score can show how far away a particular value is from the average, providing a precise measure of position within a normal distribution.

A high Z-score indicates that the data point is many standard deviations above the mean, while a low Z-score signifies many deviations below the mean. For example, a Z-score of 1.28, which we see in our task for the 90th percentile, tells us that this value is 1.28 standard deviations above the mean.

This standardization process allows for comparison between different data sets, or within different parts of the same data set, regardless of the original scale or units.

Standard deviation is a measure that tells us how much variation or dispersion exists from the average (mean) in a set of data. It gives a sense of the 'spread' of the scores. A low standard deviation means that the data points tend to be very close to the mean, whereas a high standard deviation indicates that the data points are spread out over a wider range of values.

In the context of our exercise, the standard deviation \(\sigma\) is the square root of the variance, which is calculated to be 5. This numerical value is vital in calculating the Z-score and the subsequent percentiles of the distribution. It's one of the three key ingredients along with the Z-score and the mean in the recipe for finding a desired percentile in a normally distributed data set.