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A percentile is a measure that indicates the value below which a given percentage of observations fall. In the context of a normal distribution, percentiles help us understand the spread and ranking within the data set. For example, the 90th percentile indicates the value below which 90% of the data falls, meaning 10% of the data falls above this value.
Percentiles are crucial for understanding data distribution and making comparisons. They provide a way to express how a particular value stands relative to the rest of the distribution.
To find percentiles in a normal distribution, we often use the Z-score, a standardized value that relates to the standard normal distribution.

The Z-score, also called the standard score, is a measure that describes how many standard deviations a data point is from the mean. It allows us to compare scores from different normal distributions by converting them to a common scale.
We calculate the Z-score with the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
where *X* is the value, *μ* is the mean, and *σ* is the standard deviation.
In the normal distribution, Z-scores are used to find the percentile rank of a data point. For instance, if we want to find the 90th percentile, we first determine the Z-score that corresponds to 90%. The Z-score for the 90th percentile is approximately 1.28. This means the value is 1.28 standard deviations above the mean.
Converting the Z-score back to the original distribution involves multiplying the Z-score by the standard deviations and adding the mean using:
\[ X = Z \cdot \sigma + \mu \].

Standard deviation is a fundamental concept in statistics that measures the amount of variability or dispersion in a set of data. It tells us how spread out the values in a data set are around the mean.
In a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations, following the Empirical Rule.
For the given problem, the standard deviation (σ) is 7. It helps in determining how much the values deviate from the mean (μ), which is 50. By understanding the standard deviation, we can convert between different percentiles and values in the original data set.

The mean, often known as the average, is a measure of central tendency that sums all the values in a data set and divides by the number of values. It represents the center point of a data set.
In the context of our exercise, the mean (μ) is 50. This is the central value around which the values are distributed.
When calculating percentiles, the mean plays a key role as it serves as the reference point for measuring standard deviations. For instance, the mean helps convert the Z-score back to the value in the original distribution:
\[ X = Z \cdot \sigma + \mu \]
Understanding the mean allows us to better interpret how individual values or percentiles relate to the overall distribution.