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The Z-score, also known as a standard score, is a measure that describes a value's relation to the mean of a group of values. It is expressed as a number that indicates how many standard deviations an element is from the mean. The formula to calculate the Z-score of an individual data point is:
\[ Z = \frac{X - \mu}{\sigma} \]Where:

• \( X \) is the value under consideration.
• \( \mu \) is the mean of the distribution.
• \( \sigma \) is the standard deviation.

Z-scores are a useful way of standardizing data, allowing us to compare data points from different distributions. When dealing with a normal distribution, the Z-score helps in determining the position of a data point within the distribution, such as perceiving how extreme a data point is compared to the average.

In statistics, the mean is the average value of a data set. It is calculated by adding all the numbers in a dataset and dividing by the count of those numbers:
\[ \mu = \frac{\sum X}{N} \]

• \( \mu \) represents the mean.
• \( \sum X \) is the sum of all data points.
• \( N \) is the number of data points.

Standard deviation, on the other hand, is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that values tend to be close to the mean, whereas a high standard deviation indicates that values are spread out over a wider range:
\[ \sigma = \sqrt{\frac{\sum (X_i - \mu)^2}{N}} \]Both mean and standard deviation are essential in describing the nature of a normal distribution, helping to understand the central tendency and spread of data.

Percentile calculation is a statistical measurement that provides the relative position of a value within a dataset. In terms of a normal distribution, the percentile determines the percentage of scores that fall below a particular value. This helps in understanding how a specific score compares to all other scores:
To calculate the time for a particular percentile, like the 90th percentile in a normal distribution, we can use the Z-score. A Z-score table or calculator helps find the Z-score that corresponds to a desired percentile, such as a Z-score of approximately 1.28 for the 90th percentile. Once the Z-score is determined, it can be converted to a raw score (actual value) using the mean and standard deviation.
This calculation is done by rearranging the Z-score formula to solve for the value of interest:\[ X = Z \cdot \sigma + \mu \]This formula allows us to find the specific score within a normal distribution that corresponds to any given percentile, making it a powerful tool for interpreting and understanding data.