Statistics and Probability

In statistics, standardization is the process of converting an actual data value to a z-score, which represents the number of standard deviations the data value is from the mean. This is particularly useful when comparing scores from different normal distributions. The formula for standardization is given by:

$$ z = \frac{(x - \mu)}{\sigma} $$

where \( x \) is the data value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Inverse normal calculations are used to find the data value associated with a particular cumulative probability.

The key formula for inverse normal calculations is:

$$ x = \mu + z \cdot \sigma $$

For instance, consider a student's score \( x \) on a test to be 85, with the test scores normally distributed with a mean \( \mu \) of 75 and a standard deviation \( \sigma \) of 10. To standardize this score, we calculate the z-value:

$$ z = \frac{(85 - 75)}{10} \\ z = \frac{10}{10} \\ z = 1 $$

This z-score of 1 indicates that the student's score is 1 standard deviation above the mean. This standardized score can now be compared across different normal distributions or used in further statistical analyses.

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