Statistics and Probability

Furthermore

Official Guidance, clarification and syllabus links:

Probabilities and values of the variable must be found using technology.

The standardized value (z) gives the number of standard deviations from the mean.

Use of z-values to calculate unknown means and standard deviations.

Formula Booklet:

Standardized normal variable

\( z=\dfrac{x-\mu}{\sigma} \)

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.