How is the z value calculated in the standard normal distribution?

The z value in the standard normal distribution is calculated using the formula ( Z = \frac{(x - \text{mean})}{\text{standard deviation}} ). This formula serves a crucial purpose in statistics; it transforms a given x value (an observation from a dataset) into a standardized score that reflects how far and in what direction it deviates from the mean of the distribution, measured in terms of standard deviations.

To break down the components:

• x represents the value from the dataset.
• mean is the average of the dataset, serving as a reference point.
• standard deviation quantifies the dispersion of the dataset, indicating how much the individual data points typically deviate from the mean.

By subtracting the mean from x, you find the deviation of x from the mean. Dividing this value by the standard deviation then normalizes this deviation, yielding a z score that can be used to compare observations from different normal distributions or to determine probabilities associated with specific outcomes in the context of the standard normal distribution.

This formula is central to z-scores because it allows statisticians and researchers to easily interpret how unusual or typical a particular observation is in relation to the overall distribution of the data. This