In the study of normal distributions, standardization allows us to compare and analyze different normal distributions on a common scale. This is achieved through the use of z-values, also known as z-scores or standard scores.
A z-value represents the number of standard deviations a data point is from the mean of its distribution. The formula for calculating a z-value is:
$$ z = \frac{x - \mu}{\sigma} $$
Where:
Suppose we have a normal distribution with a mean of 100 and a standard deviation of 15. If we want to find the z-score for a value of 130, we would calculate:
$z = \frac{130 - 100}{15} = 2$
This means that 130 is 2 standard deviations above the mean.
Z-scores can be positive or negative. A positive z-score indicates that the value is above the mean, while a negative z-score indicates that the value is below the mean.
Z-scores provide a standardized way to understand how far a data point is from the mean in terms of standard deviations. This is particularly useful when comparing values from different normal distributions.
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