Standardization of Normal Variables (Z-values)
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:
- $x$ is the raw score (data point)
- $\mu$ is the mean of the distribution
- $\sigma$ is the standard deviation of the distribution
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.
NoteZ-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.
Interpreting Z-scores
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.
- A z-score of 0 means the data point is exactly at the mean.
- A z-score of 1 means the data point is one standard deviation above the mean.
- A z-score of -1 means the data point is one standard deviation below the mean.
Students often forget that z-scores represent distances from the mean in terms of standard deviations, not raw units. A z-score of 2 doesn't mean "2 units above the mean," but rather "2 standard deviations above the mean."
Finding Probabilities
To find the probability that a value falls below a certain point in a normal distribution, we use the cumulative distribution function (CDF). On most graphing calculators, this is represented by the normalcdf function.
ExampleUsing a normal distribution with mean 70 and standard deviation 5, find the probability that a randomly selected value is less than 75.
On a TI-84 calculator: normalcdf(-99999, 75, 70, 5) ≈ 0.8413
This means there's about an 84.13% chance that a randomly selected value is less than 75.
