5.2:
Introduction to z Scores
The z score, or standardized score, is the number of standard deviations that a given value is away from the mean. It is one of the commonly used measures of relative standing.
Using the standardization formula, data can be converted into corresponding z scores. The standardization formula for a population and a sample differ in the mean and standard deviation notations.
z scores provide the relative position of a data point. A positive z score means the data point is above the mean and a negative z indicates below the mean. The mean value always has a zero z score.
The z score is also used to compare data measured on different scales, such as comparing a student’s height and weight with classmates. Since data are measured on different scales, they are standardized into z scores.
A z score of 1.5 indicates that the student is taller than most of his classmates, while a z score of minus 0.5 suggests that his weight is very close to the class average.
5.2:
Introduction to z Scores
A z score (or standardized value) is measured in units of the standard deviation. It tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z scores, and values of x that are smaller than the mean have negative z scores. If x equals the mean, then x has a zero z score. It is important to note that the mean of the z scores is zero, and the standard deviation is one.
z scores help to find the outliers or unusual values from any data distribution. According to the range rule of thumb, outliers or unusual values have z scores less than -2 or greater than +2.
This text is adapted from Openstax, Introductory Statistics, 6.1 The Standard Normal Distribution