The normalized deviation is a measure of variation that expresses the deviation of an individual feature value from the mean value in the number of standard deviations. It is also called a standard score or z-score.
The standard deviation can be calculated for both a sample and a population.
The standard deviation of a population is calculated using the formula
The standard deviation of a sample is calculated using the formula
The z-score has no unit of measure.
Expressing dispersion in standardized measures, i.e. in standard deviations, allows you to compare two data sets.
Example: Standardization of deviations from the arithmetic mean
In the example of the height of girls (in cm) in the 52 class of the "X" elementary school, which we used as an example of calculating the arithmetic mean, median, mode, rank, variance and standard deviation, we had the following data: 140, 141, 138, 140, 122, 160, 154, 132, 148, 135, 140. We calculated that the arithmetic mean value is 140.91 cm, and the standard deviation 10.34.
Let's assume that we measured the height of a boy in the same class and got 
=152.36cm, with a standard deviation s=7.25.
Let's assume that we are interested in whether a girl who is 135cm tall or a boy who is 161cm tall deviates more from the average.
Answer: it is necessary to standardize the deviations from the average height of boys and girls, and then compare the resulting z-scores.
z(135, girl)=(135-140.91)/10.34=-0.57
z(161, boy)=(161-152.36)/7.25=1.19
From the result obtained, we conclude that a boy who is 161cm tall deviates from the average of boys by 1.19 standard deviations, while a girl who is 135 cm tall deviates from the average of girls by 0.57 standard deviations. This means that the deviation of that boy is more significant than the deviation of the height of the observed girl from the average of the groups to which they belong.
It should be emphasized that the negative value we obtained for the girl means that the girl's height is lower than the average height, i.e. the sign tells us about the direction of the deviation, and the calculated number about the intensity of the deviation.
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