STANDARDIZE

Converts a value in a normal distribution to its equivalent in a standard normal distribution.

Syntax:

STANDARDIZE(n, μ, σ)

n is a number in a normal distribution with known mean μ and known standard deviation σ.

STANDARDIZE converts n to a corresponding value in a standard normal distribution, which has mean 0 and standard deviation 1.

STANDARDIZE calculates and returns (n - μ)/ σ.

Example:

STANDARDIZE(14, 10, 4)

returns 1. The original value 14 lies 1 standard deviation above the original mean 10; the new value lies 1 standard deviation above the new mean 0.

Application:

The STANDARDIZE function, often referred to as the z-score formula, is a statistical function that converts a value from a normal distribution into a standardized value (a z-score). This standardized value indicates how many standard deviations a given value is from the mean of the distribution. The formula for the STANDARDIZE function is:

$Z = \frac{X - μ}{σ}$

where:

Z is the standardized value (z-score)
X is the value from the normal distribution
μ is the mean of the normal distribution
σ is the standard deviation of the normal distribution

Student Test Scores

A common application of the STANDARDIZE function is in education, where it can be used to compare a student's performance on one test to their performance on another, or to compare one student's performance to the rest of the class.

Let's consider a national standardized math test where the scores are normally distributed. The distribution has a mean (μ) of 75 and a standard deviation (σ) of 8.

The table below shows four students' test scores (X) and their corresponding STANDARDIZE values (z-scores).

	Student	Test Score (X)	Mean (μ)	Standard Deviation (σ)	STANDARDIZE Value ( $Z = \frac{X - μ}{σ}$ )
	A	B	C	D	E
1	A	85	75	8	1.25
2	B	68	75	8	-0.875
3	C	75	75	8	0
4	D	91	75	8	2

Interpretation of the Z-Scores:

Student A (Z-score = 1.25): A positive z-score indicates the value is above the mean. Student A's score of 85 is 1.25 standard deviations above the average test score.
Student B (Z-score = -0.875): A negative z-score indicates the value is below the mean. Student B's score of 68 is 0.875 standard deviations below the average test score.
Student C (Z-score = 0.0): A z-score of 0.0 means the value is exactly equal to the mean. Student C's score of 75 is the same as the average test score.
Student D (Z-score = 2.0): This positive z-score shows that Student D's score of 91 is 2.0 standard deviations above the mean, indicating a very strong performance relative to the rest of the group.