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Glacius
ProbabilityConcept reference

Standardization

Standardization maps a normal threshold x to z=(x-mu)/sigma, where sigma is the positive standard deviation.

2 min read1 figureSources & further reading
On this page 9 sections
  1. Overview
  2. Standardization expresses a value as a signed number of standard deviations from its mean
  3. A threshold below the mean has a negative score
  4. If a normal model supplies variance 9, use SD 3 in the denominator
  5. A standardized score of 2 means two standard deviations above the mean
  6. Key takeaway
  7. Further questions
  8. Sources & further reading
  9. Concept connections

01Standardization expresses a value as a signed number of standard deviations from its mean#

Standardization expresses a value as a signed number of standard deviations from its mean. Subtracting the mean changes the origin; dividing by the standard deviation changes the unit. This makes values from different location-and-scale settings comparable.

With mean 1010 and SD 44, a raw value of 1414 is one SD above the center. Its standard score is 11.

Two number lines: raw normal units have mean 10, with values 6, 10 and 14. Standard units put them at -1, 0 and 1. The standard deviation is 4; subtract 10 and divide by 4.Two number lines: raw normal units have mean 10, with values 6, 10 and 14. Standard units put them at -1, 0 and 1. The standard deviation is 4; subtract 10 and divide by 4.
Figure 1Two number lines: raw normal units have mean 10, with values 6, 10 and 14. Standard units put them at -1, 0 and 1. The standard deviation is 4; subtract 10 and divide by 4.
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Subtract the mean to measure displacement from the center. Divide by the positive SD to count that displacement in SD units.

z=x−μσ,σ>0z=\frac{x-\mu}{\sigma},\qquad \sigma>0

For value 16, mean 10, and standard deviation 3, first find the deviation 16−10=616-10=6. Dividing by 3 gives standardized value 2. A value of 7 gives −1-1. If you are supplied variance 9 instead, take its square root before using the same calculation.

Check your reasoning

Normal model. Mean 33, SD 22; threshold 88. Give the standard score.

Show answer and explanation
2.5

(8−(3))/2=2.5(8-(3))/2=2.5.

02A threshold below the mean has a negative score#

A threshold below the mean has a negative score. At the mean, the score is zero. Standardizing changes the unit of measurement; it does not turn a negative displacement positive.

Check your reasoning

Normal model. Mean 66, SD 22; threshold 22. Repair reported z=−1z=-1.

  1. A−1-1
  2. B−2-2
  3. C22
Show answer and explanation
−2-2

(2−(6))/2=−2(2-(6))/2=-2.

03If a normal model supplies variance 9, use SD 3 in the denominator#

If a normal model supplies variance 99, use SD 33 in the denominator. For mean 44 and threshold 1010, the score is (10−4)/3=2(10-4)/3=2.

A standardized normal variable has mean 00 and SD 11. Its threshold is now ready for a standard-normal probability table.

Check your reasoning

Latency. Normal model. Mean 2020, SD 55; threshold 1212. Give the standard score.

Show answer and explanation
-1.6

(12−(20))/5=−1.6(12-(20))/5=-1.6.

04A standardized score of 2 means two standard deviations above the mean#

A standardized score of 2 means two standard deviations above the mean. It does not have a universal percentile unless the distribution is specified. Under a normal model, standardization lets one standard-normal table serve many different means and scales.

Key takeaway

Subtract the mean, then divide by SD. The sign shows which side of the mean contains the threshold.

  • Convert a normal threshold to standard-normal units.

Further questions

How can standard scores locate interval probabilities in a CDF?
For a normal variable, standardize each endpoint, then use the standard-normal CDF to find the interval probability.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Standardization.” Math behind ML. /learn/p-standardize