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ProbabilityConcept reference

Standard deviation

Standard deviation is the nonnegative square root of variance.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. Variance has squared measurement units
  3. Compare quantities in the same form and units
  4. Use the nonnegative root
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01Variance has squared measurement units#

Variance has squared measurement units. Standard deviation takes its nonnegative square root, putting spread back in the units of the original variable.

SD⁡(X)=Var⁡(X)\operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}

If variance is 99 seconds squared, standard deviation is 33 seconds. The square root changes both the numerical scale and the units.

Variance 9 seconds squared corresponds to standard deviation 3 seconds. The displayed span extends one SD to each side of a reference mean; it makes no claim about probability within the span.Variance 9 seconds squared corresponds to standard deviation 3 seconds. The displayed span extends one SD to each side of a reference mean; it makes no claim about probability within the span.
Figure 1Variance 9 seconds squared corresponds to standard deviation 3 seconds. The displayed span extends one SD to each side of a reference mean; it makes no claim about probability within the span.
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Check your reasoning

Variance is 6464 cm². Give the standard deviation with units.

  1. A6464 cm
  2. B3232 cm
  3. C88 cm
Show answer and explanation
88 cm

Take the nonnegative square root: 88 cm.

02Compare quantities in the same form and units#

Compare quantities in the same form and units. A standard deviation of 4 meters is greater than the spread represented by a variance of 9 meters squared, because its standard deviation is 3 meters.

Comparing the raw numbers 4 and 9 would mix different quantities.

Check your reasoning

A has standard deviation 33 s. B has variance 1616 s². Whose spread is greater?

  1. AB
  2. BA
  3. CEqual
Show answer and explanation
B

B: 16=4\sqrt{16}=4 s. A: 33 s.

03Use the nonnegative root#

Use the nonnegative root. Standard deviation describes a size of spread, so it cannot be negative.

It is not a guaranteed distance for each observation. A zero variance gives zero standard deviation; all probability is at the mean.

Check your reasoning

A report labels variance 0.090.09 kg² as standard deviation. Give the corrected standard deviation.

  1. A0.090.09 kg
  2. B0.30.3 kg
  3. C0.0450.045 kg
Show answer and explanation
0.30.3 kg

0.09=0.3\sqrt{0.09}=0.3 kg.

Key takeaway

Take the nonnegative square root and return to the variable’s original units.

  • Express spread in the original variable's units.

Sources & further reading

  1. [1]
    Pishro-Nik, Introduction to Probability: Variance ↗Pishro-Nik: Introduction to Probability · Article

Reference this concept

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

Glacius. “Standard deviation.” Math behind ML. /learn/p-standard-deviation