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Standard Deviation

TL;DR

Section titled “TL;DR”
  • Measures how spread out a set of values is around the mean.
  • Computed as the square root of the variance (the average of squared deviations from the mean).
  • Two datasets can have different means but the same standard deviation if their values are equally dispersed.

Definition

Section titled “Definition”

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of data. It is calculated as the square root of the variance, where variance is the average of the squared differences between each value and the mean.

Explanation

Section titled “Explanation”

The mean (average) of a dataset is computed by summing all values and dividing by the number of values. Standard deviation uses the mean to measure how far individual values deviate from that average. Variance is the average of the squared differences between each value and the mean; the standard deviation is the square root of that variance. A larger standard deviation indicates values are more widely dispersed around the mean; a smaller standard deviation indicates values are more tightly clustered.

Examples

Section titled “Examples”

Example — first set (2, 4, 6, 8, 10)

Section titled “Example — first set (2, 4, 6, 8, 10)”

Mean:

mean=2+4+6+8+105=6\text{mean} = \frac{2+4+6+8+10}{5} = 6

Variance:

(2−6)2+(4−6)2+(6−6)2+(8−6)2+(10−6)25\frac{(2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2}{5}

This simplifies to:

4+4+0+4+165=8\frac{4+4+0+4+16}{5} = 8

Standard deviation:

8=2.83\sqrt{8} = 2.83

Example — second set (1, 3, 5, 7, 9)

Section titled “Example — second set (1, 3, 5, 7, 9)”

Mean:

mean=1+3+5+7+95=5\text{mean} = \frac{1+3+5+7+9}{5} = 5

Variance:

(1−5)2+(3−5)2+(5−5)2+(7−5)2+(9−5)25\frac{(1-5)^2 + (3-5)^2 + (5-5)^2 + (7-5)^2 + (9-5)^2}{5}

This simplifies to:

16+4+0+4+165=8\frac{16+4+0+4+16}{5} = 8

Standard deviation:

8=2.83\sqrt{8} = 2.83

These examples show that two datasets can have different means yet the same standard deviation because standard deviation measures dispersion, not the absolute level of the values.

Use cases

Section titled “Use cases”
  • Estimating how likely a value is to be above or below a certain threshold in a group (example given: heights of a group of people).
  • Interpreting whether a group’s values are widely dispersed (large standard deviation) or closely clustered around the mean (small standard deviation).

Notes or pitfalls

Section titled “Notes or pitfalls”
  • Standard deviation measures dispersion around the mean, not the overall magnitude of the mean itself.
  • Equal standard deviations can occur for datasets with different means if their values are equally spread from their respective means.
Section titled “Related terms”
  • Mean
  • Variance