Standard deviation (Ο) quantifies how spread out data points are from the average. A low standard deviation means data points cluster near the mean; a high standard deviation means they are spread out.
Steps to Calculate
- Find the mean (average)
- Subtract the mean from each data point and square the result
- Average those squared differences (variance)
- Take the square root of the variance
Example
Data: 4, 8, 6, 5, 3. Mean = 5.2. Squared differences: 1.44, 7.84, 0.64, 0.04, 4.84. Variance = 2.96. SD = β2.96 = 1.72.
The 68-95-99.7 Rule
In a normal distribution: 68% of data falls within 1 SD, 95% within 2 SD, 99.7% within 3 SD of the mean.
Population vs. Sample Standard Deviation
When your data represents an entire population, divide by n. When your data is a sample used to estimate a larger population, divide by (nβ1) instead (Bessel's correction) β this "sample standard deviation" slightly increases the result to correct for the tendency of samples to underestimate true variability. Most statistical software defaults to the sample formula.
Standard Deviation vs. Variance
Variance is standard deviation squared. Variance is mathematically convenient for combining independent sources of variability, but its units are squared (e.g., dollarsΒ²), which is hard to interpret β that's why standard deviation, in the original units, is what's typically reported and discussed.