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Standard Deviation Explained
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Standard deviation (SD or σ) measures how spread out a set of numbers is around their mean. A small SD means the values cluster tightly around the average; a large SD means they are widely scattered. It is one of the most commonly used statistics in science, finance, and quality control.
Step-by-Step Calculation
Dataset: 4, 7, 13, 2, 1
1. Find the mean: (4+7+13+2+1) ÷ 5 = 27 ÷ 5 = 5.4
2. Subtract the mean from each value and square the result:
(4−5.4)² = 1.96
(7−5.4)² = 2.56
(13−5.4)² = 57.76
(2−5.4)² = 11.56
(1−5.4)² = 19.36
3. Find the mean of these squared differences: (1.96+2.56+57.76+11.56+19.36) ÷ 5 = 93.2 ÷ 5 = 18.64 (this is the variance)
4. Square root of the variance: √18.64 ≈ 4.32
Population SD ≈ 4.32
Population vs. Sample Standard Deviation
When calculating from a complete dataset (population), divide by n.
When calculating from a sample to estimate the population, divide by n−1. This is called Bessel's correction.
Most statistics software and our calculator allow you to choose. For general data analysis where you have a sample, use n−1 (sample SD).
The 68-95-99.7 Rule
For data that follows a normal distribution:
• About 68% of values fall within 1 SD of the mean.
• About 95% fall within 2 SDs.
• About 99.7% fall within 3 SDs.
Example: Test scores with mean 70 and SD 10.
• 1 SD range: 60–80 contains about 68% of scores.
• 2 SD range: 50–90 contains about 95% of scores.
This rule helps you quickly judge whether a data point is typical or unusual.
Real-World Uses
• Finance: SD of investment returns measures volatility (risk).
• Manufacturing: SD of product dimensions indicates consistency.
• Education: Standardized test scores use SD to assign letter grades.
• Science: Error bars on graphs often represent one SD, showing measurement precision.
Try the Standard Deviation Calculator.
Frequently Asked Questions
What is variance vs. standard deviation?
Variance is the mean of squared deviations (step 3 above). Standard deviation is the square root of variance. SD is in the same units as the original data, making it easier to interpret—which is why it is more commonly reported.
Can standard deviation be negative?
No. Standard deviation is always zero or positive. A SD of zero means all values are identical.