Standard Deviation Calculator
Calculate standard deviation, variance, and other statistical measures for a set of numbers.
How to Use This Calculator
Enter your comma-separated numbers in the field above and click "Calculate." The calculator returns mean, population standard deviation (σ), sample standard deviation (s), variance, minimum, maximum, and range.
Formulas
Step-by-Step Calculation
Example: data set 9
- Count: N = 8
- Mean: μ = (2+4+4+4+5+5+7+9) / 8 = 40/8 = 5
- Squared differences from mean: (2−5)²=9, (4−5)²=1, (4−5)²=1, (4−5)²=1, (5−5)²=0, (5−5)²=0, (7−5)²=4, (9−5)²=16
- Sum of squared differences: 9+1+1+1+0+0+4+16 = 32
- Population variance: σ² = 32/8 = 4
- Population standard deviation: σ = √4 = 2
- Sample standard deviation: s = √(32/7) ≈ 2.138
Population vs. Sample Standard Deviation
The population standard deviation (σ) divides by N and is used when your data set is the entire population you care about. The sample standard deviation (s) divides by N−1 (Bessel's correction) and is used when your data is a sample drawn from a larger population — the N−1 corrects for the bias introduced by estimating the mean from the same sample. In practice: if you measured every student in one class, use σ. If you measured 30 students and want to estimate the SD for all students at the school, use s.
Interpreting Standard Deviation
Standard deviation quantifies the typical spread around the mean. In a normal distribution (bell curve): about 68% of values fall within 1σ of the mean; about 95% within 2σ; about 99.7% within 3σ (the "68-95-99.7 rule" or empirical rule). A low σ means values cluster tightly around the mean; a high σ means values are widely spread. In finance, standard deviation of returns is used as a measure of investment risk. In quality control, σ is used in process capability analysis (Six Sigma targets less than 3.4 defects per million opportunities — 6σ from the mean).
Related Measures of Spread
Variance = σ² (units are squared, making it less intuitive but mathematically convenient). Range = max − min (simple but sensitive to outliers). Interquartile Range (IQR) = Q3 − Q1 (the middle 50% spread, robust to outliers). Mean Absolute Deviation (MAD) = average of |xᵢ − μ| (an alternative that is easier to understand intuitively than σ). For skewed distributions or data with outliers, IQR and MAD are often preferred over standard deviation.
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population std dev divides by N; sample std dev divides by N-1 (Bessel correction) to provide an unbiased estimate.
What does standard deviation tell us?
Standard deviation measures how spread out the data is from the mean. A low value means data is clustered near the mean.