Standard Deviation Calculator
Calculate standard deviation, variance, and other statistics for a dataset.
Ctrl+Enter to calculate
What Is the Standard Deviation Calculator?
The Standard Deviation Calculator computes full descriptive statistics from a dataset you enter. It calculates standard deviation (population σ or sample s), variance, mean, median, mode, min, max, range, Q1, Q3, IQR, coefficient of variation, standard error of the mean, and skewness. A histogram SVG shows the distribution shape.
- ›Population σ: use when you have all data points in the group
- ›Sample s: use when your data is a subset, applies Bessel's correction (÷ n−1)
- ›IQR (interquartile range) = Q3 − Q1, resistant to outliers
- ›CV = (σ/mean)×100%, compare spread across datasets with different units
- ›Skewness: positive = tail extends right; negative = tail extends left
Formula
Standard Deviation Formulas
Population σ
σ = √(Σ(xᵢ−μ)² / N)
Sample s
s = √(Σ(xᵢ−x̄)² / (n−1))
Variance
σ² = Σ(xᵢ−μ)² / N
Mean μ
μ = Σxᵢ / N
Coeff. of Var.
CV = (σ/|μ|) × 100%
Std Error
SEM = s / √n
How to Use
- 1Select Population (σ) or Sample (s) depending on your data type
- 2Type or paste your numbers, separated by commas, spaces, or newlines
- 3Press Ctrl+Enter or click Calculate Statistics
- 4View the primary stats (std dev, variance, mean, count) in the top tiles
- 5Scroll down for the full distribution table: median, mode, quartiles, IQR, skewness
- 6The histogram shows your data distribution with bars within 1σ highlighted
Example Calculation
Dataset: 4, 8, 15, 16, 23, 42 (population)
Mean μ = (4+8+15+16+23+42) / 6 = 108/6 = 18
Deviations²: 196, 100, 9, 4, 25, 576
Variance σ² = 910/6 = 151.667
σ = √151.667 = 12.316
Median = (15+16)/2 = 15.5
IQR = Q3−Q1 = 23−8 = 15
Same data as sample:
s = √182 = 13.491
The 68-95-99.7 Rule (Empirical Rule)
For a normal distribution: 68% of data falls within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. This rule is the foundation of quality control (Six Sigma targets 99.99966% within ±6σ), grading curves, and confidence intervals in statistics.
Understanding Standard Deviation
Standard Deviation Interpretation Guide
| Statistic | Formula | Interpretation |
|---|---|---|
| Mean (μ) | Σxᵢ / N | Central tendency, average value |
| Median | Middle value | Robust to outliers; better for skewed data |
| Mode | Most frequent | Useful for categorical / discrete data |
| Range | max − min | Sensitive to outliers; simple spread measure |
| Variance (σ²) | Σ(xᵢ−μ)²/N | Average squared deviation; additive |
| Std Dev (σ) | √variance | Spread in original units; most-used |
| IQR | Q3 − Q1 | Middle 50% spread; outlier-resistant |
| CV | (σ/|μ|)×100% | Relative variability; dimensionless |
| SEM | s/√n | Precision of the mean estimate |
| Skewness | Third standardised moment | Distribution asymmetry |
Frequently Asked Questions
When should I use population vs sample standard deviation?
Bessel's correction compensates for the fact that a sample mean is derived from the data itself (not an independent estimate), causing the sample variance to systematically underestimate the population variance when dividing by n.
- ›All students in a class → population; their heights → use σ
- ›Survey of 200 voters from millions → sample → use s
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- ›As n → ∞, s → σ (the correction becomes negligible)
What is variance and how does it relate to standard deviation?
- ›Variance in squared units → harder to interpret directly
- ›σ = √variance → same units as data → easier to contextualise
- ›Variance is additive: Var(X+Y) = Var(X) + Var(Y) for independent X, Y
- ›This additivity makes variance fundamental to ANOVA and regression analysis
What does the coefficient of variation (CV) measure?
CV is dimensionless, unlike σ, which is in the same units as your data, CV can compare datasets measured in completely different units.
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- ›CV 10–30%: moderate variability (typical biological measurements)
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- ›Not meaningful when mean ≈ 0 or data contains negatives
What is the standard error of the mean (SEM)?
- ›SEM decreases as √n, quadruple the sample size to halve the SEM
- ›95% CI ≈ x̄ ± 1.96×SEM for large samples (normal approximation)
- ›SEM vs σ: σ describes data spread; SEM describes precision of the mean estimate
- ›Error bars on scientific graphs usually show SEM (not σ)
What is skewness and what does it tell me?
- ›Symmetric (normal): skewness ≈ 0
- ›Right-skewed (positive): income distributions, wait times, stock returns
- ›Left-skewed (negative): exam scores near ceiling, age at retirement
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- ›For skewed data, median is often a better measure of central tendency
What is the interquartile range (IQR)?
- ›Q1 = 25th percentile; Q3 = 75th percentile; IQR = Q3 − Q1
- ›Outlier lower fence: Q1 − 1.5×IQR; upper fence: Q3 + 1.5×IQR
- ›Robust to outliers, preferred over range for skewed distributions
- ›Box-and-whisker plot: box = IQR, whiskers = fence, dots = outliers
Is this standard deviation calculator free?
Yes, completely free with no registration required. All calculations run locally in your browser, no data is sent to any server.