Calculate standard deviation instantly with step-by-step breakdown. Supports sample (n-1) and population (n) SD, variance, mean, coefficient of variation, standard error, IQR, outlier detection, and z-scores with complete formulas.
This standard deviation calculator computes both sample and population standard deviation with complete step-by-step calculations. Enter your dataset to instantly calculate the mean, median, variance, coefficient of variation, standard error, quartiles, IQR, skewness, and outliers — with all formulas and working shown.
Use Sample when analyzing a subset of data (most common in research and exams).
| Symbol | Name | Description |
|---|---|---|
| σ | Sigma | Population standard deviation |
| s | s | Sample standard deviation |
| σ² | Sigma squared | Population variance |
| s² | s squared | Sample variance |
| μ | Mu | Population mean |
| x̄ | x-bar | Sample mean |
| ∑ | Sigma (capital) | Summation operator |
| n | n | Number of observations |
| CV | CV | Coefficient of Variation (σ/μ × 100) |
| SE | SE | Standard Error of the Mean (s/√n) |
| IQR | IQR | Interquartile Range (Q3 − Q1) |
| z | z-score | Standardized value: (x − μ) / σ |
First compute the average of all values.
Population
Sample
Average squared deviation from the mean.
Population
Sample
Square root of variance.
Population
Sample
CV expresses standard deviation as a percentage of the mean, enabling comparison of variability across datasets with different units or scales. A CV below 15% generally indicates low variability; above 35% indicates high variability.
Standard error quantifies how accurately your sample mean estimates the true population mean. Unlike standard deviation (which describes spread within your dataset), SE decreases as sample size increases — a larger sample gives a more precise mean estimate.
The same formula produces radically different interpretations depending on context. Here are worked real-world examples across six domains — with sample datasets you can paste directly into the calculator above.
📈 Finance & Investing
Daily returns (%) for a large-cap equity. A sample SD of ~1.2% per day is typical for S&P 500 stocks. Annual volatility ≈ daily SD × √252. Crypto assets often show SD > 4% per day.
0.5, -1.2, 2.1, -0.3, 1.8, -2.5, 0.9, 1.1, -0.7, 0.4
🎓 Education
Class exam scores. SD of ~10 points is typical for a well-calibrated exam. SD < 5 suggests the test is too easy; SD > 20 suggests extreme preparation differences or poor question design.
72, 85, 91, 68, 78, 84, 92, 70, 75, 88, 95, 62
⚙️ Manufacturing (Six Sigma)
Bolt diameter measurements (mm). Target 10.00 mm. SD of 0.017 mm gives a CV of ~0.17%. Six Sigma quality requires the process spread to fit within specification limits at 6σ. Check Cpk = (spec limit − mean) / (3σ).
10.02, 9.98, 10.01, 10.00, 9.99, 10.03, 9.97, 10.00, 10.01, 9.99
🔬 Medical Research
Systolic blood pressure (mmHg) in a trial group. Normal resting BP range is 90–120 mmHg. SD here represents biological variability within the cohort. SE = SD/√n is what appears in research papers as ±SE in reported means.
120, 118, 125, 122, 119, 117, 124, 121, 123, 120, 126, 118
🏃 Sports Science
Sprint times (seconds) for an elite sprinter across 10 trials. SD of ~0.025s is excellent — top sprinters typically show CV < 0.5% in repeat sprint tests. CV > 1% may indicate fatigue, injury, or wind variation.
10.82, 10.79, 10.85, 10.80, 10.77, 10.83, 10.81, 10.84, 10.78, 10.82
📊 Survey Research
Likert-scale ratings (1–5). SD of ~1.1–1.3 on a 5-point scale is common for mixed-satisfaction products. SD > 1.5 signals polarization — customers are split into fans and detractors. Consider NPS analysis alongside SD.
4, 3, 5, 2, 4, 5, 3, 4, 1, 5, 4, 3, 5, 4, 2
Population (σ) and sample (s) standard deviation
Both formulas with correct Bessel's correction for sample SD
Coefficient of Variation (CV)
Compare variability across different scales and units
Standard Error of the Mean (SE)
Essential for research — how precisely your mean estimates the population
Five-number summary + IQR
Min, Q1, Median, Q3, Max — the complete descriptive statistics picture
Dual outlier detection
Z-score method (±2σ, ±3σ) and IQR fence method shown side by side
Domain-specific interpretation
Benchmark your SD against real norms in finance, education, QC, research, and sports
Step-by-step table with z-scores
Every value shows its deviation, squared deviation, and z-score — with outlier flagging
Export to CSV and JSON
Download full results for Excel, Python, R, or any analysis tool
Population standard deviation (σ) uses n in the denominator and applies when your data includes every member of the complete group. Sample standard deviation (s) uses n−1 (Bessel's correction) to correct for the bias that arises when estimating population variance from a subset. In most real-world cases — surveys, research, classroom tests — you should use sample standard deviation.
The coefficient of variation (CV = σ/|μ| × 100%) expresses standard deviation as a percentage of the mean. Its main advantage is scale-independence — you can compare variability between datasets measured in different units. For example, comparing consistency of rainfall (in mm) with temperature (in °C) requires CV rather than raw SD. A CV below 15% generally indicates low variability; above 35% indicates high variability.
Standard deviation (SD) measures the spread of individual values within your dataset. Standard error (SE = SD/√n) measures how precisely your sample mean estimates the true population mean. SE shrinks as sample size grows. In published research, means are typically reported as mean ± SE, not mean ± SD. Use SD to describe your data's variability; use SE to describe the reliability of your mean estimate.
An outlier is a value that differs significantly from the rest of the dataset. Two standard detection methods exist: (1) Z-score method — values with |z| > 2 are mild outliers; |z| > 3 are extreme outliers. Best for approximately normal distributions. (2) IQR fence method — values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are flagged. Best for skewed data because it does not assume normality. This calculator applies both methods automatically.
A high standard deviation means data points are widely spread around the mean — high variability, less consistency. What counts as 'high' depends entirely on context. In manufacturing, a CV above 5% may signal process problems. In financial returns, daily SD above 3% indicates high volatility. Always interpret SD relative to the mean (using CV) and against domain benchmarks.
Use IQR when your data is skewed, contains outliers, or does not follow a normal distribution. IQR covers the middle 50% of the data and is unaffected by extreme values. Standard deviation is the better choice for symmetric, approximately normal data and for use in parametric statistical tests like t-tests, ANOVA, and regression analysis.
Yes — for grouped data (frequency tables), use the midpoint of each class interval as the representative value and weight it by its frequency. This calculator currently supports raw (ungrouped) data. For grouped data, multiply each midpoint by its frequency before entering the values, then adjust by dividing appropriately. We recommend entering the full raw dataset whenever available for maximum accuracy.
The Empirical Rule states that for a normal (bell-shaped) distribution: approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. The interactive bell curve visualization above shows exactly where your actual values fall within these bands using your dataset's real mean and SD.
This calculator implements the standard formulas defined in ISO 3534-1 (Statistics — Vocabulary and symbols). Sample standard deviation uses Bessel's correction (n−1) as specified by the unbiased estimator convention used in statistics, data science, and academic research worldwide.
Results are intended for educational, analytical, and informational purposes. Minor rounding differences may occur for very large datasets.
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