Last updated: July 16, 2026
Standard Deviation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
Standard deviation quantifies the spread of a dataset around its mean. The sample formula divides by n−1 (Bessel's correction) for an unbiased estimate from a subset of data; the population formula divides by N when the full dataset is available. The result is in the same units as the original data, making it more intuitive than variance. In a normal distribution, ±1σ covers 68% of values, ±2σ covers 95%, and ±3σ covers 99.7%.
Standard deviation measures how spread out your data is. For a sample, divide the sum of squared differences from the mean by n minus 1, then take the square root. For a population, divide by N instead. A larger standard deviation means your data is more spread out.
Key Takeaways
- Standard deviation measures how spread out values are around the mean — a higher SD means more variability.
- Use sample SD (divide by n−1) when your data is a subset of a larger population; use population SD (divide by N) when you have all the data.
- The empirical rule states that ≈68%, 95%, and 99.7% of data lies within ±1σ, ±2σ, and ±3σ of the mean in a normal distribution.
- Standard deviation is the square root of variance; both measure dispersion but SD is in the original units, making it more interpretable.
- Standard deviation is sensitive to outliers because squaring deviations amplifies extreme values — consider median absolute deviation for skewed data.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Population: σ = √(Σ(xᵢ − μ)² / N) | Sample: s = √(Σ(xᵢ − x̄)² / (n−1))
Where:
- σ=Population Standard Deviation
- s=Sample Standard Deviation
- μ=Population Mean
- x̄=Sample Mean
- N=Population Size(count)
- n=Sample Size(count)
Worked Examples
Sample Standard Deviation
Calculate the sample standard deviation for a dataset of seven test scores.
- 1Collect the data: {2, 4, 5, 6, 6, 9, 10}, n = 7
- 2Compute the mean: (2+4+5+6+6+9+10) / 7 = 42 / 7 = 6
- 3Compute squared deviations: (2−6)²=16, (4−6)²=4, (5−6)²=1, (6−6)²=0, (6−6)²=0, (9−6)²=9, (10−6)²=16
- 4Sum of squared deviations = 16+4+1+0+0+9+16 = 46
- 5Sample variance = 46 / (7−1) = 46 / 6 ≈ 7.6667
- 6Sample standard deviation s = √7.6667 ≈ 2.7689
Population Standard Deviation
Compute the population standard deviation for the ages of all employees in a small department.
- 1Dataset: {10, 12, 23, 23, 16, 23, 21, 16}, N = 8
- 2Compute the mean: (10+12+23+23+16+23+21+16) / 8 = 144 / 8 = 18
- 3Squared deviations: (10−18)²=64, (12−18)²=36, (23−18)²=25, (23−18)²=25, (16−18)²=4, (23−18)²=25, (21−18)²=9, (16−18)²=4
- 4Sum of squared deviations = 64+36+25+25+4+25+9+4 = 192
- 5Population variance σ² = 192 / 8 = 24
- 6Population standard deviation σ = √24 ≈ 4.8990
Small Sample Dataset
Find the sample standard deviation for five measurements from a laboratory experiment.
- 1Data: {4, 8, 15, 16, 23}, n = 5
- 2Mean = (4+8+15+16+23) / 5 = 66 / 5 = 13.2
- 3Squared deviations: (4−13.2)²=84.64, (8−13.2)²=27.04, (15−13.2)²=3.24, (16−13.2)²=7.84, (23−13.2)²=96.04
- 4Sum of squared deviations = 218.8
- 5Sample variance = 218.8 / (5−1) = 218.8 / 4 = 54.7
- 6Sample SD s = √54.7 ≈ 7.3959
Introduction
Standard deviation is the most widely used measure of statistical dispersion — it tells you, on average, how far each data point deviates from the mean. A small standard deviation means values cluster tightly around the average; a large standard deviation signals high variability. It is used in finance for portfolio risk, in science for measurement precision, in manufacturing for quality control, and in education for grading analysis. This calculator supports both the sample standard deviation (using Bessel's correction, dividing by n−1) and the population standard deviation (dividing by N), so you can choose the formula appropriate for your dataset.
Sample vs. Population Standard Deviation
The key distinction lies in the denominator. When your data represent the entire population (every member), divide by N to get σ. When your data are a random sample from a larger population, divide by n−1 (Bessel's correction) to get s. Bessel's correction compensates for the tendency of a sample to underestimate the true population variance — a bias that grows larger for small samples. For large n the two values converge. Use the variance calculator to explore variance directly, or the descriptive statistics calculator for a full summary including skewness and kurtosis.
Step-by-Step Calculation Method
1. Find the mean (x̄ or μ): sum all values and divide by n. 2. Compute deviations: subtract the mean from each value (xᵢ − x̄). 3. Square each deviation: (xᵢ − x̄)². 4. Sum the squared deviations: Σ(xᵢ − x̄)². 5. Divide by the appropriate denominator: n−1 (sample) or N (population) to get variance. 6. Take the square root of the variance to obtain the standard deviation. The squaring step in (3) is critical — it eliminates negative deviations and weights larger deviations more heavily, making standard deviation sensitive to outliers.
The 68-95-99.7 Empirical Rule
For any approximately normal distribution, about 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. This rule, sometimes called the '3-sigma rule', is the backbone of statistical process control and quality management. Learn more about normal distributions at Wikipedia — Normal distribution and from the NIST Engineering Statistics Handbook.
Real-World Applications
Finance: portfolio volatility is expressed as annualised standard deviation of returns. A stock with σ = 20% is riskier than one with σ = 5%. Quality control: Six Sigma aims for processes that operate within ±6σ of specifications, yielding fewer than 3.4 defects per million opportunities. Academic testing: IQ scores are scaled to μ = 100, σ = 15; SAT scores to μ = 500, σ = 100 per section. Science: measurement uncertainty is routinely reported as ±1σ or 95% confidence intervals (±1.96σ). The coefficient of variation calculator expresses σ as a percentage of the mean, enabling comparison across different units.
Limitations and Alternatives
Standard deviation assumes data are roughly symmetric and that outliers are rare. Heavily skewed distributions (income, house prices) are often better described by the median absolute deviation (MAD), which is more robust to extreme values. For comparing variability across different units or scales, the coefficient of variation (CV = σ/μ) is preferable. When working with grouped data, see the grouped data standard deviation calculator. For details on the mathematical properties, see MathWorld — Standard Deviation and the formal definition at Wikipedia — Standard deviation.
Quick Reference Card
Standard Deviation Quick Reference
Quick reference • Standard Deviation Calculator
s = √(Σ(xᵢ−x̄)²/(n−1)) | σ = √(Σ(xᵢ−μ)²/N)Valid range: SD ≥ 0 always; SD = 0 only when all values are identical
Common Values
⚠ Watch Out
- •Always choose sample SD (n−1) when your dataset is a sample from a larger population.
- •Outliers inflate standard deviation significantly — inspect your data for extreme values before reporting.
- •Standard deviation assumes approximate symmetry; for skewed distributions consider IQR or MAD instead.
- •Do not compare standard deviations across datasets measured in different units without normalising (use CV).
Pro Tips
- →Use the coefficient of variation (CV = SD/mean × 100%) to compare variability across datasets with different means.
- →For quick mental estimation: in a symmetric dataset, range ÷ 4 gives a rough approximation of SD.
- →When n < 30, the difference between sample and population SD is non-trivial — always use the correct formula.
- →To reduce SD, reduce measurement noise, collect more data, or control for confounding variables in your experiment.
FAQs
What is the difference between sample and population standard deviation?
Population standard deviation (σ) divides by N (the total number of observations) and is used when your dataset contains every member of the group. Sample standard deviation (s) divides by n−1, applying Bessel's correction to produce an unbiased estimate of the true population σ when you only have a subset of the data. For large samples the difference is negligible; for small samples (n < 30) it is important.
Why do we square the deviations instead of just taking the absolute values?
Squaring deviations has three mathematical advantages: it eliminates negative values (preventing cancellation), it weights larger deviations more heavily than smaller ones, and it produces variance — a quantity with many useful algebraic properties such as additivity for independent random variables. The absolute-deviation approach yields the mean absolute deviation (MAD), which is simpler but lacks these properties.
What does a high standard deviation mean?
A high standard deviation means the data points are spread far from the mean, indicating high variability. In finance this means higher risk; in manufacturing it means less consistent production. Whether a standard deviation is 'high' depends on context: σ = 5 is large for heights measured in centimetres but tiny for national populations measured in millions.
Can standard deviation be negative or zero?
Standard deviation is always non-negative (≥ 0). It equals exactly zero only when every value in the dataset is identical, meaning there is no variability at all. It can never be negative because it is computed as a square root of a sum of squares, which is always non-negative.
What is Bessel's correction and why is n−1 used?
Bessel's correction (n−1 instead of n in the denominator) corrects for the fact that a sample mean is computed from the same data used to estimate variance, introducing a downward bias. The n−1 denominator produces an unbiased estimator: on average across many samples, s² equals the true population variance σ². The correction was formalised by Friedrich Bessel in the 19th century.
How does standard deviation relate to the normal distribution?
In a normal (Gaussian) distribution, the empirical rule states that approximately 68% of data lies within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. Standard deviation is one of the two parameters that fully define a normal distribution (the other being the mean). This relationship makes it an essential tool in statistical inference, hypothesis testing, and confidence interval construction.