Standard Deviation Calculator
Standard Deviation Calculator
Your Data
Results
| ±1σ expect 68% | 4.61 to 16.59 | 60.0% |
| ±2σ expect 95% | -1.37 to 22.57 | 100.0% |
| ±3σ expect 99.7% | -7.36 to 28.56 | 100.0% |
| Sum | 106.0000 |
| Minimum | 3.0000 |
| Q1 (25th pct) | 6.2500 |
| Median | 10.0000 |
| Q3 (75th pct) | 14.2500 |
| Maximum | 21.0000 |
| Range | 18.0000 |
| IQR | 8.0000 |
| Mode | none |
| Std. Error of Mean | 1.8927 |
| Coeff. of Variation | 56.46% |
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 12.00 | 1.400 | 1.960 |
| 7.00 | -3.600 | 12.960 |
| 3.00 | -7.600 | 57.760 |
| 15.00 | 4.400 | 19.360 |
| 9.00 | -1.600 | 2.560 |
| 21.00 | 10.400 | 108.160 |
| 4.00 | -6.600 | 43.560 |
| 18.00 | 7.400 | 54.760 |
| 11.00 | 0.400 | 0.160 |
| 6.00 | -4.600 | 21.160 |
Based on your results — what to do next:
Turn volatility into a projection
A 6.0 standard deviation around a 10.6 mean return is the risk band — model what that compounds to.
Work with the percentages
Your coefficient of variation is 56.5% — convert between percentage change and percentage points cleanly.
Risk-adjust your returns
Mean 10.60 with 5.99 of dispersion is the raw input for any risk-adjusted return measure.
Standard deviationmeasures how far a set of numbers typically sits from its own average. A small standard deviation means the values cluster tightly around the mean; a large one means they are spread out. Paste any list of numbers — comma, space, or line separated — and this calculator returns the sample or population standard deviation along with variance, mean, median, quartiles, IQR, standard error, coefficient of variation, Tukey outlier flags, and a full step-by-step deviation table.
The Standard Deviation Formula
Both versions share the same five steps. They differ only in what you divide by.
| Symbol | Meaning | Notes |
|---|---|---|
| x | Each individual value | Your data points |
| x̄ / μ | Sample mean / population mean | Sum ÷ count |
| n | Number of values | Count of the data set |
| Σ | Sum of | Add across all values |
| n − 1 | Degrees of freedom | Bessel's correction |
① Variance is the intermediate step
Variance = Σ(x − x̄)² ÷ (n − 1)
Standard deviation is just the square root of variance. Variance is in squared units (dollars², points²), which is why the square root is taken — it restores the original units.
② Standard error of the mean
SEM = s ÷ √n
How precisely your sample mean estimates the true mean. Quadrupling your sample size halves the standard error.
③ Coefficient of variation
CV = (s ÷ |x̄|) × 100
A unitless spread measure that lets you compare variability across data sets with different scales — the only honest way to say one set is "more variable" than another.
④ Tukey outlier fence
Lower = Q1 − 1.5 × IQR | Upper = Q3 + 1.5 × IQR
Quartile-based, so unlike standard deviation it is not itself distorted by the outliers it is trying to find.
Worked Example — data set: 2, 4, 4, 4, 5, 5, 7, 9
- Mean: (2+4+4+4+5+5+7+9) ÷ 8 = 40 ÷ 8 = 5
- Deviations: −3, −1, −1, −1, 0, 0, 2, 4
- Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 → sum = 32
- Population variance: 32 ÷ 8 = 4 → σ = 2.000
- Sample variance: 32 ÷ 7 = 4.571 → s = 2.138
- Note the 7% gap between the two — with small n, the choice matters a great deal
💡 Pro-Tip
Standard deviation quietly assumes your data is roughly symmetric — check before you trust it. Because deviations are squared, a single extreme value moves the result far more than it moves the median or the IQR. Investment returns are the classic trap: they have fat tails, so the "95% within 2σ" promise badly understates how often a portfolio has a catastrophic year. Two quick diagnostics are built into the results panel above. First, compare the actual percentages inside ±1σ, ±2σ, and ±3σ against the expected 68/95/99.7 — a large gap means the distribution is skewed. Second, compare the mean to the median: if they diverge meaningfully, report the median and IQR alongside the standard deviation instead of relying on it alone.
Standard Deviation FAQ
Should I use sample or population standard deviation?
Use sample(n − 1) unless your data literally contains every member of the group you care about. Test scores for one specific class of 30 students is a population; 30 students surveyed to learn about the whole school is a sample. When in doubt, use sample — it is the conservative choice and the default in Excel's STDEV.S, R, and virtually all statistical software.
Why divide by n − 1 instead of n?
Because the sample mean is computed from the same data, it sits slightly closer to your points than the unknown true mean does. That makes the raw squared deviations systematically too small. Dividing by n − 1 rather than n inflates the result just enough to correct the bias — this is Bessel's correction. Its effect shrinks as n grows: it matters enormously at n = 5 and is negligible at n = 1,000.
What is a good standard deviation?
There is no universal answer, because standard deviation inherits the units of your data. Use the coefficient of variationinstead: CV under 15% is low variability, 15–30% is moderate, and above 30% is high. In investing, a fund with an 18% standard deviation of annual returns is substantially more volatile than one at 9%, regardless of which returned more.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root. Variance is mathematically convenient — variances of independent variables add — but it is in squared units and therefore uninterpretable. Standard deviation is in the same units as your data, so it can be read directly: "typically ±$4,200 from the average."
How is standard deviation used in finance?
It is the definition of volatility. The standard deviation of annual returns is the headline risk number on every fund fact sheet, and it is the denominator of the Sharpe ratio= (return − risk-free rate) ÷ standard deviation, which is how investments get compared on a risk-adjusted rather than raw-return basis. The S&P 500 has a long-run annual standard deviation of roughly 15–18%.
Can standard deviation be negative?
No. It is the square root of a sum of squares, so it is always zero or positive. A standard deviation of exactly 0 means every value in the set is identical. If you compute a negative one, you have made an arithmetic error.