Free Standard Deviation Calculator Online
This free online standard deviation calculator computes standard deviation, variance, mean, and more — for both population and sample formulas.
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Standard Deviation Formulas
| Statistic | Formula | Description |
|---|---|---|
| Mean (x̄) | Σx / n | Sum of all values divided by count |
| Population Variance (σ²) | Σ(x − x̄)² / n | Average squared deviation from the mean |
| Sample Variance (s²) | Σ(x − x̄)² / (n − 1) | Unbiased estimator using Bessel's correction |
| Population Std Dev (σ) | √(σ²) | Square root of population variance |
| Sample Std Dev (s) | √(s²) | Square root of sample variance |
How Standard Deviation Works
Standard deviation is the most widely used measure of spread in statistics. This free standard deviation calculator follows four steps to produce the result — calculate online and see the full breakdown instantly:
- Calculate the mean (average) of the dataset.
- For each value, compute the squared difference from the mean: (x − mean)².
- Find the average of those squared differences. Divide by n for population, by n−1 for sample. This gives the variance.
- Take the square root of the variance. That is the standard deviation.
For the dataset {2, 4, 4, 4, 5, 5, 7, 9}: mean = 5, population variance = 4, population standard deviation = 2. This is the classic textbook example — you can confirm it with the standard deviation calculator above.
Population vs. Sample Standard Deviation
Population Standard Deviation (σ)
Use this when your dataset IS the entire population — every member is included. Formula divides by n. Examples: the heights of all 30 students in a specific class, all scores from a completed test, all temperatures recorded this week. The standard deviation calculator uses σ = √(Σ(x−x̄)²/n).
Sample Standard Deviation (s)
Use this when your data is a sample from a larger population — you're trying to estimate the true population spread. Formula divides by n−1 (Bessel's correction), which corrects for the fact that samples tend to underestimate spread. The standard deviation calculator uses s = √(Σ(x−x̄)²/(n−1)).
If you're unsure which to use, the sample standard deviation (s) is the safer default in most statistical and research contexts.
Where Standard Deviation Is Used
Finance & Investing
Standard deviation is the primary measure of investment risk (volatility). A stock with a daily return standard deviation of 3% fluctuates more than one with 0.5%. This free online standard deviation calculator helps quantify risk when comparing assets or building a portfolio.
Science & Research
Experiments report results as mean ± standard deviation. If a drug reduced blood pressure by 10 ± 2 mmHg in a trial, the standard deviation tells you how consistent the effect was across participants. This free online tool produces both values from raw trial data.
Education & Testing
SAT scores are reported with a mean and standard deviation. Standard deviation tells you how far a student's score is from the average. One standard deviation above the mean puts you roughly in the top 16% of test-takers on a normal distribution.
Using the Standard Deviation Calculator Effectively
Paste Data Directly
You can copy a column of numbers from Excel or Google Sheets and paste it directly into this free standard deviation calculator. It handles numbers separated by line breaks without needing to add commas manually. Calculate online in seconds.
Check Both Modes
The standard deviation calculator shows both population (σ) and sample (s) values. If you have the full population, use σ. If your data is a subset, use s. The difference is small for large datasets but matters for n < 30.
Apply the 68-95-99.7 Rule
Once you have the standard deviation, apply the empirical rule: ~68% of data falls within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ. Values beyond 3 standard deviations are statistical outliers.
Outliers Inflate Standard Deviation
A single extreme value can significantly raise the standard deviation. If your standard deviation seems unexpectedly large, check the range and min/max values — an outlier may be the cause. The standard deviation calculator shows range and extremes alongside σ for this reason.
Frequently Asked Questions
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