Standard Deviation Calculator
Calculate the standard deviation, variance, and mean from any list of numbers. Shows both population and sample standard deviation.
What Is a Standard Deviation Calculator?
Standard deviation tells you how spread out a set of numbers is around the mean. A small standard deviation means most values cluster tightly around the average; a large one means they're scattered widely. It's the most commonly used measure of variability in statistics, science, finance, and data analysis.
There are two versions: population standard deviation (σ) for when your dataset is the entire population, and sample standard deviation (s) for when it's a sample drawn from a larger group. The difference is in the denominator — population uses n, sample uses n-1 (Bessel's correction). Most real-world use cases involve sample standard deviation.
Enter your numbers separated by commas or spaces. The calculator returns both versions alongside the variance and mean, so you have everything needed for further statistical work.
How Do You Use This Standard Deviation Calculator?
Type or paste your numbers into the input box, separated by commas or spaces. Click Calculate to see the sample standard deviation, population standard deviation, variance, mean, and count.
- Type or paste your numbers into the input box.
- Separate values with commas, spaces, or a combination.
- Click Calculate.
- Sample standard deviation is the primary result for most use cases.
- Population SD is shown if your dataset represents an entire population.
- Variance is the square of the standard deviation.
How Does the Standard Deviation Calculator Formula Work?
The formula used: Sample SD: s = √(Σ(x - x̄)² / (n-1)). Population SD: σ = √(Σ(x - μ)² / n)
Standard deviation measures the average distance of each value from the mean.
Step 1: Calculate the mean: x̄ = Σx / n
Step 2: Find each value's squared difference from the mean: (x - x̄)²
Step 3: Sum the squared differences: Σ(x - x̄)²
Step 4 (sample): Divide by n-1 to get variance, then take the square root: s = √(Σ(x - x̄)² / (n-1))
Step 4 (population): Divide by n instead: σ = √(Σ(x - x̄)² / n)
What Are Some Example Calculations?
Dataset: 2, 4, 4, 4, 5, 5, 7, 9. Mean = 5. Sample SD = 2.0. Population SD = 1.871. Variance (sample) = 4.0.
Exam scores: 55, 72, 68, 90, 45
Mean = 66. Deviations: -11, 6, 2, 24, -21. Squared: 121, 36, 4, 576, 441. Sum = 1178. Sample variance = 1178/4 = 294.5.
Sample SD = 17.16. Population SD = 15.35.
Consistent dataset: 10, 10, 10, 10, 10
Mean = 10. All deviations = 0. Sum of squared deviations = 0.
Standard deviation = 0. No spread — all values identical.
When Should You Use a Standard Deviation Calculator?
In data analysis and research when you need to describe how variable a dataset is. A lower standard deviation means more consistent results — useful for quality control, comparing test score variability across schools, or assessing investment risk.
In finance, standard deviation is the standard measure of volatility. A stock with a high standard deviation has more unpredictable returns than one with a low standard deviation. Most portfolio analysis tools and risk assessments are built on this calculation.
What Do These Terms Mean?
How Do the Options Compare?
| Measure | Formula | When to Use |
|---|---|---|
| Population SD (σ) | √(Σ(x-μ)²/n) | You have data on the entire population |
| Sample SD (s) | √(Σ(x-x̄)²/(n-1)) | You have a sample from a larger group |
| Variance | SD² | Intermediate step; used in further calculations |
What Are the Best Tips to Know?
- Use sample SD (s) when your data is a sample; use population SD (σ) only if you have data for the entire population.
- Standard deviation is in the same units as your original data — if data is in kg, SD is in kg.
- 68% of data falls within one SD of the mean in a normal distribution; 95% within two SDs.
- Outliers inflate standard deviation significantly — check your data for errors if SD seems unusually high.
What Mistakes Should You Avoid?
- Using population SD when you have a sample — this underestimates true variability.
- Confusing variance with standard deviation — variance is SD squared and is in squared units.
- Ignoring outliers that may indicate data entry errors rather than genuine variation.
Frequently Asked Questions
What is a good standard deviation?
It depends entirely on the context. For exam scores out of 100, an SD of 10 means most students scored within 10 marks of the average. For a precision manufacturing process, an SD of 0.001mm might be acceptable. 'Good' means low relative to the scale and purpose of the measurement.
What is the difference between population and sample standard deviation?
Population SD (σ) divides by n and is used when you have data on every member of a group. Sample SD (s) divides by n-1 and is used when your data is a sample drawn from a larger population. Sample SD gives a slightly larger value, which compensates for the fact that samples tend to underestimate variability.
Why does standard deviation use squared differences?
Squaring the differences removes negative signs (so deviations don't cancel each other out) and gives extra weight to values far from the mean. Taking the square root at the end converts the result back into the original units.
What does a standard deviation of 0 mean?
It means all values in the dataset are identical — there is no variation whatsoever.
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