Standard Deviation Calculator: Variance & SD
Calculate Variance and Standard Deviation (Population & Sample).
Standard Deviation
Calculate standard deviation, variance, and data spread
Quick Tips
- • Standard deviation measures data spread
- • Variance is the square of standard deviation
- • Low std dev = values close to mean
- • High std dev = values spread out from mean
What is Standard Deviation?
Standard deviation measures how spread out your numbers are from the mean.
- Low SD: numbers cluster near the mean
- High SD: numbers are more spread out
It’s one of the most common ways to describe variability in statistics.
Variance vs Standard Deviation
| Measure | Symbol | Unit | What it means |
|---|---|---|---|
| Variance | σ² (pop) / s² (sample) | squared units | average squared distance from mean |
| Standard Deviation | σ (pop) / s (sample) | same units as data | typical distance from mean |
Population vs Sample (Quick Explanation)
Population means you have all data points in the group you care about. Sample means your data is only a part of a larger group.
That’s why sample variance divides by (n − 1) instead of n. It reduces bias when estimating the population variance.
Formulas
Mean: μ = (Σx) / n
Population Variance: σ² = Σ(x − μ)² / n
Population SD: σ = √(σ²)
Sample Variance: s² = Σ(x − x̄)² / (n − 1)
Sample SD: s = √(s²)
How to Calculate Standard Deviation (Step-by-Step)
- 1Find the mean of the dataset.
- 2Subtract the mean from each value (deviation).
- 3Square each deviation.
- 4Add the squared deviations.
- 5Divide by n (population) or n−1 (sample) to get variance.
- 6Take the square root of variance to get standard deviation.
Worked Example (Same Data, Both Results)
| Dataset | Mean | Population Variance (σ²) | Population SD (σ) | Sample Variance (s²) | Sample SD (s) |
|---|---|---|---|---|---|
| 5, 10, 15, 20 | 12.5 | 31.25 | 5.5902 | 41.6667 | 6.4550 |
Which one should you use? If your dataset is the complete group you care about, use population. If it’s a subset and you want to estimate a larger group, use sample.
How to Interpret SD (In Plain Words)
SD tells you the “typical distance” from the mean.
- If SD is close to 0, values are very consistent.
- If SD is large, values vary a lot.
Two datasets can have the same mean but very different standard deviations.
Visual: Tight vs Spread-Out Data
Common Uses of Standard Deviation
- Quality control: consistency of product measurements
- Finance: volatility of returns
- Education: spread of test scores
- Science/engineering: measurement error and reliability
- Comparing datasets with the same mean
Standard Deviation FAQs
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean. Standard deviation is the square root of variance, so it uses the same unit as your data.
When should I use population vs sample standard deviation?
Use population when you have the full group. Use sample when your data is a subset and you want to estimate the population’s spread.
Why does sample variance divide by (n − 1)?
Dividing by (n − 1) helps correct underestimation of population variance when you only have a sample.
Can standard deviation be negative?
No. It’s based on squared distances and a square root, so it’s always 0 or positive.
What does a standard deviation of 0 mean?
All values are identical. There is no variability.
Does standard deviation handle decimals and negatives?
Yes. SD works for any real numbers, including negatives and decimals.
Is SD affected by outliers?
Yes. Outliers can increase variance and SD a lot because deviations are squared.
What’s the fastest way to reduce SD in a process?
In real-world processes, you reduce SD by reducing sources of variation (inputs, measurement noise, or process instability).