Standard Deviation Calculator
Calculate standard deviation, variance, mean & more instantly. Enter data and get sample or population SD with step-by-step workings.
What is Standard Deviation?
Standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range.
Standard Deviation Formulas
Sample Standard Deviation
s = √[Σ(x - x̄)² / (n - 1)]
Use this when your data represents a sample from a larger population.
Population Standard Deviation
σ = √[Σ(x - μ)² / N]
Use this when your data represents the entire population.
Understanding the Results
- 68% of data falls within 1 standard deviation of the mean
- 95% of data falls within 2 standard deviations
- 99.7% of data falls within 3 standard deviations
This is known as the Empirical Rule or 68-95-99.7 Rule for normally distributed data.
How to Calculate Standard Deviation by Hand
Follow these steps to compute standard deviation without a calculator:
Step 1: Find the Mean
Add all values and divide by the count.
Data: 4, 8, 6, 5, 3, 7
Mean = (4 + 8 + 6 + 5 + 3 + 7) / 6 = 33 / 6 = 5.5
Step 2: Find Each Deviation from the Mean
| Value (x) | x − x̄ | (x − x̄)² |
|---|---|---|
| 4 | −1.5 | 2.25 |
| 8 | +2.5 | 6.25 |
| 6 | +0.5 | 0.25 |
| 5 | −0.5 | 0.25 |
| 3 | −2.5 | 6.25 |
| 7 | +1.5 | 2.25 |
Step 3: Sum the Squared Deviations
Σ(x − x̄)² = 2.25 + 6.25 + 0.25 + 0.25 + 6.25 + 2.25 = 17.50
Step 4: Divide by (n − 1) for Sample or n for Population
- Sample variance: 17.50 / (6 − 1) = 17.50 / 5 = 3.50
- Population variance: 17.50 / 6 = 2.917
Step 5: Take the Square Root
- Sample SD: √3.50 = 1.871
- Population SD: √2.917 = 1.708
Worked Example: Exam Scores
A teacher records these test scores for 8 students: 72, 85, 90, 65, 78, 82, 88, 76
| Statistic | Value |
|---|---|
| Mean | 79.5 |
| Sample SD | 8.26 |
| Population SD | 7.73 |
| Variance (sample) | 68.29 |
| Range | 25 |
| Min / Max | 65 / 90 |
Interpretation: Scores vary by about 8.3 points from the class average of 79.5. Using the empirical rule, roughly 68% of students scored between 71.2 and 87.8.
Sample vs. Population: Side-by-Side Comparison
| Feature | Sample SD (s) | Population SD (σ) |
|---|---|---|
| Denominator | n − 1 | n |
| Use when | Data is a subset | Data is the entire group |
| Correction | Bessel’s correction (less biased) | No correction needed |
| Example | 50 surveyed customers | All 50 employees’ salaries |
| Most common? | ✅ Yes — almost always | Rare in practice |
Rule of thumb: If you’re unsure, use sample standard deviation. You almost never have the full population.
Sample vs. Population
Choose Sample when:
- Your data is a subset of a larger group
- You want to estimate the population variability
- Most practical applications
Choose Population when:
- You have data for the entire group
- You’re calculating for a complete, finite dataset
Frequently Asked Questions
How do I calculate standard deviation?
To calculate standard deviation: (1) Find the mean of your data, (2) Subtract the mean from each value to get deviations, (3) Square each deviation, (4) Find the mean of squared deviations (divide by n-1 for sample, n for population), (5) Take the square root. Or simply enter your data in our calculator above!
What is the difference between sample and population standard deviation?
Sample standard deviation (s) divides by n-1 (called Bessel’s correction) because samples tend to underestimate population variability. Population standard deviation (σ) divides by N because you have all the data. For most real-world applications, use sample SD.
What is a “good” standard deviation?
There’s no universal “good” standard deviation—it depends entirely on your context and what you’re measuring. A SD of 5 might be small for house prices but large for exam scores. Compare SD to the mean using the coefficient of variation (CV = SD/mean × 100%) for relative comparisons.
What does standard deviation tell you?
Standard deviation tells you how spread out values are from the average. A small SD means values cluster tightly around the mean (consistent data). A large SD means values are widely spread (variable data). It’s the most common measure of variability.
What is variance vs standard deviation?
Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance. SD is preferred because it’s in the same units as your data, making it easier to interpret. Variance = SD².
How does standard deviation relate to the normal distribution?
For normally distributed data, the 68-95-99.7 rule applies: 68% of data falls within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD. This makes SD useful for identifying outliers and understanding data spread.
When should I use standard deviation vs. range?
Standard deviation uses all data points and is more stable than range (which only uses min and max). Use SD for formal analysis and when you need a precise measure of spread. Use range for quick, rough estimates or very small samples.
Related Tools
- Mean Calculator - Calculate the average
- Z-Score Calculator - Standardize your values
- Histogram Generator - Visualize your data distribution
- Variance Calculator - Full descriptive statistics
- Sample Size Calculator - Determine required sample size
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