Standard Deviation Calculator
Calculate variation or dispersion of a set of values. Get population and sample standard deviation, variance, and mean for comprehensive statistical analysis.
Result
- Population Std Dev (σ)
- 0.0000
- Sample Std Dev (s)
- 0.0000
- Variance (σ²)
- 0.0000
- Mean (μ)
- 0.00
Formula & Guide
Formula
σ
Population SD
σ = √[Σ(x - μ)² / n]
Square root of average squared differences
s
Sample SD
s = √[Σ(x - x̄)² / (n-1)]
Uses n-1 for unbiased estimation
σ²
Variance
σ² = Σ(x - μ)² / n
Average squared difference from mean
Formula Variables
σ
Population Standard Deviation
Measures spread for the entire population, divides by n
s
Sample Standard Deviation
Measures spread for a sample, divides by n-1 for unbiased estimation
μ or x̄
Mean
The average of all values in the dataset
n
Count
The number of values in the dataset
x
Individual Value
Each value in the dataset
Step-by-Step Scenario
Example Scenario
Values
10, 20, 30, 40, 50
1
Calculate the Mean
- Mean = (10 + 20 + 30 + 40 + 50) / 5
2
Calculate Squared Differences
- (10-30)² = 400
- (20-30)² = 100
- (30-30)² = 0
- (40-30)² = 100
- (50-30)² = 400
- Sum = 1000
3
Calculate Variance and Standard Deviation
- Variance = 1000 / 5 = 200
- Population SD = √200 ≈ 14.14
Additional Examples
Low Variation
Values: 28, 29, 30, 31, 32
Mean
30
SD
≈ 1.41 (low spread)
High Variation
Values: 10, 20, 30, 40, 50
Mean
30
SD
≈ 14.14 (high spread)
Characteristics of Standard Deviation
Measure of Spread
Standard deviation quantifies how much values deviate from the mean. Higher SD means more variability in the data.
Same Units as Data
Unlike variance, standard deviation is in the same units as the original data, making it easier to interpret.
Statistical Foundation
Essential for hypothesis testing, confidence intervals, and many statistical analyses. Used in quality control, research, and data science.
Normal Distribution Rule
For normal distributions: 68% of values fall within 1 SD, 95% within 2 SDs, and 99.7% within 3 SDs of the mean.
Important Notes
- Standard deviation requires at least 2 values. With only 1 value, there's no variation to measure.
- Population SD divides by n, while sample SD divides by n-1. Use sample SD when working with samples from larger populations.
- Variance is the square of standard deviation. It's always non-negative and measured in squared units.
- Standard deviation is sensitive to outliers. A single extreme value can significantly increase the SD.
- For normal distributions, the empirical rule (68-95-99.7) helps interpret standard deviation in terms of data distribution.
Frequently Asked Questions
Find answers to common questions about standard deviation calculations.
Standard deviation measures how spread out or dispersed a set of values is from the mean. A low standard deviation means values are close to the mean, while a high standard deviation means values are spread out over a wider range.
Population standard deviation (σ) divides by n (total count) and is used when you have data for the entire population. Sample standard deviation (s) divides by n-1 and is used when you have a sample from a larger population. Sample SD is slightly larger to account for estimation uncertainty.
Standard deviation tells you how much variation exists in your data. For example, if mean is 50 and SD is 10, most values fall between 40-60 (within one SD). About 68% of values fall within one SD, and 95% within two SDs (for normal distributions).
Variance is the square of standard deviation. It measures the average squared difference from the mean. Variance = σ². Standard deviation is the square root of variance, making it easier to interpret since it's in the same units as the data.
Use population SD when you have data for every member of the group you're studying. Use sample SD when your data is a sample from a larger population. In practice, sample SD is more common since we rarely have complete population data.
No, standard deviation is always non-negative (zero or positive). It's the square root of variance, and variance is always non-negative since it's the average of squared differences.