Standard Deviation Calculator
Population vs Sample
Population Standard Deviation (σ)
Use when your data includes the entire population being studied.
σ = √(Σ(x - μ)² / N)
Sample Standard Deviation (s)
Use when your data is a sample from a larger population.
s = √(Σ(x - x̄)² / (n-1))
Understanding Standard Deviation
Standard deviation is a measure of how spread out numbers are from their average (mean). A low standard deviation means values are close to the mean, while a high standard deviation means values are more spread out.
Step-by-Step Calculation
- Calculate the mean (average) of all values
- Subtract the mean from each value (these are the deviations)
- Square each deviation
- Calculate the average of the squared deviations (this is the variance)
- Take the square root of the variance (this is the standard deviation)
Why Divide by (n-1) for Samples?
When calculating standard deviation from a sample, we divide by (n-1) instead of n. This is called Bessel's correction. It compensates for the fact that we're using the sample mean instead of the true population mean, which tends to underestimate the actual variance.
The 68-95-99.7 Rule
For normally distributed data:
- 68% of values fall within 1 standard deviation of the mean
- 95% of values fall within 2 standard deviations of the mean
- 99.7% of values fall within 3 standard deviations of the mean
Example Calculation
Data set: 2, 4, 4, 4, 5, 5, 7, 9
- Mean = (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
- Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16
- Sum of squared deviations = 32
- Population variance = 32 / 8 = 4
- Population std dev = √4 = 2
Applications
- Quality Control: Monitoring manufacturing consistency
- Finance: Measuring investment risk and volatility
- Science: Reporting experimental precision
- Education: Grading on a curve
- Weather: Climate variation analysis
- Sports: Performance consistency metrics
Frequently Asked Questions
How do I calculate the standard deviation?
To calculate standard deviation: 1) Find the mean (average) of your data, 2) Subtract the mean from each value to get deviations, 3) Square each deviation, 4) Find the average of squared deviations (variance), 5) Take the square root of the variance. Use n for population or n-1 for sample in step 4.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) is used when you have data for an entire population and divides by N. Sample standard deviation (s) is used when you have a sample from a larger population and divides by n-1 (Bessel's correction) to provide an unbiased estimate.
What does standard deviation tell you?
Standard deviation measures how spread out numbers are from the mean. A low standard deviation means values cluster close to the average, while a high standard deviation indicates values are more dispersed. It's commonly used to measure variability, risk, and consistency.
What is a good standard deviation?
There's no universal 'good' standard deviation—it depends on context. In quality control, lower is better (more consistency). In investments, it measures risk. The coefficient of variation (CV = std dev / mean × 100%) helps compare variability across different scales.