Population Standard Deviation Calculator
Calculate population standard deviation ($\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$), population variance ($\sigma^2$), sum of squares ($SS$), and population mean ($\mu$).
Calculate Population Standard Deviation (σ)
Enter entire population dataset values separated by commas, spaces, or lines.
Calculation Results
Calculated using population standard deviation equations: \sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}
Quick Summary
The Population Standard Deviation Calculator evaluates population standard deviation ($\sigma$), population variance ($\sigma^2$), sum of squares ($SS$), and population mean ($\mu$).
Formula Explanation
\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}
\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}How It Works
When you measure every member of an entire population (rather than a sample subset), degrees of freedom equal total count $N$. The Population Standard Deviation Calculator computes population mean $\mu$, squared deviations, population variance $\sigma^2$, and population standard deviation $\sigma$.
Step-by-Step Worked Example
Practical Problem: Calculate population standard deviation for all 5 employees in a small department: 12, 18, 24, 30, 36 ($N = 5$).
- Step 1: Calculate population mean ($\mu$): $(12+18+24+30+36)/5 = 120 / 5 = \mathbf{24.0000}$.
- Step 2: Calculate squared deviations $(x_i - 24)^2$: $(-12)^2=144, (-6)^2=36, 0^2=0, 6^2=36, 12^2=144$.
- Step 3: Sum squared deviations ($SS$): $144 + 36 + 0 + 36 + 144 = \mathbf{360.0000}$.
- Step 4: Divide by total population count ($N = 5$): $\sigma^2 = 360 / 5 = \mathbf{72.0000\text{ Population Variance}}$.
- Step 5: Take square root for population standard deviation ($\sigma$): $\sigma = \sqrt{72} = \mathbf{8.4853}$.
Real-World Calculation Examples
Scenario 1: Department Census (12, 18, 24, 30, 36)
Parameters: Population N = 5
Result: Population SD σ 8.4853 (Variance 72.00).
Scenario 2: Complete Census Census Data
Parameters: All 10 state regional offices
Result: Population SD σ (No sample bias correction needed).
Scenario 3: Total Factory Daily Production Batches
Parameters: All 3 shift outputs: 100, 105, 95
Result: Population SD σ 4.0825.
Scenario 4: Complete Country Olympic Medals Count
Parameters: All 4 competing teams
Result: Population Variance σ².
Key Benefits of Using This Calculator
True Population Parameter ($\sigma$)
Computes exact population parameter without sample $n-1$ inflation.
Population Variance ($\sigma^2$)
Outputs population variance ($\sigma^2$) alongside standard deviation.
Sum of Squares Breakdown
Displays total Sum of Squared Deviations ($SS$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is population standard deviation ($\sigma$)?
Population standard deviation measures dispersion when data includes every individual observation of an entire population.
When should I use population standard deviation instead of sample?
Use population standard deviation when you have data for every single member of the target group (e.g. census data).
Why is population standard deviation smaller than sample standard deviation?
Population SD divides by $N$, whereas sample SD divides by $n-1$ (a smaller denominator), making sample SD slightly larger.
How do I calculate population SD in Excel?
Use formula =STDEV.P(range).
What symbol represents population standard deviation?
The lowercase Greek letter sigma ($\sigma$).
What is population mean symbol?
The lowercase Greek letter mu ($\mu$).
Can population standard deviation be zero?
Yes — if every member of the population has the exact same value, $\sigma = 0$.
What is population variance symbol?
Sigma squared ($\sigma^2$).
How does N affect population standard deviation?
Larger $N$ increases precision of population parameter calculations.
What is standard error of population mean?
If the entire population is measured, standard error is 0 because there is no sampling error.