Population Variance Calculator
Calculate population variance ($\sigma^2 = \frac{\sum (x - \mu)^2}{N}$), population standard deviation ($\sigma$), sum of squares ($SS$), and population mean ($\mu$).
Calculate Population Variance (σ²)
Enter complete population dataset values separated by commas, spaces, or lines.
Calculation Results
Calculated using population variance equations: \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}
Quick Summary
The Population Variance Calculator evaluates population variance ($\sigma^2$), population standard deviation ($\sigma$), sum of squares ($SS$), and population mean ($\mu$).
Formula Explanation
\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}
\sigma = \sqrt{\sigma^2}How It Works
Population variance computes the mean squared distance of all population elements from the population mean $\mu$. The Population Variance Calculator sums squared deviations ($SS$) and divides directly by total population size $N$.
Step-by-Step Worked Example
Practical Problem: Calculate population variance for the complete population: 5, 10, 15, 20, 25 ($N = 5$).
- Step 1: Calculate population mean ($\mu$): $(5+10+15+20+25)/5 = 75 / 5 = \mathbf{15.0000}$.
- Step 2: Calculate squared deviations $(x_i - 15)^2$: $(-10)^2=100, (-5)^2=25, 0^2=0, 5^2=25, 10^2=100$.
- Step 3: Sum squared deviations ($SS$): $100 + 25 + 0 + 25 + 100 = \mathbf{250.0000}$.
- Step 4: Divide by total population count ($N = 5$): $\sigma^2 = 250 / 5 = \mathbf{50.0000\text{ Population Variance}}$.
- Step 5: Calculate Population SD ($\sigma$): $\sigma = \sqrt{50} = \mathbf{7.0711}$.
Real-World Calculation Examples
Scenario 1: Population Dataset (5 to 25)
Parameters: N = 5 population
Result: Population Variance σ² 50.0000 ($\sigma = 7.0711$).
Scenario 2: Complete Census Office Salaries
Parameters: All 4 executives
Result: Population Variance σ².
Scenario 3: Machine Sensor Calibration Test
Parameters: 100, 102, 98, 100
Result: Population Variance σ² 2.0000.
Scenario 4: Complete City District Rainfall
Parameters: 12, 14, 16, 18
Result: Population Variance σ² 5.0000.
Key Benefits of Using This Calculator
Unadjusted Population Variance
Divides by $N$ without sample bias inflation for complete census datasets.
Sum of Squares Output
Displays total Sum of Squared Deviations ($SS$).
Population Standard Deviation
Outputs population standard deviation ($\sigma$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is population variance ($\sigma^2$)?
Population variance measures average squared distance of all population items from the population mean $\mu$.
How do I calculate population variance in Excel?
Use formula =VAR.P(range).
Why does population variance divide by N?
Because every member of the target population is measured, there is no sample estimation bias.
What is symbol for population variance?
Sigma squared ($\sigma^2$).
Can population variance be 0?
Yes — if all population observations are identical, $\sigma^2 = 0$.
What is relationship between $\sigma^2$ and $\sigma$?
$\sigma = \sqrt{\sigma^2}$ (standard deviation is the square root of variance).
Is population variance larger or smaller than sample variance?
Population variance ($\sigma^2 = SS/N$) is smaller than sample variance ($s^2 = SS/(n-1)$) for the same numerical dataset.
What is population mean symbol?
Greek letter mu ($\mu$).
What is degrees of freedom for population variance?
Degrees of freedom equal total population size $N$.
How is population variance used in probability distributions?
Variance quantifies second central moment $E[(X - \mu)^2]$ of probability density functions.