Sample Variance Calculator
Calculate sample variance ($s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$), sample standard deviation ($s$), degrees of freedom ($df = n-1$), and sample mean ($\bar{x}$).
Calculate Sample Variance (s²)
Enter sample dataset values separated by commas, spaces, or lines.
Calculation Results
Calculated using Bessel's corrected sample variance formula: s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}
Quick Summary
The Sample Variance Calculator evaluates sample variance ($s^2$), sample standard deviation ($s$), degrees of freedom ($n-1$), and sample mean ($\bar{x}$).
Formula Explanation
s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}
s = \sqrt{s^2}How It Works
Sample variance estimates population dispersion from a sample dataset. The Sample Variance Calculator subtracts sample mean $\bar{x}$ from each observation, squares deviations, sums them up ($SS$), and divides by $n-1$.
Step-by-Step Worked Example
Practical Problem: Calculate sample variance for sample: 10, 15, 20, 25, 30 ($n = 5$).
- Step 1: Calculate sample mean ($\bar{x}$): $(10+15+20+25+30)/5 = 100 / 5 = \mathbf{20.0000}$.
- Step 2: Calculate squared deviations $(x_i - 20)^2$: $(-10)^2=100, (-5)^2=25, 0^2=0, 5^2=25, 10^2=100$.
- Step 3: Sum squared deviations ($SS$): $100 + 25 + 0 + 25 + 100 = \mathbf{250.0000}$.
- Step 4: Divide by $n - 1 = 4$ for sample variance ($s^2$): $s^2 = 250 / 4 = \mathbf{62.5000\text{ Sample Variance}}$.
- Step 5: Calculate Sample SD ($s$): $s = \sqrt{62.50} = \mathbf{7.9057}$.
Real-World Calculation Examples
Scenario 1: Sample Dataset (10 to 30)
Parameters: n = 5 sample
Result: Sample Variance s² 62.5000 (Sample SD 7.9057).
Scenario 2: Clinical Lab Trial Sample
Parameters: 4.0, 4.2, 3.8, 4.0
Result: Sample Variance s² 0.0267.
Scenario 3: Stock Price Daily Return Variance
Parameters: +2%, -1%, +3%, -2%
Result: Sample Variance s² 5.6667.
Scenario 4: Customer Survey Satisfaction Ratings
Parameters: 7, 8, 9, 10
Result: Sample Variance s² 1.6667.
Key Benefits of Using This Calculator
Unbiased Sample Variance Estimator
Uses Bessel's correction ($n-1$) to provide an unbiased estimate of population variance.
Sum of Squares ($SS$) Display
Displays total Sum of Squared Deviations ($SS$).
Sample Standard Deviation ($s$)
Outputs sample standard deviation ($s$) alongside variance.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is sample variance ($s^2$)?
Sample variance measures dispersion in a sample dataset using $n-1$ degrees of freedom.
How do I calculate sample variance in Excel?
Use formula =VAR.S(range).
Why is sample variance divided by (n - 1)?
Dividing by $n-1$ corrects for the fact that sample values tend to be closer to sample mean $\bar{x}$ than population mean $\mu$.
What is symbol for sample variance?
Lowercase $s^2$.
What is symbol for sample standard deviation?
Lowercase $s$.
Can sample variance be 0?
Yes — if all numbers in the sample are identical, $s^2 = 0$.
What is degrees of freedom ($df$)?
$df = n - 1$.
How are $s^2$ and $s$ related?
$s = \sqrt{s^2}$.
What is sample size $n$?
Sample size $n$ is the total number of observations in the sample dataset.
Is sample variance used in t-tests and ANOVA?
Yes — sample variance is fundamental to hypothesis testing in t-tests and ANOVA.