Sample Variance Calculator: Easy-to-Use Tool for Statistical Analysis

How to use the tool

1. Enter your data in the input box, separating values with commas, spaces, or line breaks.
   Sample input 1: 2 9 4 12 7
   Sample input 2: 18, 23, 19, 27, 25, 30
2. (Optional) Tick “Show Calculation Steps” to display every deviation, square, and subtotal.
3. Press “Calculate.” The screen updates instantly with mean and sample variance, plus the full step list if selected.
4. Interpret the numbers. A larger variance signals wider data spread; a variance of 0 means all values are identical.

Formulas applied by the JavaScript

Mean

$$ \bar{x}=rac{\sum_{i=1}^{n}x_i}{n} $$

Sample variance (Bessel-corrected)

$$ s^{2}=rac{\sum_{i=1}^{n}(x_i-\bar{x})^{2}}{n-1} $$

Worked example

• Data set: 3, 5, 7, 9
• Mean: $$\bar{x}=rac{3+5+7+9}{4}=6$$
• Squared deviations: 9, 1, 1, 9 (sum = 20)
• Variance: $$s^{2}=rac{20}{4-1}=6.6667$$

Quick-Facts

• Needs at least two observations to compute variance (Hogg & Tanis, 2019).
• Dividing by n – 1 yields an unbiased estimator—Bessel’s correction (Rice, 2006).
• Units are squared (e.g., cm² if input is cm) (NIST e-Handbook, 2023).
• A sample size ≥ 30 stabilises variance estimates (Montgomery & Runger, 2020).

FAQ

What is sample variance?

Sample variance measures how far your data points spread around their mean; higher numbers indicate more dispersion (NIST e-Handbook, 2023).

Why divide by n – 1 instead of n?

Subtracting one degree of freedom corrects downward bias in variability estimates—“Bessel’s correction makes the estimator unbiased” (Rice, 2006).

How much data do I need?

Two points are the minimum, but statisticians recommend 30 + observations for stable results (Montgomery & Runger, 2020).

How do I convert variance to standard deviation?

Take the square root: $$s=rac{\sqrt{\sum (x_i-\bar{x})^{2}/(n-1)}}{}$$ (NIST, 2023).

Can variance be negative?

No. Squaring deviations guarantees non-negative results (Hogg & Tanis, 2019).

How does variance appear in ANOVA?

ANOVA partitions total variance into within-group and between-group components to test mean differences (Fisher, 1925).

Do outliers affect variance?

Yes—variance increases sharply with extreme values because squared deviations amplify large gaps (Barnett & Lewis, 1994).

When should I use population variance?

Use population variance when every member of the group is measured; divide by n because no estimation bias exists (ISO 3534-1, 2006).

Create Your Own Web Tool for Free

Tool Description:

Create a Web Tool!