Free Variance Calculator

What Is Variance?

Variance measures how far a set of numbers spreads out from their average value.
A high variance indicates that the data points are widely scattered, while a low variance means they are closely clustered around the mean.
Understanding variance is the first step toward calculating standard deviation, relative standard deviation, and confidence intervals—key quantities in statistics.
A free online Variance Calculator can handle both population and sample datasets instantly, giving you fast and accurate results.

Variance Formulas

Population Variance

For an entire population of $N$ values, the variance $\sigma^2$ (sigma‑squared) is defined as the average of the squared differences from the mean:

$$\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2$$

• $x_i$: the $i$-th data point
• $\mu$: the population mean
• $N$: the total number of data points

The formula first squares each deviation from the mean, then takes the arithmetic mean of those squares.

Sample Variance

When only a sample of the population is available, the same formula would produce a biased estimate (underestimating the true variance).
Bessel’s correction removes this bias by dividing by $N-1$ instead of $N$:

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

• $\bar{x}$: the sample mean
• $s^2$: the unbiased estimate of the population variance
• $N$: the number of observations in the sample

Step‑by‑Step Calculation Example

Consider eight quiz scores: 5, 5, 5, 7, 8, 8, 9, 9.

1. Calculate the mean

$$\bar{x} = \frac{5+5+5+7+8+8+9+9}{8} = \frac{56}{8} = 7$$

2. Compute deviations and squared deviations

Score	Deviation (Score – 7)	Squared Deviation
5	–2	4
5	–2	4
5	–2	4
7	0	0
8	1	1
8	1	1
9	2	4
9	2	4

Sum of squared deviations $= 4+4+4+0+1+1+4+4 = 22$.

3. Apply the variance formula

• If the data represent a population: $\displaystyle \sigma^2 = \frac{22}{8} = 2.75$
• If the data are a sample: $\displaystyle s^2 = \frac{22}{7} \approx 3.1429$

The two results differ because the sample formula uses Bessel’s correction to avoid underestimating the true population spread.

Hand‑Calculation Shortcut

When using a basic calculator, the computational form of the variance avoids computing each deviation individually:

• Population:

  $$\sigma^2 = \frac{\sum x_i^2 - \frac{(\sum x_i)^2}{N}}{N}$$

• Sample:

  $$s^2 = \frac{\sum x_i^2 - \frac{(\sum x_i)^2}{N}}{N-1}$$

Example with a sample {1, 2, 4, 6}

$$\sum x = 13,\quad \sum x^2 = 1+4+16+36 = 57$$ $$s^2 = \frac{57 - \frac{13^2}{4}}{3} = \frac{57 - 42.25}{3} = \frac{14.75}{3} \approx 4.9167$$

This result is identical to what you would get with the definitional formula, but the computation is simpler to perform on a handheld device.

Summary of Key Formulas

Quantity	Population	Sample
Mean	$\displaystyle \mu = \frac{\sum x_i}{N}$	$\displaystyle \bar{x} = \frac{\sum x_i}{N}$
Sum of squares (SS)	$\sum (x_i - \mu)^2$	$\sum (x_i - \bar{x})^2$
Variance	$\displaystyle \sigma^2 = \frac{SS}{N}$	$\displaystyle s^2 = \frac{SS}{N-1}$
Standard deviation	$\sigma = \sqrt{\sigma^2}$	$s = \sqrt{s^2}$

Using the Online Variance Calculator

To automate the whole process, enter your numbers into this free Variance Calculator. Select “population” or “sample” depending on your data, and the tool will output the variance, standard deviation, sum of squares, and other helpful statistics. Whether you are learning statistics or analysing real‑world data, the calculator saves time and minimises arithmetic mistakes.