Variance Calculator

How to Calculate Variance

Variance is a measure of variability in statistics that quantifies how far a set of numbers are spread out from their average value. It's an essential concept in data analysis, providing insights into the dispersion of data points.

Formulas and Their Meanings

The formula for variance depends on whether you're calculating for a population or a sample:

1. Population Variance:

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

2. Sample Variance:

$$s^2=\frac{\sum _{i=1}^n(x_i-\overline{x}{)}^2}{n-1}$$

Where:

• ${\sigma }^2$ or $s^2$ is the variance
• $x_i$ are the individual values in a dataset
• $\mu$ is the population mean
• $\overline{x}$ is the sample mean
• $N$ is the total number of values in a population
• $n$ is the number of values in a sample

Calculation Steps

1. Calculate the mean of the dataset.
2. Subtract the mean from each data point to get the deviations.
3. Square each deviation.
4. Sum all the squared deviations.
5. Divide the sum by N (for population) or n-1 (for sample).

Example Calculation

Let's calculate the sample variance for the dataset: 2, 4, 4, 4, 5, 5, 7, 9

1. Calculate the mean:
   $\overline{x}=\frac{2+4+4+4+5+5+7+9}{8}=5$
2. Calculate deviations from the mean and square them:
   (2 - 5)² = 9, (4 - 5)² = 1, (4 - 5)² = 1, (4 - 5)² = 1,
   (5 - 5)² = 0, (5 - 5)² = 0, (7 - 5)² = 4, (9 - 5)² = 16
3. Sum the squared deviations:
   9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32
4. Divide by (n-1) = 7:
   $s^2=\frac{32}{7}=4.57$

The sample variance is approximately 4.57.

Visual Representation

This diagram illustrates the distribution of the example dataset. Each blue dot represents a data point, and the red line represents the mean. The vertical spread of the points from the mean line visually represents the variance in the data.