Variance Calculator

variance calculator

Sample Variance (s²)Population Variance (σ²)

Enter numbers separated by commas, spaces, or new lines.

Answer:

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Count (n):

Mean (x̄):

Sum:

Sum of Squares (Σ(x-x̄)²):

Standard Deviation:

Variance Calculator Use

The variance calculator is a specialized statistical tool designed to measure the spread or dispersion of a data set. In simple terms, variance tells you how far each number in the set is from the mean (average) and thus from every other number in the set. Whether you are a student, researcher, or financial analyst, calculating variance is a fundamental step in understanding data variability.

This tool allows you to input raw data and instantly receive the variance, standard deviation, and mean for either a whole population or a sample of that population.

Choose Data Type

Select "Sample Variance" if your data is a subset of a larger group. Select "Population Variance" if your data represents the entire group being studied.

Data Set

Enter your numerical values here. You can separate them using commas (1, 2, 3), spaces (1 2 3), or by placing each number on a new line.

Show Sum of Squares

Check this box if you need the intermediate calculations, such as the sum of all values and the sum of the squared deviations from the mean.

How It Works

Variance is calculated by taking the differences between each number in the set and the mean, squaring those differences (to make them positive), and then averaging those squared differences. However, the "average" depends on whether you are working with a sample or a population.

The Population Variance Formula

When you have data for every member of a group, use this formula:

σ² = Σ (xᵢ – μ)² / N

• σ² = Population Variance
• Σ = Summation symbol (add them all up)
• xᵢ = Each individual value
• μ = Population mean
• N = Total number of values in the population

The Sample Variance Formula

When you only have a subset of a group, you use Bessel's correction (dividing by n-1 instead of n) to provide an unbiased estimate of the population variance:

s² = Σ (xᵢ – x̄)² / (n – 1)

• s² = Sample Variance
• x̄ = Sample mean
• n = Number of values in the sample

Calculation Example

Example: Suppose you have a sample of exam scores: 70, 80, and 90. You want to find the sample variance.

Step-by-step solution:

1. Find the Mean (x̄): (70 + 80 + 90) / 3 = 80
2. Subtract Mean from each score and square the result:
   (70 – 80)² = (-10)² = 100
   (80 – 80)² = (0)² = 0
   (90 – 80)² = (10)² = 100
3. Sum the squared differences: 100 + 0 + 100 = 200
4. Divide by (n – 1): Since n = 3, we divide by (3 – 1) = 2.
5. Result: 200 / 2 = 100. The sample variance is 100.

Common Questions

What is the difference between variance and standard deviation?

Variance is the average of squared deviations, while standard deviation is the square root of the variance. Standard deviation is often preferred because it is expressed in the same units as the original data (e.g., dollars or inches), making it easier to interpret.

Why do we square the differences?

If we didn't square the differences, the positive and negative deviations from the mean would cancel each other out, resulting in a sum of zero. Squaring ensures all values are positive so we can measure the total magnitude of the spread.

When should I use the sample variance calculator?

You should use the sample variance whenever you are drawing conclusions about a larger population based on a smaller group. Most scientific experiments and polls use sample variance because it is usually impossible or impractical to measure every single member of a population.