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Variance Calculator

Calculate population and sample variance, standard deviation, mean, and coefficient of variation. Free statistical analysis tool for data sets of any size.

Last revised 2 September 2026

By Team GlobalCalqulate · About our editorial standards

Data Input & Options

Data Values (comma separated)

Enter numbers separated by commas.

Data Type

Sample for a subset; population for complete data.

Decimal Places

Methodology

Formula:: Mean
Content last updated:: September 2026
Scheduled re-verification:: September 2027

Calculation type: Exact calculation.

Formula & methodology

Formula used

Mean

x̄ = Σxᵢ / n

The arithmetic average of the values you enter.

Variance

s² = Σ(xᵢ − x̄)² / (n − 1)   [sample]     σ² = Σ(xᵢ − x̄)² / n   [population]

The mean squared distance from the mean. The sample form divides by n − 1 (Bessel's correction) to avoid underestimating the spread of the wider population.

Standard deviation

s = √s²

The square root of variance, expressed in the original units.

Variable definitions

Symbol	Meaning	Unit
xᵢ	Each value. one data point from your set	—
n	Count. how many values you entered	count
x̄	Mean. the arithmetic average	—

Step-by-step calculation

1. Enter your data
Comma-separated values; at least two are required.
2. Choose sample or population
This selects the divisor: n − 1 or n.
3. Read variance and its square root
Variance is in squared units; the standard deviation returns to the original units.

Limitations

Variance is expressed in squared units (dollars squared, metres squared), which is why the standard deviation is usually the more interpretable figure.
Squaring makes the measure highly sensitive to outliers and to data-entry errors.
A variance computed from a small sample is itself imprecise; it estimates the population value rather than establishing it.

This page is educational. It applies the formula described above to the figures you enter and shows the arithmetic behind the result. The output is only as good as the inputs and the assumptions listed here, and it is general information rather than advice about your particular situation.

Help & FAQs

Frequently Asked Questions

Clear answers to common questions to help you use this calculator confidently.

What is variance in simple terms?

Tap to view the answer

Variance measures how spread out your data is from the average. If all values are similar, variance is low. If values differ widely, variance is high. Think of it as 'average squared distance from the mean'.

What is the difference between variance and standard deviation?

Tap to view the answer

Variance is measured in squared units, making it hard to interpret directly. Standard deviation is the square root of variance, measured in the original units. Use variance for further calculations, standard deviation for understanding real-world spread.

When should I use population variance vs. sample variance?

Tap to view the answer

Use population variance when you have all the data (a complete customer database, all product batches). Use sample variance when you have a sample that represents a larger population (a survey of 100 out of 10,000 customers).

Why is sample variance divided by (n−1) instead of n?

Tap to view the answer

This is Bessel's correction. Samples tend to underestimate true population variance, so dividing by (n−1) instead of n provides an unbiased estimate — the smaller denominator slightly increases the sample variance to better represent the population.

What is the 68-95-99.7 rule?

Tap to view the answer

In a normal distribution: about 68% of data falls within ±1 standard deviation of the mean, 95% within ±2, and 99.7% within ±3. This rule helps identify outliers — values beyond 3 SDs are extremely rare and often signal errors or special cases.

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Understanding Your Results

Variance
The average of squared differences from the mean. Measures spread in squared units.
Standard Deviation
The square root of variance. In the same units as the original data, making it more interpretable.
Sample vs Population
Sample uses n-1 (Bessel's correction) for unbiased estimation. Population uses n.

Key Tips

✓
Variance = average of squared deviations. SD = square root of variance. SD is in original units.
✓
Use sample variance (n-1) when working with a subset. Use population (n) when you have complete data.
✓
A low variance means data clusters near the mean. High variance means wide spread.
✓
CV = SD/Mean x 100% compares variability across data sets with different scales.

How Variance and Standard Deviation Work

Why Squared Deviations?

Variance uses squared differences from the mean rather than raw differences. Without squaring, negative and positive deviations cancel to zero. Squaring ensures all deviations contribute positively to the spread measure. The tradeoff: variance is in squared units (dollars^2, inches^2), which is why standard deviation takes the square root to return to original units.

Sample vs Population - When n-1 Matters

When you have data for an entire population (e.g., test scores for every student in a class), population variance divides by n. When you work with a sample (e.g., surveying 100 people to estimate a city's opinion), sample variance divides by n-1. Bessel's correction accounts for the sample mean being estimated from the same data, making variance slightly too low without correction. The effect is largest with small samples.

Real Applications of Variance

In finance, variance measures investment volatility - a stock with high variance has unpredictable returns. In manufacturing, variance from target specifications indicates quality problems. In scientific research, variance analysis (ANOVA) determines whether experimental groups differ significantly. A low CV (< 10%) in lab results indicates precise measurements.

Data Quality Checklist

• All values are numeric (no text, symbols, or units in cells). • No duplicate entries (unless intentional repetition). • All values in same units (cm, not mixed with inches). • No obviously wrong entries (typos checked). • Decided: population or sample variance?. • Documented: where data came from and what it represents.

What is Variance and Standard Deviation?

Variance measures how spread out data points are from the average (mean). It quantifies the degree of variability or dispersion in a dataset. Standard deviation is the square root of variance and represents the average distance of data points from the mean. Both metrics are fundamental to statistics, data science, finance, and quality control. Understanding variance is critical for risk assessment, quality management, and data-driven decision making. High variance signals inconsistency; low variance indicates stability. For example, investment portfolios use standard deviation to measure risk, manufacturers use it to ensure product consistency, and researchers use it to validate experimental accuracy.

Real-World Applications:

• Finance: Measure investment volatility and portfolio risk. • Quality Control: Monitor manufacturing process consistency. • Medical Research: Analyze patient data variability and treatment consistency. • Education: Evaluate test score distribution and student performance. • Data Science: Feature scaling, anomaly detection, and model validation. • Weather Forecasting: Assess temperature and precipitation variability. The key distinction: population variance uses all available data points (accurate for complete datasets), while sample variance adjusts for sample bias using (n-1) instead of n (correct for samples drawn from larger populations). Choosing the right metric ensures accurate analysis and reliable conclusions.

Population Variance Formula

σ² = Σ(xᵢ - μ)² / N. σ² = Population variance. xᵢ = Each individual data value. μ = Population mean (average of all values). N = Total number of data points. Σ = Sum (add all values).

Sample Variance Formula

s² = Σ(xᵢ - x̄)² / (n - 1). s² = Sample variance. x̄ = Sample mean. n = Number of samples (note: n-1, not n).

Standard Deviation Formula

σ = √(σ²) or s = √(s²). Standard deviation is simply the square root of variance. It converts variance (squared units) back to original data units, making it more interpretable.

Step-by-Step Calculation Example

Data: 10, 12, 15, 18, 20. Step 1: Calculate Mean. Mean = (10 + 12 + 15 + 18 + 20) ÷ 5 = 75 ÷ 5 = 15. Step 2: Find Differences from Mean. 10 - 15 = -5. 12 - 15 = -3. 15 - 15 = 0. 18 - 15 = +3. 20 - 15 = +5. Step 3: Square Each Difference. (-5)² = 25. (-3)² = 9. (0)² = 0. (3)² = 9. (5)² = 25. Step 4: Average the Squared Differences. Population Variance = (25 + 9 + 0 + 9 + 25) ÷ 5 = 68 ÷ 5 = 13.6. Sample Variance = 68 ÷ 4 = 17 (using n-1). Step 5: Calculate Standard Deviation. Population Std Dev = √13.6 = 3.69. Sample Std Dev = √17 = 4.12.

Why Use (n-1) for Sample Variance?

When calculating sample variance, we divide by (n-1) instead of n. This is called Bessel's correction and compensates for sample bias. Samples tend to slightly underestimate true population variance, so (n-1) produces an unbiased estimate. Rule: Use population formula only when analyzing ALL data. Use sample formula when your data is a sample representing a larger population.

Enter Your Data Values

Input numbers separated by spaces, commas, or line breaks. Example: "10 20 30 40 50" or "10, 20, 30, 40, 50". You need at least 2 values to calculate variance. 2.

Choose the Right Metric

Use Population Variance when analyzing your entire dataset (all customers, all products). Use Sample Variance when working with a sample that represents a larger population (survey of 500 people from 100,000). 4.

Interpret Standard Deviation

Standard deviation shows spread in original data units. If mean income is $50K with SD=$10K, most employees earn between $40K-$60K. Larger SD = more diversity/inconsistency.

Formula & methodology

Formula used

Mean

x̄ = Σxᵢ / n

The arithmetic average of the values you enter.

Variance

s² = Σ(xᵢ − x̄)² / (n − 1)   [sample]     σ² = Σ(xᵢ − x̄)² / n   [population]

The mean squared distance from the mean. The sample form divides by n − 1 (Bessel's correction) to avoid underestimating the spread of the wider population.

Standard deviation

s = √s²

The square root of variance, expressed in the original units.

Variable definitions

Symbol	Meaning	Unit
xᵢ	Each value. one data point from your set	—
n	Count. how many values you entered	count
x̄	Mean. the arithmetic average	—

Step-by-step calculation

1. Enter your data
Comma-separated values; at least two are required.
2. Choose sample or population
This selects the divisor: n − 1 or n.
3. Read variance and its square root
Variance is in squared units; the standard deviation returns to the original units.

Limitations

Variance is expressed in squared units (dollars squared, metres squared), which is why the standard deviation is usually the more interpretable figure.
Squaring makes the measure highly sensitive to outliers and to data-entry errors.
A variance computed from a small sample is itself imprecise; it estimates the population value rather than establishing it.

How Variance and Standard Deviation Work

Why Squared Deviations?

Sample vs Population - When n-1 Matters

Real Applications of Variance

Data Quality Checklist

What is Variance and Standard Deviation?

Real-World Applications:

Population Variance Formula

σ² = Σ(xᵢ - μ)² / N. σ² = Population variance. xᵢ = Each individual data value. μ = Population mean (average of all values). N = Total number of data points. Σ = Sum (add all values).

Sample Variance Formula

s² = Σ(xᵢ - x̄)² / (n - 1). s² = Sample variance. x̄ = Sample mean. n = Number of samples (note: n-1, not n).

Standard Deviation Formula

σ = √(σ²) or s = √(s²). Standard deviation is simply the square root of variance. It converts variance (squared units) back to original data units, making it more interpretable.

Step-by-Step Calculation Example

Why Use (n-1) for Sample Variance?

Enter Your Data Values

Input numbers separated by spaces, commas, or line breaks. Example: "10 20 30 40 50" or "10, 20, 30, 40, 50". You need at least 2 values to calculate variance. 2.

Choose the Right Metric

Interpret Standard Deviation

Standard deviation shows spread in original data units. If mean income is $50K with SD=$10K, most employees earn between $40K-$60K. Larger SD = more diversity/inconsistency.

Calculator inputs

Data Input & Options

Your results

Methodology

Formula & methodology

Formula used

Mean

Variance

Standard deviation

Variable definitions

Step-by-step calculation

Limitations

What is variance in simple terms?

What is the difference between variance and standard deviation?

When should I use population variance vs. sample variance?

Why is sample variance divided by (n−1) instead of n?

What is the 68-95-99.7 rule?

Related calculators

More everyday calculators

Calculator inputs

Data Input & Options

Your results

Methodology

How to Use This Calculator

Example: 5, 7, 9, 11, 13 (Sample)

Understanding Your Results

Key Tips

How Variance and Standard Deviation Work

Why Squared Deviations?

Sample vs Population - When n-1 Matters

Real Applications of Variance

Data Quality Checklist

What is Variance and Standard Deviation?

Real-World Applications:

Population Variance Formula

Sample Variance Formula

Standard Deviation Formula

Step-by-Step Calculation Example

Why Use (n-1) for Sample Variance?

Enter Your Data Values

Choose the Right Metric

Interpret Standard Deviation

Formula & methodology

Formula used

Mean

Variance

Standard deviation

Variable definitions

Step-by-step calculation

Limitations

How to Use This Calculator

Example: 5, 7, 9, 11, 13 (Sample)

Understanding Your Results

Key Tips

How Variance and Standard Deviation Work

Why Squared Deviations?

Sample vs Population - When n-1 Matters

Real Applications of Variance

Data Quality Checklist

What is Variance and Standard Deviation?

Real-World Applications:

Population Variance Formula

Sample Variance Formula

Standard Deviation Formula

Step-by-Step Calculation Example

Why Use (n-1) for Sample Variance?

Enter Your Data Values

Choose the Right Metric

Interpret Standard Deviation