Calculate sample and population variance with step-by-step solutions. Includes standard deviation, mean, sum of squares, and detailed statistical analysis.
Enter at least 2 numbers separated by commas or spaces
Formula:
Enter data values and click Calculate to see results
Choose between sample variance (s²) or population variance (σ²)
Variance is a fundamental statistical measure that quantifies the spread or dispersion of a data set. Our comprehensive variance calculator helps you quickly compute both sample variance (s²) and population variance (σ²) with detailed step-by-step solutions, making it perfect for students, researchers, data analysts, and anyone working with statistical data.
Understanding variance is crucial for data analysis because it tells you how much your data points differ from the mean. A low variance indicates that data points are clustered closely around the mean, while a high variance indicates greater spread. This calculator not only computes the variance but also provides the standard deviation, mean, sum of squares, and a complete breakdown of all calculations.
Whether you're analyzing scientific experiments, financial data, quality control measurements, or academic research, this tool provides accurate results with full transparency into the calculation process. The step-by-step solutions help you understand not just the answer, but the methodology behind variance calculations.
Variance is a numerical measure of how data points in a data set are spread out from their average value (mean). It's calculated by taking the average of the squared differences from the mean. The larger the variance, the more spread out the data points are; the smaller the variance, the closer the data points are to the mean.
For example, if you measure the heights of students in a class and the variance is 25 cm², this tells you that the typical squared deviation from the mean height is 25 square centimeters. The standard deviation (square root of variance) would be 5 cm, which is more interpretable in the original units.
Understanding the difference between sample variance and population variance is crucial for correct statistical analysis. Here's a comprehensive comparison:
| Aspect | Sample Variance (s²) | Population Variance (σ²) |
|---|---|---|
| Definition | Variance of a subset of data | Variance of entire population |
| Symbol | s² | σ² (sigma squared) |
| Divisor | n - 1 (Bessel's correction) | N (population size) |
| When to Use | Working with a sample from larger population | Have data for entire population |
| Purpose | Estimate population variance | Describe actual population spread |
⚠️ Important Note:
Sample variance uses (n-1) instead of n as the divisor to provide an unbiased estimate of the population variance. This correction, known as Bessel's correction, compensates for the fact that sample data tends to be less spread out than the full population. Most real-world statistical analyses use sample variance unless you have data for the entire population.
Choose "Sample" if you're working with a subset of data, or "Population" if you have data for the entire population.
Input your data values separated by commas or spaces. You need at least 2 numbers to calculate variance.
The calculator will compute variance, standard deviation, mean, sum of squares, and provide a complete deviation breakdown.
Examine the variance, standard deviation, summary statistics, deviation table, and step-by-step solution to fully understand the calculations.
Use the variance to understand data spread. Higher variance means more spread; lower variance means data is closer to the mean.
Example: Calculate sample variance for data set: 10, 12, 15, 16, 20
Result: Sample variance = 14.8, Sample standard deviation ≈ 3.847
Example: Calculate population variance for the same data set
Result: Population variance = 11.84, Population standard deviation ≈ 3.441
Measure investment risk and volatility. Higher variance in stock returns indicates greater risk. Portfolio managers use variance to balance risk and return in investment strategies.
Manufacturing uses variance to ensure product consistency. Low variance in product measurements indicates consistent quality, while high variance suggests inconsistent production processes.
Researchers use variance to assess experimental reliability and measurement precision. Low variance in repeated measurements indicates reliable, reproducible results.
Analyze test score distributions and student performance variability. High variance suggests diverse skill levels, while low variance indicates uniform performance across students.
Meteorologists use variance to analyze temperature fluctuations, rainfall variability, and climate patterns. Helps predict weather extremes and climate change impacts.
Medical professionals use variance to assess treatment effectiveness, patient response variability, and diagnostic test reliability. Critical for clinical trials and medical research.
Wrong: Using population variance (dividing by n) when you have sample data.
Correct: Use sample variance (dividing by n-1) when working with a subset of data to get an unbiased estimate.
Wrong: Adding up deviations without squaring them first.
Correct: Always square each deviation before summing. This ensures all values are positive and emphasizes larger deviations.
Wrong: Treating variance and standard deviation as interchangeable.
Correct: Standard deviation = √(variance). Variance is in squared units; standard deviation is in original units.
Wrong: Using an estimated or rounded mean for variance calculations.
Correct: Always calculate the exact mean from your data set first. Rounding the mean can introduce significant errors in variance.
Wrong: Forgetting that variance is in squared units of the original data.
Correct: If data is in meters, variance is in meters². For interpretation, use standard deviation (meters) instead.
Wrong: Rounding intermediate calculations (mean, deviations) before final result.
Correct: Keep full precision throughout all calculations. Only round the final variance and standard deviation.
Use the computational formula s² = [Σx² - (Σx)²/n] / (n-1) for easier calculation with large datasets. It's algebraically equivalent but requires less intermediate steps.
Verify that Σ(deviations) ≈ 0. The sum of deviations from the mean should be zero (or very close due to rounding). If not, you've made an error in calculating the mean or deviations.
Compare variance to the mean. For meaningful interpretation, calculate the coefficient of variation (CV = standard deviation / mean × 100%) to compare variability across different datasets.
Variance is sensitive to outliers because deviations are squared. One extreme value can dramatically increase variance. Always check for and consider outliers in your analysis.
While variance is useful for calculations, standard deviation (√variance) is easier to interpret because it's in the same units as your data. Report both when appropriate.
Variance equals zero only when all values are identical. If you get zero variance, all your data points are the same. If you get negative variance, you've made a calculation error.
Var(X) ≥ 0 - Variance is always non-negative because it's based on squared deviations. Zero variance means no variability (all values are identical).
Var(X + c) = Var(X) - Adding a constant to all data points doesn't change the variance. The spread remains the same; only the location shifts.
Var(cX) = c²Var(X) - Multiplying all data by a constant multiplies the variance by the square of that constant. This is why variance has squared units.
For independent variables X and Y: Var(X + Y) = Var(X) + Var(Y). Variances add for independent variables, which is fundamental to many statistical analyses.
Test your understanding with these practice problems. Use our calculator to check your answers!
Problem 1: Beginner
Calculate the sample variance for the data set: 5, 7, 9, 11, 13Hint: First find the mean, then calculate squared deviations.
Problem 2: Beginner
Find the population variance for: 2, 4, 6, 8Hint: Population variance divides by N, not (N-1).
Problem 3: Intermediate
A sample has variance 16. What is the standard deviation? If each value in the dataset is multiplied by 3, what is the new variance?Hint: Remember the relationship between variance and standard deviation, and how variance changes with multiplication.
Problem 4: Intermediate
The mean of a dataset is 50 and the variance is 25. All values are increased by 10. What are the new mean and variance?Hint: Think about how adding a constant affects mean vs. variance.
Problem 5: Advanced
Sample A: 10, 20, 30, 40, 50. Sample B: 28, 29, 30, 31, 32. Which sample has greater variance? Why does this make sense intuitively?Hint: Both have the same mean, but different spreads.
Problem 6: Challenge
Test scores have a mean of 75 and standard deviation of 8. If scores are converted using the formula (New = 2 × Old + 10), what are the new mean and standard deviation?Hint: Apply both constant multiplication and addition properties.
Variance (s² or σ²) is the average of squared deviations from the mean, while standard deviation (s or σ) is the square root of variance. Standard deviation is in the same units as the original data, making it more interpretable. Variance is in squared units. For example, if data is in meters, variance is in meters² and standard deviation is in meters. Both measure spread, but standard deviation is generally easier to understand.
Use sample variance (s²) when you're working with a subset of data from a larger population and want to estimate the population variance. Use population variance (σ²) only when you have data for the entire population. In practice, most analyses use sample variance because we rarely have complete population data. The key difference is the divisor: sample variance divides by (n-1) while population variance divides by N.
This is called Bessel's correction. When we use sample data to estimate population variance, dividing by n tends to underestimate the true population variance because the sample mean is closer to the sample data than the population mean would be. Dividing by (n-1) instead of n corrects this bias and provides an unbiased estimate of the population variance. This is why sample variance using (n-1) is also called "unbiased variance."
No, variance cannot be negative. Since variance is calculated by squaring deviations from the mean and then averaging them, and squared numbers are always non-negative, variance must be non-negative (≥ 0). The minimum variance is zero, which occurs only when all data points are identical. If you calculate a negative variance, you've made an error in your calculations—check your arithmetic, especially the mean calculation and the divisor.
High variance indicates that data points are spread far from the mean, showing high variability or diversity in the dataset. For example, in finance, high variance in stock returns indicates high volatility and risk. In quality control, high variance suggests inconsistent production. However, "high" is relative—compare variance to the mean or to variances in similar datasets. The coefficient of variation (CV = standard deviation/mean) helps make this comparison.
Low variance indicates that data points cluster closely around the mean, showing consistency and homogeneity. For example, low variance in test scores suggests students performed similarly. In manufacturing, low variance indicates consistent, high-quality production. Zero variance means all values are identical. Low variance isn't always desirable—in some contexts like investment portfolios, you might want some variability for potential returns.
In a normal distribution, variance (or standard deviation) determines the spread of the bell curve. About 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is the empirical rule (68-95-99.7 rule). Higher variance means a wider, flatter bell curve; lower variance means a narrower, taller curve. Variance is one of the two parameters (along with mean) that completely define a normal distribution.
Sum of squares (SS) is the total of all squared deviations from the mean: SS = Σ(xᵢ - x̄)². It's the numerator in the variance formula. SS represents the total variation in the dataset. Variance is essentially the average squared deviation, calculated by dividing SS by n (for population) or (n-1) (for sample). SS is fundamental to many statistical techniques including ANOVA and regression analysis.
Variance is very sensitive to outliers because deviations are squared. A single extreme value can dramatically increase variance. For example, the dataset (1, 2, 3, 4, 5) has variance ≈2.5, but (1, 2, 3, 4, 100) has variance ≈1,555. If outliers are errors or anomalies, consider removing them. If they're legitimate data, report both the variance with and without outliers, or use robust measures of spread like interquartile range (IQR) that are less sensitive to outliers.
Yes, but use caution. Variance magnitude depends on both spread and scale. A dataset with values 1000-1100 will have much larger variance than 1-10 even with similar relative spread. For meaningful comparison across datasets with different means or units, use the coefficient of variation (CV = standard deviation / mean × 100%), which expresses variability as a percentage of the mean. This allows you to compare relative variability independent of scale.
Our Variance Calculator provides a comprehensive, user-friendly tool for calculating both sample and population variance with complete transparency and educational value. Whether you're a statistics student learning about variability measures, a researcher analyzing experimental data, a business analyst examining performance metrics, or a data scientist working on machine learning models, this calculator delivers accurate results with detailed explanations.
Understanding variance is fundamental to statistics, data analysis, and research. By mastering variance calculations and interpretations, you develop critical analytical skills for understanding data spread, variability, and distribution. The step-by-step solutions provided by our calculator help you learn the methodology, not just obtain the answer, building deeper statistical understanding.
Start using our calculator today to compute variance and standard deviation quickly, verify your homework, prepare for exams, or analyze your research data. With proper mathematical notation, detailed deviation breakdowns, summary statistics, and comprehensive explanations, understanding and applying variance concepts has never been easier. Make this calculator your essential tool for all statistical variance calculations and build confidence in your data analysis skills!