Variance Calculator — Step-by-Step Solutions

The Variance Calculator computes the spread of a dataset by measuring how far each value lies from the mean. It supports population variance (dividing by N), sample variance with Bessel's correction (dividing by n-1), variance from a frequency table, and grouped data variance using class midpoints.

Enter your data values to see the variance, standard deviation, mean, and sum of squares with detailed step-by-step working. The calculator complements the Standard Deviation Calculator by focusing on variance-specific features and input formats.

Data values (comma-separated)

Your calculations will appear here

How to Use the Variance Calculator

Select a calculation mode: Population Variance, Sample Variance, From Frequency Table, or Grouped Data Variance.
For Population and Sample Variance, enter your data values separated by commas (e.g. 4, 8, 6, 5, 3, 7). Population variance divides by N; sample variance divides by n-1.
For Frequency Table, enter the values and their corresponding frequencies in separate fields. Choose whether to treat the data as a population or sample.
For Grouped Data, enter the lower bounds, upper bounds, and frequencies for each class interval. The calculator uses class midpoints to estimate the variance.
Results include variance, standard deviation, mean, and a full step-by-step breakdown of the calculation.

Formula and Theory

Variance measures the average squared deviation of each data point from the mean. It quantifies how spread out the values in a dataset are. A variance of zero means all values are identical; larger variance indicates greater dispersion.

Population variance uses the formula sigma squared = sum of (xi - mu) squared divided by N, where mu is the population mean and N is the total number of values. This is used when you have data for the entire population.

Sample variance uses Bessel's correction, dividing by n-1 instead of n: s squared = sum of (xi - x bar) squared divided by (n-1). This corrects the downward bias that occurs when estimating population variance from a sample. The denominator n-1 represents the degrees of freedom.

For grouped data, variance is estimated using class midpoints as representative values. Each midpoint is weighted by its frequency. This approach is common when data is presented in frequency distributions or histograms rather than as individual observations.

Worked Examples

Example 1: Population variance of a small dataset

Problem: Find the population variance of {2, 4, 4, 4, 5, 5, 7, 9}.

Solution: Mean = 40/8 = 5. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16. Sum = 32. Variance = 32/8 = 4.

Answer: Variance = 4, SD = 2

Example 2: Sample variance

Problem: Find the sample variance of {3, 7, 7, 19}.

Solution: Mean = 36/4 = 9. Squared deviations: 36, 4, 4, 100. Sum = 144. Sample variance = 144/3 = 48.

Answer: Variance = 48, SD = 6.93

Example 3: Variance from a frequency table

Problem: Values: 1, 2, 3, 4, 5 with frequencies: 3, 5, 8, 4, 2. Find the population variance.

Solution: Weighted mean = (3+10+24+16+10)/22 = 63/22 = 2.864. Weighted SS = 3(1-2.864) squared + 5(2-2.864) squared + ... = 28.409. Variance = 28.409/22 = 1.291.

Answer: Variance = 1.291

Example 4: Grouped data variance

Problem: Classes: 0-10, 10-20, 20-30 with frequencies: 5, 12, 3. Find the population variance.

Solution: Midpoints: 5, 15, 25. Weighted mean = (25+180+75)/20 = 14. SS = 5(5-14) squared + 12(15-14) squared + 3(25-14) squared = 405+12+363 = 780. Variance = 780/20 = 39.

Answer: Variance = 39

Frequently Asked Questions

What is the difference between variance and standard deviation?

Standard deviation is the square root of variance. Both measure spread, but standard deviation is in the same units as the original data, making it easier to interpret. Variance is in squared units.

When should I use population vs sample variance?

Use population variance when your data includes every member of the group you are studying. Use sample variance when your data is a subset drawn from a larger population, which is the more common scenario in research and statistics.

Why does sample variance divide by n-1 instead of n?

Dividing by n-1 (Bessel's correction) corrects a bias in the estimate. When you use the sample mean instead of the true population mean, the squared deviations tend to be slightly too small on average. Dividing by n-1 compensates for this, giving an unbiased estimate of the population variance.

Can variance be negative?

No. Variance is always zero or positive because it is a sum of squared differences. A variance of zero means all data points are identical.

How does this differ from the Standard Deviation Calculator?

The Standard Deviation Calculator computes basic statistics from raw data. This Variance Calculator adds frequency table input, grouped data input with class intervals, and focuses specifically on variance as the primary output.