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.
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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.
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
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
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
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
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