Variance and standard deviation measure how spread out data is from the average. To calculate them by hand, start by finding the mean (average) of all values, then measure how far each value deviates from that mean.
For Variance:
- Calculate the mean: add all values and divide by the count
- Find the deviation for each value: subtract the mean from each data point
- Square each deviation
- Add all the squared deviations together
- Divide by the count (for population variance) or by count minus 1 (for sample variance)
Example: For values 2, 4, 6:
- Mean = (2+4+6)/3 = 4
- Deviations: (2-4)=-2, (4-4)=0, (6-4)=2
- Squared: 4, 0, 4
- Sum of squares: 4+0+4 = 8
- Variance = 8/3 ≈ 2.67 (population) or 8/2 = 4 (sample)
For Standard Deviation: Simply take the square root of the variance. Using the example above, standard deviation ≈ √2.67 ≈ 1.63 (population) or √4 = 2 (sample).
Use population variance (divide by n) when analyzing an entire group. Use sample variance (divide by n-1) when your data represents a sample from a larger population. The sample method corrects for bias when estimating from incomplete data. Standard deviation is preferred in practice because it's in the same units as your original data, making it more interpretable.