Classic population
For 2,4,4,4,5,5,7,9: mean=5, squared-deviation sum=32, variance=4, and SD=2.
Calculate population or sample standard deviation and variance, then inspect every deviation, squared deviation, and z-score.
Standard deviation measures spread around the mean in the original data units. Population variance divides the sum of squared deviations by N; sample variance divides by n−1 when estimating population variance from a sample. Standard deviation is the square root of the selected variance.
A value of zero means every observation is identical. Standard deviation is never negative.
σ² = Σ(xᵢ−μ)²/NThe n−1 denominator is Bessel's correction.
s² = Σ(xᵢ−x̄)²/(n−1)σ=√σ² or s=√s²For 2,4,4,4,5,5,7,9: mean=5, squared-deviation sum=32, variance=4, and SD=2.
For the same data, sample variance is 32/7 and sample SD is about 2.13809.
For 1,2,3: population variance is 2/3 and sample variance is 1.
Choose population for the entire group and sample when estimating a larger population.
Because deviations are squared, extreme observations can substantially increase variance and standard deviation.
The calculator uses a two-pass method: first compute the mean, then compute squared deviations. It does not subtract two nearly equal large quantities.
A z-score divides a value's deviation by the selected standard deviation. Z-scores are undefined when standard deviation is zero.
It measures dispersion around the mean in the original data units.
It is the sum of squared deviations from the mean divided by N for a population or n−1 for a sample estimate.
Sample variance divides by n−1; population variance divides by N.
Bessel's correction reduces bias when a sample estimates population variance.
Yes, when every value is identical.
No. It is a square root of non-negative variance.
Yes. Any finite numeric values are valid.
It is a deviation expressed in standard-deviation units.