What does standard deviation actually tell me?
How spread out values are around the mean. A small standard deviation means the data clusters tightly; a large one means it is dispersed. Two data sets can share an identical mean and be completely different — exam scores averaging 70 with a standard deviation of 3 describe a uniform class, while the same mean with a deviation of 25 describes a split one. The average alone hides that entirely.
What is the difference between population and sample standard deviation?
The population version divides by n; the sample version divides by n minus 1. That adjustment, Bessel's correction, exists because a sample's mean is itself estimated from the sample, which systematically underestimates the true spread. Use the sample formula whenever your data is a subset of something larger — which is nearly always. The difference is negligible for large n and substantial below about 20 values.
Why square the deviations instead of just taking absolute values?
Squaring makes all deviations positive, weights large deviations more heavily, and produces a quantity that is mathematically well behaved — differentiable, and additive across independent variables. Mean absolute deviation exists and is more intuitive, but variance and standard deviation underpin far more statistical theory, which is why they became standard despite being harder to explain.
What is the 68-95-99.7 rule?
For normally distributed data, roughly 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. It is what makes standard deviation interpretable rather than abstract. The important caveat is that it only holds for approximately normal distributions — apply it to skewed data such as income and it will mislead badly.
How does standard deviation relate to variance?
Variance is the standard deviation squared. Variance has better mathematical properties, particularly that variances of independent variables add. Standard deviation is more useful for reporting because it is in the same units as the data — a spread of 5 cm rather than 25 square centimetres, which means nothing physically. Both describe the same spread.
Is standard deviation affected by outliers?
Heavily, because squaring amplifies large deviations. A single extreme value can inflate the standard deviation dramatically while barely moving the median. If your data contains genuine outliers, consider the interquartile range instead, which uses the middle 50% and is robust to extremes. Reporting a standard deviation from outlier-heavy data implies a tidiness the data does not have.