How standard deviation works
The mean (average) tells you where the data sits; the standard deviation tells you how far from that centre the values typically fall. The calculation squares each deviation from the mean, sums them, and takes a square root:
σ = √[ Σ(x − x̄)² ÷ N ] (population) · s = √[ Σ(x − x̄)² ÷ (N − 1) ] (sample)
The population form (divide by N) is right when you have every member of the group. The sample form (divide by N − 1) is right when your list is a sample of a larger population — dividing by N − 1 corrects the downward bias that the plain average would give. The variance is simply the standard deviation squared, which is why it is reported in squared units and standard deviation in the original units.
Worked example
Take the classic set 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5. Squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, summing to 32. Population variance = 32 ÷ 8 = 4, so σ = √4 = 2. As a sample, variance = 32 ÷ 7 ≈ 4.571 and s ≈ 2.138. The interpretation: values typically sit about 2 units from the average, and the 9 sits at z = +2 — a clear outlier worth a look.
How to read it
For a roughly bell-shaped (normal) distribution, about 68% of values lie within 1σ, 95% within 2σ and 99.7% within 3σ. That is the empirical rule behind quality control limits, risk bands and "how many standard deviations from the mean" reporting. A large standard deviation relative to the mean (high coefficient of variation) means the numbers are highly variable relative to their size.