What standard deviation measures
Standard deviation summarises how far values typically sit from their mean. Two classes can both average 70% on a test, but a class with stdev 5 is tightly clustered while a class with stdev 20 has both stars and strugglers.
sample: s = √( Σ(x − x̄)² ÷ (n − 1) ) · population: σ = √( Σ(x − μ)² ÷ n )
Sample vs population — the choice that matters
The two formulas differ only in the divisor, but picking the right one matters. Use population (divide by n) when your list is everything you care about — all 6 machines in your shop, every match a team played this season. Use sample (divide by n − 1) when your list is a subset used to estimate a larger group — 50 measured customers standing in for all customers. The n − 1, called Bessel's correction, compensates for the fact that deviations measured from a sample's own mean are systematically a bit too small. With big n the two answers converge; with small n the difference is real. When in doubt, sample is the safer default — which is why it is the headline number here.
Worked example
For the defaults 4, 8, 6, 5, 3, 7: the mean is 33 ÷ 6 = 5.5. The deviations are −1.5, 2.5, 0.5, −0.5, −2.5, 1.5; squared and summed they give Σ(x − x̄)² = 17.5. Sample variance = 17.5 ÷ 5 = 3.5, so s = √3.5 ≈ 1.8708. Population variance = 17.5 ÷ 6 ≈ 2.9167, so σ ≈ 1.7078.
Variance vs standard deviation
Variance is the squared version and lives in squared units (points², kg²), which is awkward to interpret. Taking the square root brings it back to the data's own units — that is all standard deviation is. You need at least two numbers for a sample standard deviation, since with one number there is nothing to deviate from.