Mean, spread and confidence

Analysts measure the same sample several times (replicates) and report the mean, the plain average. Five titrations of 10.1, 10.3, 9.9, 10.1 and 10.1 mL have a mean of 10.1 mL.

The standard deviation, s, says how spread out the replicates are. Take each value's difference from the mean, square it, add them up, divide by one less than the number of values, and take the square root. For those five: differences 0, 0.2, −0.2, 0, 0; squares add to 0.08; 0.08 ÷ 4 = 0.02; √0.02 = 0.14 mL. A calculator's s or STDEV function does it for you.

To compare spreads across different sizes of number, use the relative standard deviation, RSD = s ÷ mean × 100%. Here 0.14 ÷ 10.1 = 1.4%. Many lab methods require an RSD under 2% or so.

A confidence interval puts a range round the mean that very likely holds the true value: mean ± t × s ÷ √n, where n is the number of replicates and t comes from a table (for 95% confidence and five replicates, t is 2.78). More replicates and less scatter give a tighter range. Saying 10.10 ± 0.18 mL is more honest than just 10.10.

A value far from the others may be an outlier, from a spill or a bubble in the burette. Don't drop it just because it's inconvenient: use a written test (such as Grubbs' test) or a known cause, and record what you did.