Standard deviation use cases

Education and grading

A teacher's class of ten scores 72, 85, 90, 64, 78, 88, 95, 70, 82, 76 on a test. The mean is 80 and the sample standard deviation is 9.76. A student who scored 95 sits (95 − 80) ÷ 9.76 = 1.54 standard deviations above the mean. If the scores are roughly normal, that is about the 94th percentile.

The SD also shows how well the test separated students. An SD of 2 points would mean nearly everyone scored the same, so the test told the teacher little. An SD of 25 would suggest part of the class was lost. Standardising with z-scores also lets marks from two tests with different spreads be compared or combined fairly.

Tools: test score SD, z-score, grade average, weighted average for coursework weighted by credit.

Manufacturing and quality control

A line machines shafts to a specification of 9.95–10.05 mm. Measurements give a process mean of 10.01 mm and a standard deviation of 0.012 mm.

• Control limits sit at the mean ± 3σ: 9.974 to 10.046 mm. A reading outside them signals that something changed in the process, not just ordinary variation.
• Capability. Cp = (10.05 − 9.95) ÷ (6 × 0.012) = 1.39, so the spread would fit the specification. But the process is running high. Cpk = (10.05 − 10.01) ÷ (3 × 0.012) = 1.11, below the common 1.33 target. Assuming normal output, about 430 parts per million will fall above the upper limit.

Centring the process at 10.00 mm would raise Cpk to Cp. Reducing σ is usually harder and more expensive. The standard deviation is what tells you which of the two problems you have.

Tools: Cpk and process capability, empirical rule, large dataset SD for a full batch export, outliers.

Laboratory and analytical chemistry

Six replicate injections of a standard read 10.12, 10.08, 10.21, 10.15, 10.05, 10.18 mg/mL. The mean is 10.132 and the sample SD is 0.0605 mg/mL. Labs report this as a percentage of the mean, the relative standard deviation: 100 × 0.0605 ÷ 10.132 = 0.60% RSD.

%RSD is used because it compares precision across concentrations. A 0.06 mg/mL SD is excellent at 10 mg/mL and poor at 0.1 mg/mL. A method's acceptance criterion is usually written as a maximum %RSD, such as 2% for replicate injections.

Tools: %RSD, coefficient of variation, pooled SD across runs, CSV data for instrument exports.

Research and psychology

A study compares a new teaching method (M = 78.4, SD = 9.2, n = 25) with the usual one (M = 72.1, SD = 10.1, n = 27). The means differ by 6.3 points, but whether that matters depends on the spread. Divide by the pooled SD of 9.68 to get Cohen's d = 0.65, a medium effect by Cohen's benchmarks. The 95% interval for d, 0.09 to 1.21, is wide: with around 25 per group the true effect could be small or large.

Papers report every group as M and SD so readers can compute effect sizes and meta-analyses can pool results. A t-test on the same numbers tests whether the difference could be chance; d says how big it is. You need both.

Tools: Cohen's d, t-test, pooled SD, SD in psychology.

Finance and investing

In finance the standard deviation of returns is called volatility, the standard measure of risk. A fund whose daily returns have an SD of 1.2% has an annualised volatility of about 1.2% × √252 = 19.0%, since there are roughly 252 trading days a year. Scaling by the square root of time assumes returns are independent from day to day, which is a simplification, but it is the market convention.

Combining assets reduces risk when they are not perfectly correlated. A portfolio of 60% in an asset with 18% volatility and 40% in one with 7%, correlated at 0.25, has a standard deviation of 11.8%. The weighted average of the two SDs is 13.6%. The 1.8-point gap is the diversification benefit.

Tools: portfolio SD, standard deviation of a return series, CV for risk per unit of return, normal distribution.

Sports

A hitter with 152 hits in 541 at-bats bats .281. Each at-bat is a hit or not, so the binomial standard error, √(0.281 × 0.719 ÷ 541) = 0.019, measures how much of that is luck. A true .281 hitter would finish anywhere from about .243 to .319 over a season this long by chance alone. That is why one season's batting title is weak evidence of who is really best.

The SD also measures consistency. Two players both average 1.3 hits per game over ten games, but one's game-by-game SD is 0.95 and the other's is 0.48. The second player is the steadier one. Same mean, very different reliability.

Tools: batting average, standard deviation of game scores, binomial distribution.

Surveys and A/B testing

A poll of 1,000 people finds 52% support. The margin of error at 95% confidence is 1.96 × √(0.52 × 0.48 ÷ 1000) = ±3.1 percentage points, so the poll cannot tell 52% from an even split. To get ±3 points whatever the true proportion, you need 1,068 respondents. Halving the margin to ±1.5 points needs four times as many.

A/B tests work the same way. The standard deviation of the metric sets how many users you need to detect a given lift, and Cohen's d puts the result on a scale that is comparable across experiments.

Tools: margin of error, sample size, confidence interval, Cohen's d.

Weather and everyday data

Two cities have the same average daily high over ten days, 23.5 °C. In the coastal city the highs range from 22 to 25 °C (SD 1.0 °C). In the inland city they range from 14 to 35 °C (SD 6.2 °C). The mean alone would call their weather identical. The standard deviation shows that one needs a jacket and the other does not.

One caution: do not turn a temperature SD into a coefficient of variation. Celsius has an arbitrary zero, so a percentage of the mean means nothing there. Commute times, grocery bills and sleep hours all work the same way. Wherever an average is quoted, the SD says how much to trust it for any single day.

Tools: standard deviation, range, standard deviation graph, histogram maker.