1sigma

1sigma, or one sigma, refers to the standard deviation, a measure of dispersion in a dataset. It is denoted by the Greek letter sigma (σ) and can describe either the population standard deviation or the sample standard deviation. The standard deviation quantifies how spread out the values are around the mean.

In a normal distribution, about 68.27% of values lie within one standard deviation of the mean, about 95.45% within two standard deviations, and about 99.73% within three. Consequently, the interval formed by mean ±1σ is commonly used to express a typical uncertainty range around a central value. In measurement reporting, a result may be written as x ± σ, indicating the estimated mean x and its standard deviation.

Calculation: for a population, σ = sqrt( (1/N) ∑ (xi − μ)² ). For a sample, s = sqrt( (1/(n−1)) ∑ (xi − x̄)² ). When combining independent error sources, variances add; the total standard deviation is the square root of the sum of the individual variances.

Applications: 1σ is a conventional way to express measurement precision in science and engineering. It is also used in discussions of statistical significance, where statements such as "5σ" refer to a result five standard deviations away from a null hypothesis, corresponding to a very small probability under Gaussian assumptions.

Limitations: the interpretation of 1σ assumes a roughly normal distribution of errors. For skewed or heavy-tailed data, the ±1σ interval may not reflect the true central probability; other intervals or nonparametric methods may be appropriate.