The standard deviation symbol: σ, s, and everything around them

Sx is the sample standard deviation of the x values, the one that divides by n − 1. It is what TI and Casio calculators print for s. The σx shown next to it is the population standard deviation, which divides by n. If your data is a sample from a larger group, which it usually is in a class exercise, report Sx.

Both are. Sx is the sample standard deviation (n − 1) and σx the population one (n). For the same data Sx is always slightly larger; the gap shrinks as n grows. Use σx only when the list is the entire group you care about. On a keyboard without a σ key, σx is often typed as "ox"; it is the same population standard deviation.

It depends on which standard deviation you mean. The symbol that represents the population standard deviation is the lowercase Greek sigma, σ. The sample standard deviation is a plain italic s. SD and "std dev" are written abbreviations for the words, not symbols, and do not say which of the two you mean: the SD symbol is σ for a population and s for a sample. Using σ for a sample statistic is the single most common notation mistake in introductory statistics.

σ is a parameter: a fixed, usually unknown property of a whole population. s is a statistic: a number computed from a sample, which varies from sample to sample and is used to estimate σ.

The convention runs throughout statistics: Greek letters for the truth you cannot see, Roman letters for the estimates you can compute. μ and x̄ follow the same pattern.

Windows: hold Alt and type 229 on the numeric keypad, or use Win+. and search "sigma".
Mac: Option+w gives ∑ (summation); for σ use Ctrl+Cmd+Space and search "sigma".
Word: type 03C3 then press Alt+X.
LaTeX: \sigma for σ, \Sigma for Σ.
HTML: σ and Σ.

Capital sigma, Σ, is the summation operator: it means "add up everything that follows". It is a different symbol from lowercase σ despite sharing a name. In Σ(x − x̄)² it instructs you to compute (x − x̄)² for every value and total the results.

σ² is the variance: the standard deviation squared. σx̄ (sometimes written σ_M or SE) is the standard error of the mean, the standard deviation of the sampling distribution of the mean, equal to σ/√n.

Sx is the sample standard deviation that your calculator computes when you run the statistics mode. The "x" tells you it is working on the x-list (the data you entered), and the "S" tells you it divides by n − 1. On TI and Casio calculators it appears in the stats summary screen alongside σx, the population version that divides by n. If your data is a sample, which it almost always is in a class exercise, Sx is the one to report.

The symbol for population standard deviation is the lowercase Greek sigma, σ. It follows the convention that Greek letters describe whole populations. On a calculator, population standard deviation is shown as σx or sometimes just σ.

The symbol for sample standard deviation is a plain italic s. On TI and Casio calculators it appears as Sx. The italic s follows the convention that Roman letters describe samples. It divides by n − 1, which is why it is slightly larger than the population value for the same data.

Sx is the sample standard deviation of the x-values (your primary dataset). Sy is the sample standard deviation of the y-values, used in regression. Both divide by n − 1. The corresponding population symbols are σx and σy.

For a normal distribution, 99.73% of values lie within ±3σ of the mean, so 0.27% (about 1 in 370) fall outside on one side or the other, and 0.135% (about 1 in 741) beyond +3σ alone. Real data with heavy tails produces 3-sigma events more often than that.

Subtract the sample mean x̄ from each value x. For 2, 4 and 9 the mean is 5, so the deviations x − x̄ are −3, −1 and 4. They always add up to zero, which is why the standard deviation squares them before adding.

The sample variance of the variable x: the square of its sample standard deviation Sx. If a calculator shows Sx = 3, then s²ₓ = 9.

s_p is the pooled standard deviation: one estimate of spread that combines several groups assumed to share the same σ. It is not better or worse than a single group's Sx; it answers a different question. See the pooled standard deviation calculator.

In regression, s (or σ̂) is the residual standard error: the typical distance of the data points from the fitted line, in the units of y. It is the square root of the residual sum of squares divided by its degrees of freedom.