Standard deviation calculator (3 numbers)

Definition

Standard deviation is the square root of variance. For a finite set of three numbers a, b, c the arithmetic mean is m = (a + b + c) / 3, the sum of squared deviations is SSD = (a−m)² + (b−m)² + (c−m)², the population variance is σ² = SSD / 3, and σ = √σ². This calculator implements exactly that chain: the divisor is always n = 3, never n − 1.

The population-versus-sample distinction matters. The symbol σ with divisor n describes the spread of the whole population (here: the three entered values treated as the complete set). The symbol s with divisor n − 1 (Bessel’s correction) is an unbiased estimator of an unknown population variance from a sample. This page returns population σ; the result panel also shows s with divisor 2 for comparison, but the primary result is always σ with /3.

The result unit matches the data unit: if a, b, c are in meters, σ is in meters too. Variance has a squared unit; the square root restores the original scale.

Historical and conceptual context

Karl Pearson popularized the term “standard deviation” around the turn of the twentieth century; earlier least-squares work by Gauss and Legendre already used sums of squared deviations from the mean. Variance as the second central moment (μ₂) and deviation as √μ₂ entered the canon of descriptive statistics and inference. The NIST/SEMATECH e-Handbook presents measures of scale, including standard deviation, as basic descriptors of spread.

Conceptually σ measures a typical distance from the mean on the data scale, unlike the range (max − min), which looks only at extremes. For three numbers the range is simple, but σ uses all three squared deviations. The neighboring two-number standard deviation page divides by 2 (also population /n); the variance-of-n-numbers page leaves σ² from a list — here there are always three fields and the square root at once.

When teaching the topic, name honestly what the software computes: spreadsheets and libraries differ by divisor (Excel STDEV.P versus STDEV.S, R sd() with /(n−1), NumPy with ddof). This page matches STDEV.P / population with n = 3.

Formal properties

For any a, b, c ∈ ℝ: σ ≥ 0. σ = 0 if and only if a = b = c. Shift: σ(a+c₀, b+c₀, c+c₀) = σ(a, b, c). Scaling: σ(λa, λb, λc) = |λ| σ(a, b, c). Field order does not change the result because the formula is symmetric under permutation.

Identity: SSD = a² + b² + c² − 3m² for m = (a+b+c)/3, so σ² = (a²+b²+c²)/3 − m². The calculator still computes deviations from m directly, by the formula: v = ((a−m)²+(b−m)²+(c−m)²)/3.

Relation to the sample estimator: s² = SSD / (n−1) = (3/2) σ² when n = 3, so s = σ √(3/2). For 1, 2, 3: σ ≈ 0.816497 and s ≈ 1.

Full derivation

Step 1. Enter three real numbers a, b, c (period or comma in a decimal).

Step 2. Compute the mean m = (a + b + c) / 3.

Step 3. Compute deviations dₐ = a − m, dᵦ = b − m, dₑ = c − m and SSD = dₐ² + dᵦ² + dₑ².

Step 4. Population variance σ² = SSD / 3; deviation σ = √σ² (nonnegative root).

Step 5. Example 1, 2, 3: m = 2; SSD = 1+0+1 = 2; σ² = 2/3; σ = √(2/3) ≈ 0.816497 (six decimals for a non-integer). Example 2, 2, 2: SSD = 0, σ = 0. Example 0, 6, 12: m = 6; SSD = 36+0+36 = 72; σ² = 24; σ = √24 ≈ 4.898979.

Formal correctness follows from the second central moment of the empirical uniform measure on three points (mass 1/3 each). The square root is uniquely defined as a nonnegative function.

Special cases

Three equal values → σ = 0. Arithmetic progression 1, 2, 3 → σ = √(2/3). Symmetric values around zero −2, 0, 2: m = 0, σ² = (4+0+4)/3 = 8/3, σ = √(8/3) ≈ 1.632993. Decimals 1.5 / 2.5 / 3.5 behave like a shifted 1, 2, 3 sequence: the same σ ≈ 0.816497.

An empty field is not allowed — all three values are required. Invalid text yields an error message. Zero is a number: 0, 0, 0 gives σ = 0; 0, 6, 12 gives ≈ 4.898979.

This page does not compute standard deviation from a variable-length list, does not apply weights, does not drop outliers, and does not switch the primary result to divisor n − 1.

Numerical notes

Computation uses IEEE-754 floating-point arithmetic in the browser (√). For typical school and engineering inputs the result matches paper arithmetic after rounding to six decimals. An integer result (for example 0) displays without trailing zeros; a non-integer uses six decimal places, as on the other statistics calculators.

With extremely large |a|, |b|, |c|, cancellation when subtracting the mean can lose precision. In educational practice three moderate-scale fields are fine. The number-line diagram marks a, b, c and m with a dashed line.

Related concepts and sources

Related: two-number standard deviation (divisor 2), variance of n numbers (σ² from a list), arithmetic mean, range (max − min), median. The coefficient of variation CV = σ / |m| is not computed here.

Sources: NIST/SEMATECH e-Handbook — Measures of Scale; Wikipedia — Standard deviation (population vs sample); MathWorld — Standard Deviation; Excel STDEV.P vs STDEV.S documentation. Educational tool.