Definition D1
Let dᵢ = xᵢ − x̄. Then σ² = (Σdᵢ²)n⁻¹ is called the variance and σ = √(σ²) the standard deviation.
Proof.
(1)Σaᵢ denotes a₁ + ⋯ + aₙ, u − v denotes u + (−v), and t² denotes t·t. By associativity and commutativity of addition, this sum is uniquely determined regardless of the order and grouping of terms.convention, A1, A2
(2)Since n ≥ 1, we have n ≠ 0 and n⁻¹ exists.A10, A6
(3)The square root is taken as the non-negative root, so σ is uniquely determined and 0 ≤ σ.A11
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Definition D2
For a constant c, the data obtained by yᵢ = c xᵢ are called the data x scaled by c.
Proof.
Just as lengths are rewritten from m to cm, changing units is precisely scaling the data by a constant.convention
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Lemma L1
If yᵢ = c xᵢ, then ȳ = c x̄.
Proof.
(1)n ȳ = Σyᵢ = Σ(c xᵢ)A9, D2
(2)Σ(c xᵢ) = c Σxᵢ(1), A8
(3)Since Σxᵢ = n x̄, n ȳ = c(n x̄)(2), A9, E
(4)c(n x̄) = n(c x̄)A4, A5
(5)Since n ≠ 0, cancelling n gives ȳ = c x̄(3), (4), A10, A7
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Theorem T1
If yᵢ = c xᵢ, then σ_y² = c² σ_x².
Proof.
(1)yᵢ − ȳ = c xᵢ + (−(c x̄)) = c xᵢ + c(−x̄) = c(xᵢ − x̄)D2, L1, A15, A3
(2)(yᵢ − ȳ)² = (c dᵢ)² = c² dᵢ²(1), D1, A4, A5
(3)Σ(yᵢ − ȳ)² = Σ(c² dᵢ²) = c² Σdᵢ²(2), A8
(4)σ_y² = (c² Σdᵢ²)n⁻¹ = c²((Σdᵢ²)n⁻¹) = c² σ_x²(3), D1, A4, A5
(5)The data 1, 1, 3, 3 have x̄ = 2, dᵢ = −1, −1, 1, 1, Σdᵢ² = 4, so σ_x² = 1.A9, D1, N
(6)If c = 100, the data are 100, 100, 300, 300 with ȳ = 200, Σ(yᵢ − ȳ)² = 40000, σ_y² = 10000.(5), D2, L1, D1, N
(7)100² × 1 = 10000, consistent with (4).(4), (5), (6), N
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Theorem T2
If yᵢ = c xᵢ, then σ_y = |c| σ_x.
Proof.
(1)σ_y = √(σ_y²) = √(c² σ_x²)D1, T1
(2)c² σ_x² = (c σ_x)²A4, A5
(3)√((c σ_x)²) = |c σ_x|(2), A12
(4)|c σ_x| = |c| |σ_x|(3), A13
(5)Since 0 ≤ σ_x, |σ_x| = σ_xD1, A11, A14
(6)Therefore σ_y = |c| σ_x(1), (3), (4), (5)
(7)For the data 1, 1, 3, 3, σ_x = √1 = 1, and if c = 100 then σ_y = √10000 = 100 = |100| × 1.T1, D1, A11, N
(8)If c = −1, the data are −1, −1, −3, −3 with σ_y² = 1, σ_y = 1. Since c σ_x = −1, the factor is |c|, not c.T1, D1, A11, N
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Corollary C
The only quantity that scales by the same factor as the data is the standard deviation.
Proof.
(1)Scaling the data by c scales the mean by c and the variance by c².L1, T1
(2)The standard deviation scales by |c|, the same factor as the data.T2
(3)Rewriting lengths from m to cm means c = 100. Then the variance grows by a factor of 10000 and the standard deviation by a factor of 100.D2, T1, T2, N
(4)Therefore the variance moves by a different factor than the data, and cannot be read side by side with them.(1), (3)
(5)Squaring changes the factor from c to c², and the square root returns c² to |c|. The place it returns to is the place of the data.(1), (2), D1
(6)That the returned factor is |c| rather than c also has meaning: flipping the sign of all the data leaves the spread unchanged.T2, N
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