Change of Origin and Scale

Variance and standard deviation respond in specific, predictable ways when the data is shifted or rescaled, and this property is used to make hand computation easier, especially for grouped data with large or awkward mid-values.

1. Change of origin only: define $d = x - A$, where A is a constant (commonly the class mark of the middle class — if the number of classes is odd, A is the mid-value of the single middle class; if even, A is the mid-value of whichever of the two middle classes has the greater frequency). Since every value is shifted by the same constant amount, the spread of the data does not change: $\sigma_d^2 = \sigma_x^2, \qquad \sigma_d = \sigma_x$ In other words, the standard deviation of $x_1-A, x_2-A, \ldots, x_n-A$ is exactly the same as the standard deviation of $x_1, x_2, \ldots, x_n$ — variance and S.D. are independent of a change of origin.

2. Change of scale: define $u = \dfrac{x-A}{h}$, where h is the class width (or, when the data has no class intervals, h is the common difference between consecutive values of $x_i$), with $h\neq 0$. Unlike a pure shift, dividing by h does change the spread proportionally: $\sigma_x = h\,\sigma_u, \qquad \sigma_x^2 = h^2\,\sigma_u^2$ So the standard deviation of $\frac{x_1-A}{h}, \frac{x_2-A}{h}, \ldots, \frac{x_n-A}{h}$ is exactly $\frac{1}{h}$ times the standard deviation of the original values — variance and S.D. are NOT independent of a change of scale; they scale with h.

In practice this substitution converts the original x (or class mid-value) column into a column of small, easy integers u (…, −2, −1, 0, 1, 2, …), whose variance $\sigma_u^2$ is computed the ordinary way using $\text{Var}(u)=\sigma_u^2=\frac{\sum f_iu_i^2}{N}-\bar{u}^2$, after which the answer is converted back to the original scale using $\sigma_x = h\sigma_u$ and $\text{Var}(X)=h^2\text{Var}(u)$.

Worked Example (Ex.4): Values 15, 20, 25, 30, 35, 40, 45 with frequencies 13, 12, 15, 18, 17, 10, 15 (N=100). Using $u=(x-30)/5$, the u-values run from −3 to 3. Building the f.u and f.u² columns gives $\sum f_iu_i = 4$ and $\sum f_iu_i^2 = 372$, so $\bar{u}=4/100=0.04$ and $\text{Var}(u) = 372/100 - 0.04^2 = 3.72-0.0016=3.7184$, giving $\sigma_u \approx 1.928$. Converting back with $h=5$: $\sigma_x = 5\times1.928 \approx 9.64$.

Worked Example (Ex.5): A grouped distribution with 8 classes of width 10 from 45-55 to 115-125, frequencies 7, 20, 27, 23, 13, 6, 3, 1 (N=100), mid-values 50 to 120. Using $u=(x-90)/10$, the f.u and f.u² columns give $\sum f_iu_i = -150$ and $\sum f_iu_i^2 = 450$, so $\bar{u}=-150/100=-1.5$ and $\text{Var}(u) = 450/100 - (-1.5)^2 = 4.5-2.25 = 2.25$. Converting back with $h=10$: $\text{Var}(X) = h^2\text{Var}(u) = 100\times2.25 = 225$, so $\text{S.D.} = \sigma_x = \sqrt{225} = 15$. …