Why Adding a Constant Does Not Change Standard Deviation

In the HKDSE Mathematics syllabus (Compulsory Part), understanding how transformations affect statistical measures is a core competency. When we add a constant to every data point in a set, we are performing a translation of the data distribution.

Measures of dispersion—such as Range, Interquartile Range (IQR), and Standard Deviation (SD)—describe the "spread" or "scatter" of a data set. They measure how far apart data points are from each other or from the mean.

Imagine a group of students standing in a line. If every student takes 5 steps to the right, their individual positions change, and the "average position" (mean) shifts 5 steps to the right. However, the distance between the first and last student (Range) and the way they are clustered (SD) remains exactly the same.

Consider a data set x_1, x_2, \dots, x_n. Let y_i = x_i + k be the new data set after adding a constant k.

A. Change in Mean \begin{align*} \bar{y} &= \frac{\sum (x_i + k)}{n} \\ &= \frac{\sum x_i + nk}{n} \\ &= \bar{x} + k \end{align*} The mean increases by the same constant k.

B. Change in Range Let x_{max} and x_{min} be the maximum and minimum values of the original set. \begin{align*} \text{New Range} &= y_{max} - y_{min} \\ &= (x_{max} + k) - (x_{min} + k) \\ &= x_{max} - x_{min} \\ &= \text{Original Range} \end{align*}

C. Change in Standard Deviation ( \sigma) The standard deviation measures the deviation from the mean: (x_i - \bar{x}). \begin{align*} \text{New Deviation} &= y_i - \bar{y} \\ &= (x_i + k) - (\bar{x} + k) \\ &= x_i - \bar{x} \end{align*} Since the deviation for every point is unchanged, the sum of squares and the resulting standard deviation remain identical to the original.

For a transformation y = x + k:

In the visual below, you can use the slider to add a constant k to the data points. Observe that while the cluster moves along the axis, the lengths of the "Range Bar" and "SD Bar" do not change.