This tutorial explains how mean can be viewed as an orthogonal projection onto a subspace defined by the span of an all 1’s vector (i.e., basis vector).
Suppose that \( \vec{y} \in \mathbb{R}^n \) and \( L \subset \mathbb{R}^n\) is the span defined by the space of vector \( \vec{x} \), namely,
\( \vec{x}=\left[\begin{array}{ccc}
1\\
1\\ …\\ 1 \end{array}
\right], L ={c \vec{x}; c \in \mathbb{R}}\)
The question arises as to what the value of \( c \) is, if we want to have a minimal distance between \( \vec{x}\) and the subspace \( L \).
As discussed in another tutorial on orthogonal projection, we know that the shortest distance between a vector and a subspace is via an orthogonal projection.
Thus, we can get the following,
\( c=\frac{\vec{x} \cdot \vec{y}}{\vec{x} \cdot \vec{x}} \)
Thus,
\( c=\frac{\vec{x} \cdot \vec{y}}{\vec{x} \cdot \vec{x}} =\frac{\sum_{i=1}^n y_i}{\sum_{i=1}^n 1}= \frac{\sum_{i=1}^n y_i}{n} =\bar{y}\)
Further, it is worth pointing out that we can also just write out the proof process to show that the shortest distance between vector \( \vec{y} \) and the subspace defined by the \( c \vec{x} \) is the mean.
Based on the tutorial on orthogonal projection, we can get the following to calculate the distance between the vector and the space.
\( \sum_i (cx_{i}-y_{i})^2 \)
We can then calculate the partial derivative with respect to \( c \) as follows.
\( \frac{d}{dc} \sum_i (cx_{i}-y_{i})^2 =0 \)
Since all \( x_i \) is 1, we can get the following.
\( \frac{d}{dc} \sum_i (c-y_{i})^2 =0 \)
Next,
\( 2\sum_i (c-y_{i})=0 \)
Next,
\( \sum_i (c-y_{i})=0 \)
Next,
\( nc -\sum_i y_{i}=0 \)
Next,
\( c =\frac{\sum_i y_{i}}{n} =\bar{y}\)
Thus, we can see it is not difficult to prove that a vector (e.g., \( \vec{y}\) ) projecting onto a constant (i.e., all 1s such as \( \vec{x}\) ) subspace is actually the mean of the vector (i.e., \( \vec{y}\) ).
Reference and Further Reading
It is beneficial to read the following two tutorials. One is from a webpage version of a tutorial on Projections (Github) and one is my own tutorial on Orthogonal Projection.