8 Mean

The starting point in multivariate analysis consists in computing various summary measures—such as means, and variances—to get an idea of the common or central values, and the amount of variability of each variable. In this chapter you will learn how concepts like means can be expressed in terms of vector-matrix operations.

8.1 Mean of a variable

To measure variation, we usually begin by calculating a “typical” value. The idea is to summarize the values of a variable with one or two representative values. You will find this notion under several terms like measures of center, location, central tendency, or centrality.

The prototypical summary value of center is the mean, sometimes referred to as average. The mean of a variable \(X = (x_1, x_2, \dots, x_n)\), represented by \(\bar{x}\), is obtained by adding all the \(x_i\) values and then dividing by their total number \(n\):

\[ \bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} \]

Using summation notation we can express \(\bar{x}\) in a very compact way as:

\[ \bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_i \]

If we want to compute the mean value of weight

\[ \overline{\texttt{weight}} = \frac{1}{5} (22 + 15 + 23 + 19 + 14) = 18.6 \]

If you associate a constant weight of \(1/n\) to each observation \(x_i\), you can look at the formula of the mean as a weighted sum:

\[ \bar{x} = \frac{1}{n} x_1 + \frac{1}{n} x_2 + \dots + \frac{1}{n} x_n \]

This is a slightly different way of looking at the mean that will allow you to generalize the concept of an “average” as a weighted aggregation of information. For example, if we denote the weight of the \(i\)-th individual as \(w_i\), then the average can be expressed as:

\[ \bar{x} = w_1 x_1 + w_2 x_2 + \dots + w_n x_n = \sum_{i=1}^{n} w_i x_i \]

8.2 Mean as a Balancing Point

What does the mean do? In what sense the mean is a “typical” value? How do you make sense of a mean value?

To understand what the mean is doing is helpful to visualize the values of a variable as points in a number-line, or with a dot-plot. Imagine that the number line is like a weighting scale or a teeter-totter (or see-saw). The mean plays the role of the balancing point. In other words, \(\bar{x}\) is the value that balances out the scale and keeps things on equilibrium.

Values on a number-line

Figure 8.1: Values on a number-line

The value of 5 is the balancing point:

Mean as a balancing point

Figure 8.2: Mean as a balancing point

Saying that 5 is the balancing point has a specific meaning. It means that the signed distances of all the values with respect to the mean cancel out. If you calculate the deviations from the mean: \((x_i - \bar{x})\), and add them all, the sum would be zero.

\[ (1 - 5) + (2 - 5) + (6 - 5) + (7 - 5) + (9 - 5) = 0 \]

This is a very special property of the mean, indicating that this value is optimal in the sense that it is the only number for which the sum of deviations with itself is zero. In a more algebraic way, say you are looking for a number \(a\) that makes the sum of deviations zero:

\[ \sum_{i=1}^{n} (x_i - a) = 0 \]

What would \(a\) be? Then mean \(\bar{x}\)! No other value would be able to cancel out the sum of deviations around itself.

Keep in mind that the mean is just a summary. One possible kind of “typical” value (other common measures of center are the median or the mode). As any summary statistic, it involves compressing the information of a variable into a single representative number. But remember: this number does not tell you the whole story about a variable.

8.3 Mean with Vector Notation

It is very useful to be able to compute a mean using vector-matrix notation. First, notice that the formula of the mean consists of computing a (weighted) sum. Second, recall that a sum of numbers can be expressed with an inner product by using the unit vector (or summation operator). If we denote \(\mathbf{1}_{n}\) a vector of ones of size \(n\), then the mean value of a vector \(\mathbf{x}\) can be obtained with an inner product:

\[ \bar{x} = \left(\frac{1}{n}\right) \mathbf{1}_{n}^\mathsf{T} \mathbf{x} \]

As you can tell, \(\bar{x}\) is calculated by multiplying the scalar \(1/n\) with the inner product \(\mathbf{1}_{n}^\mathsf{T} \mathbf{x}\), which is equivalent to:

\[ \bar{x} = \left(\frac{1}{n}\right) <\mathbf{1}_{n}, \mathbf{x}> = \left(\frac{1}{n}\right) <\mathbf{x}, \mathbf{1}_{n}> \]

This way of using an inner product can also be genelarized with the notation \(<\mathbf{a}, \mathbf{b}>_{M}\) which is simply an inner product in a vector space endowed with a metric matrix \(\mathbf{M}\)

\[ <\mathbf{a}, \mathbf{b}>_{M} = \mathbf{a}^\mathsf{T} \mathbf{M b} \]

If we use a metric matrix \(\mathbf{D} = diag(1/n)\) then we have that the mean is given by:

\[ \bar{x} = <\mathbf{x}, \mathbf{1}>_{D} = \mathbf{x}^\mathsf{T} \mathbf{D 1} \]

8.4 Mean as a Projection

We can gain insight to what a mean is doing if we regard a variable from the space of individuals perspective. This interpretation is less common than the balancing point, but it is equally revealing and informative.

Consider the \(n\)-dimensional space \(\mathbb{R}^n\) illustrated in the figure below. In this space, each dimension is associated to an individual. The variable \(X\) is represented by the blue vector \(\mathbf{x}\). In turn, the orange vector represents the \(n\)-element unit-vector \(\mathbf{1}\). From this point of view, the mean acquires a very interesting meaning: it turns out that the mean is the measure of the projection of \(\mathbf{x}\) onto the axis spanned by the unit vector \(\mathbf{1}\).

Mean as the measure of a projection

Figure 8.3: Mean as the measure of a projection

So what is the meaning of this orthogonal projection? It tells you that \(\bar{x} \mathbf{1}\) is the multiple of the unit-vector that is the closest to \(\mathbf{x}\) in the least squares sense:

A variable and its mean vector

Figure 8.4: A variable and its mean vector

The size of the residual vector \(\mathbf{e} = \mathbf{x} - \bar{x} \mathbf{1}\) is the smallest among all other multiples of \(\mathbf{1}\) that you can use to approximate \(\mathbf{x}\).

8.5 Centroid

If we have several variables or vectors of the same size—like in a \(n \times p\) matrix \(\mathbf{X}\)—we can get the mean vector also known as centroid. The row vector of means of \(\mathbf{X}\) is denoted by:

\[ \mathbf{\bar{x}} = \frac{1}{n} \mathbf{1}_{n}^\mathsf{T} \mathbf{X} \]