Averages vs. Means (vs. Expectations)

Averages

Averages are statistics calculated over a set of samples. If you have a set of samples $x = x_1,\ldots,x_N$, their average, often written $\bar{x}$, is defined by

$\bar{x} = \frac{1}{N} \sum_{n=1}^N x_n$.

Means

Means are properties of distributions. If $p(x)$ is a discrete probability mass function over the natural numbers $\mathbb{N}$, then its mean is defined by

$\sum_{x \in \mathbb{N}} \, x \times p(x)$.

If $p(x)$ is a continuous probability density function over the real numbers $\mathbb{R}$, then its mean, if it exists, is defined by

$\int_{\mathbb{R}} \, x \times p(x) \, dx$.

This also shows how summations over discrete probability functions, $\sum_{x \in \mathbb{N}}$ relate to integrals over continuous probability functions, $\int_{\mathbb{R}} dx$. (Distributions can also be mixed, like spike and slab priors, but the math gets more complicated due to the need to unify the notion of summation and integration.)

Expectations

To confuse matters further, there are expectations. Expectations are properties of (some) random varaibles. The expectation of a random variable is the mean of its distribution. If $X$ is a discrete random variable with probability mass function $p(x)$, then its expectation is defined to be

$\mathbb{E}[X] = \sum_{x \in \mathbb{N}} \, x \times p(x)$.

If $X$ is a continuous random variable with probability density function $p(x)$, then

$\mathbb{E}[X] = \int_{\mathbb{R}} \, x \times p(x) \, dx$.

Look familiar?

Sample Means

Samples don’t have means per se. They have averages. But sometimes the average is called the “sample mean”. Just to confuse things.

Averages as Estimates of the Mean

Gauss showed that the average of a set of independent, identically distributed (i.i.d.) samples from a distribution $p(x)$ is a good estimate of the mean.

What’s good about the average as an estimator of the mean? First, it’s unbiased, meaning the expectation of the average of a set of i.i.d. samples from a distribution is the mean of the distribution. Second, it has the lowest expected mean square error among all estimators of the mean. That’s why everyone likes square error (that, and its convexity, which I discussed in a previous blog post on Mean square error, or why committees won the Netflix Prize).

The median is a good estimator too. Laplace proved that it has the lowest expected absolute error among estimators (I just learned it was Laplace from the Wikipedia entry on median unbiased estimators). It’s also more robust to outliers.

More on Estimators

Of course, in Bayesian statistics, we’re more concerned with a full characterization of posterior uncertainty, not just a point (or even interval) estimate.

Summary

• Means are properties of distributions.
• Expectations are properties of random variables.
• Averages or sample means are statistics calculated from samples.

3 Responses to “Averages vs. Means (vs. Expectations)”

1. Johannes Goller Says:

Just to make the confusion complete, it may be worth mentioning that the example you give for ‘average’ is in fact a variety often referred to as ‘arithmetic mean’.

2. Cosma Shalizi Says:

May I ask what your sources are for these prescriptions? Because they do not conform at all to what I am used to in statistics. For instance, I am used to using “average” as a general term embracing lots of measures of central tendency or location (arithmetic mean, geometric mean, median, mode, etc.), though with a weak presumption that it refers to what you call the “average” and what we very, very often call the “sample mean” or “empirical mean”. (It is, after all, the mean under the “empirical distribution”, which puts probability 1/n at each observation.) — Of course, different notions of “average” are useful for different purposes, and refer to somewhat different aspects of samples and distributions. For some distributions, the sample median is indeed a more robust estimator of the central tendency than the sample mean, but for many others the median and the mean are just getting at different things.

I have never encountered anyone drawing the distinction you do between means of distributions and expectations of random variables.

So, in short, if there is some research community where the usages you are prescribing are common and accepted, could you point me to it?