Summary Statistics: Part I

To get started, we will review several summary statistics that will likely be familiar to you: mean, median, standard deviation, skewness and kurtosis.

1. First, a few definitions and a bit of nomenclature…

Throughout the course, we will define \(X\) as a random variable (e.g. surface air temperature). The “population” of \(X\) represents all possible values of \(X\). The true population mean and standard deviation of this variable are denoted as:

\[\begin{split} \begin{align} \mu &= \text{population mean}\\ \sigma &= \text{population standard deviation} \end{align} \end{split}\]

A sample of \(X\) is a subset of values of \(X\). We will define a random sample of \(X\) with \(N\) elements as [\(x_1\), \(x_2\), \(x_3\), …, \(x_N\)]. The sample mean and standard deviation are denoted as:

\[\begin{split} \begin{align} \overline{x} &= \text{sample mean}\\ s &= \text{sample standard deviation} \end{align} \end{split}\]

2. Moments

Moments are quantitative parameters that we will use to describe a sample of \(X\).

2.1 The Mean

The first moment is the mean. The sample mean is defined as:

\[ \overline{x} = \frac{1}{N}\sum_{i=1}^{N}x_i \]

If we assume that our sample is a measurement taken at a fixed location but at a certain frequency in time (aka, a time series), then the overbar denotes the sample time mean and the subscript \(i\) denotes the time step.

Note

The sample mean is an unbiased estimate of the population mean, \(\mu\).

In other words, for an infinite number of samples from the same time series, the population mean of all of the sample means (\(\mu_{\overline{x}}\)) is equal to the population mean (\(\mu\)).

The median is the value in the centre of a population or sample. This is a useful statistic when there are large outliers in the population/sample (e.g. house prices, salaries, rain fall rate, earthquake magnitude)

2.2 Variance and Standard Deviation

The second moment of a sample \(X\) is the variance. As you can probably guess, the variance of a sample of \(X\) describes the variability of \(X\). The sample variance is defined as:

\[ \overline{x^{\prime 2}} = \frac{1}{N-1}\sum_{i=1}^{N}(x_i - \overline{x})^2 \]

where

\[ x^{\prime}_i = x_i - \overline{x} \]

Again, assuming that our sample is a time series, then \(x^{\prime}_i\) represents the deviations about the time mean.

The variance is in squared units for whatever our variable \(X\) is. To get an estimate of the variability of \(X\) from our sample in the correct units, we use the sample standard deviation:

\[ s = \sqrt{\overline{x^{\prime 2}}} \]

Note

The sample variance is an unbiased estimate of the population variance, \(\sigma^2\).

Notice that we are dividing by \(N-1\) instead of \(N\). Since the calculation of the sample variance depends on the sample mean (which is itself an approximation and introduces some bias), then we need to divide by \(N-1\) rather than \(N\) to remove this bias. You can find out more here.