Characteristics and measures of hydrologic data

4.1   Arithmetic mean: A classical measure of central tendency

$\def\mean#1{\bar{#1}}$ \begin{equation} \mean{X}= \sum_{i=1}^n\frac{X_i}{n}= \sum_{i=1}^k\mean{X}_i\frac{n_i}{n}= \mean{X}_{(j)}\frac{n-1}{n}+X_j\frac{1}{n}= \mean{X}_{(j)}+\left(X_j-\mean{X}_{(j)}\right)\frac{1}{n} \end{equation}

Sensitive to outliers.

4.2   Median: A resistant measure of central tendency

$\def\median{\text{Median}}$ \begin{equation} \median= \begin{cases} X\left(\frac{n+1}{2}\right)&\text{if $n$ is odd}\\ \frac{1}{2}\left[X\left(\frac{n}{2}\right)+X\left(\frac{n}{2}+1\right)\right]&\text{if $n$ is even} \end{cases} \end{equation}

Less sensitive to outliers.

4.3   Mode

Occurring most often from a discrete dataset

4.4   Geometric mean

$\def\gmean{\text{GM}}$ \begin{equation} \gmean=\left(\prod_{i=1}^nX_i\right)^{1/n} \end{equation} where $X_i>0$.

Useful for positively skewed datasets.

5.1   Sample variance: A classical measure of variability

\begin{equation} s^2=\sum_{i=1}^n\frac{\left(X_i-\mean{X}\right)^2}{n-1} \end{equation}

5.2   Interquartile range (IQR): A resistant measure of variability

Percentiles $P_{X,j}$ can be calculated from a sorted dataset from smallest to largest, $X_i$ for $i=1,\cdots,n$: \begin{equation} P_j=X_{(n+1)\cdot j} \end{equation} and the interquartile range (IQR) can be calculated as follows: $\def\iqr{\text{IQR}}$ \begin{equation} \iqr=P_{0.75}-P_{0.25} \end{equation}

What if $(n+1)\cdot j$ is not an integer? Interpolation and we typically use the Weibull plotting position in hydrology (type=6 in quantile() in R).

5.3   Median absolute deviation (MAD): A resistant measure of variability

$\def\mad{\text{MAD}}$ \begin{equation} \mad(X)=\median{\left(\left|X_i-\median(X)\right|\right)} \end{equation}

5.4   Coefficient of variation (CV): A nondimensional measure of variability

$\def\cv{\text{CV}}$ \begin{equation} \cv=\frac{s^2}{\mean{X}} \end{equation}

Useful for characterizing the degree of variability in datasets.