Mean

For $n$ observed values $x$ of variable $X$, the mean equals

$$\begin{equation*} \overline{x}=\frac{ \sum_{i=1}^n\limits x_i}{n} \end{equation*}$$

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Mean of a frequency distribution

For $n$ observed values $x$ of variable $X$, with $k$ different outcomes with frequency $f$, the mean equals

$$\begin{equation*} \overline{x}=\frac{ \sum_{i=1}^k\limits f_i x_i}{n} \end{equation*}$$ For a dichotomous (binary) variable $X$ with different outcomes $x=0$ and $x=1$, the mean equals the proportion of outcomes $x=1$, referred to as $p_x$.

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Median

The median is the middle observed value of all ordered observations. The median corresponds to the $50$th percentile, $P_{50}$ (see `Percentiles' below).

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Mode

The modus is the most frequent observed value.

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Standard deviation

The standard deviation (as estimator for the population value $\sigma$) is

$$\begin{equation*} s_x = \sqrt{\frac{\sum_{i=1}^n (x_i-\overline{x})^2}{n-1}}. \end{equation*}$$
The standard deviation population value for a dichotomous (binary) variable is
$$\begin{equation*} \sigma_x = \sqrt{p_x(1-p_x)}.\end{equation*}$$

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Variance

The variance (as estimator for the population value $\sigma^2$) is

$$\begin{equation*} s_x^2 = \displaystyle\frac{\sum_{i=1}^n (x_i-\overline{x})^2}{n-1}.\end{equation*}$$
The variance population value for a dichotomous (binary) variable is
$$\begin{equation*} \sigma_x^2 = p_x(1-p_x). \end{equation*}$$

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Percentiles

The $p$th percentile is the value for which $p$ percent of observations is smaller or equal. For example, $50$th percentile is the value for which holds that half of all observations are smaller or equal. This is referred to as $P_{50}$ (which is equivalent to the median).

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Interquartile

The interquartile distance is

$$\begin{equation*} IQR=Q_3-Q_1, \end{equation*}$$ where $Q_3$ corresponds to $P_{75}$ and $Q_1$ corresponds to $P_{25}$.

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Range

The range indicates within which distance from each other with all observed values are located. It is calculated by

$$\begin{equation*} range = maximum - minimum. \end{equation*}$$

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Z-score

The z-score, or standardized score

$$\begin{equation*} z_{x_i}=\frac{x_i-\overline{x}}{s_x}. \end{equation*}$$ (This is a linear transformation with $a=-\overline{x}/s_x$ and $b=1/s_x$, see `Linear transformation' below).

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Linear transformation

For a linear transformation $y_i=a+bx_i$ the following holds

$$\begin{equation*} \overline{y} = a+b\cdot \overline{x} \end{equation*}$$ and $$\begin{eqnarray*} s_y^2 & = & b^2\cdot s_x^2 \\ s_y & = & b\cdot s_x. \end{eqnarray*}$$

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Covariance

The covariance between $x$ and $y$

$$\begin{equation*} s_{xy}=\frac{1}{n-1}\sum_{i=1}^n\limits (x_i-\overline{x})(y_i-\overline{y}). \end{equation*}$$ The following rules apply with respect to the variance and covariance: $$\begin{eqnarray*} s_{xx} & = & s_x^2 \\ s_{x+y}^2 & = & s_x^2+s_y^2+2s_{xy} \\ s_{x-y}^2 & = & s_x^2+s_y^2-2s_{xy}. \end{eqnarray*}$$
For two dichotomous (binary) variables $X$ and $Y$, where $p_{xy}$ equals the probability of a score of 1 for both $X$ and $Y$, the covariance population value equals
$$\begin{equation*} \sigma_{xy}=p_{xy}-p_xp_y. \end{equation*}$$

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Pearson's (product-moment) correlation coefficient

The correlation between $x$ and $y$

$$\begin{eqnarray*} r_{xy}& =& \frac{s_{xy}}{s_xs_y}\\ & =& \frac{1}{n-1}\sum_{i=1}^n\limits z_{x_i}z_{y_i}\\ & = & \frac{1}{n-1}\sum_{i=1}^n\limits \left(\frac{x_i-\overline{x}}{s_x}\right)\left(\frac{y_i-\overline{y}}{s_y}\right). \end{eqnarray*}$$