Central tendency measures

Mode

A value x with the highest probability \(f(x) = P(X = x)\) is called mode of the random variable X (resp. its distribution)

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Quantiles and median

Let 0 < p < 1. Every value \(\tilde{x_p}\) for which the random variable X - takes on a value less or equal \(\tilde{x_p}\) with probability of at least p an - takes on a value greater or equal \(\tilde{x_p}\) with probability of at least 1 - p
is called p-quantile of the random variable X
For p = 0.5 the 50%-quantile \(\tilde{x_p} = \tilde{x_0.5}\) is called the median

Concrete calculation

To get the median for a continuous random variable with given density function f(x) you are searching for a \(m\), so that \(\int_{-\infty}^{m} f(x) \; dx = 0.5\)

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Expected value

The expected value is the most important location measure for metric discrete random variables

\[\mu = E(x) = x_1p_1 + ... + x_kp_k + ... = \sum\limits_{i \ge 1} x_ip_i\]

Estimators: \(\overline{X} = g(x_1, \dots, x_n) = \frac{1}{n}\sum_{i = 1}^{n} xi\)
\(1-\alpha\)-confidence of a \(N(\mu, o^2)\)-distribution for known \(o^2\): \([\overline{X}-Z_{1-\alpha/2} * \frac{o}{\sqrt{n}}, \overline{X}+Z_{1-\alpha/2} * \frac{o}{\sqrt{n}}]\)

Interpretation of the expected value

Matching the analogy of expected value and arithmetic mean, we can observe the following frequency interpretation of the expected value:
Let the random trial underlying the metric random variable X be performed n-times independently, observing the n realizations \(\hat{x_1},...,\hat{x_n}\) of X. Then for large 𝑛𝑛, the arithmetic mean \(\overline{x}\) of \(\hat{x_1},...,\hat{x_n}\) is close to \(\mu = E(X)\) with high probability.

E(X) can be interpreted as the value that results on average if we repeat the random process often enough

Properties of the expected value

• \(E(X+Y) = E(X) + E(Y)\)
• \(E(a_1X_1 + ... + a_nX_n = a_1E(X_1) + ... + a_nE(X_n)\)
• For independent metric random variables X and Y it holds that \(E(X * Y) = E(X) * E(Y)\)

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