Problem — Mean, median, variance, standard deviation

Text

Compute the mean, median, variance and standard deviation of the sample $\{2,4,4,4,5,5,7,9\}$.

Solution

Set-up. The data are $n=8$ values that are already ordered.

• Sample mean: $\bar x = \dfrac{1}{n}\sum_i x_i$, i.e. the sum of the values divided by $8$.
• Median: with $n$ even it is the average of the two central values (the $4^\circ$ and the $5^\circ$ in the ordering).
• Sample variance: $s^2 = \dfrac{1}{n-1}\sum_i (x_i-\bar x)^2$ (one may use $n$ or $n-1$ in the denominator depending on the convention adopted; for a sample $n-1$ is preferred).
• Standard deviation: $s = \sqrt{s^2}$.

Follow the order: first $\bar x$, then the deviations $x_i-\bar x$, their sum of squares, and finally the square root.

Links

Topics: Probabilita
Concepts: Deviazione standard · Media campionaria · Mediana · Varianza campionaria
Skills: Calcolare · Stimare
Exercise type: Problema modellizzazione