Measures of location and dispersion

After visualising the data, we summarise them with a few numbers: measures of location (where the centre is) and measures of dispersion (how spread out they are).

Definition — Measures of location and dispersion

Given a sample $x_1,x_2,\ldots,x_n$:

• sample mean: $\bar x=\dfrac{1}{n}\sum_{i=1}^n x_i$ — the “centre of mass” of the data.
• median: the central value after ordering ($n$ odd) or the mean of the two central ones ($n$ even). Robust to outliers.
• sample variance: $s^2=\dfrac{1}{n-1}\sum_{i=1}^n (x_i-\bar x)^2$.
• sample standard deviation: $s=\sqrt{s^2}$ (same unit of measurement as the data).

Observation — Why " $n-1$" and not "$n$"

In the sample variance one divides by $n-1$ instead of $n$ (Bessel’s correction): it is the number of residual degrees of freedom after spending one degree in computing $\bar x$. It guarantees that $E[s^2]=\sigma^2$ (unbiased estimator). Dividing by $n$ would give a systematic underestimate. Reading the denominator $n-1$ as “residual degrees of freedom” is taken from Esty (Ch. 14).

Example — Test marks

Marks of a class of $10$ pupils: $4,5,6,6,7,7,7,8,9,10$. Mean: $\bar x=69/10=6,9$. Median: mean of the two central ones $(7+7)/2=7$. Variance: sum of $(x_i-6,9)^2 = 26,9$, divided by $9$: $s^2\approx 2,99$; $s\approx 1,73$.

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Topics: Probabilita
Concepts: Deviazione standard · Media campionaria · Mediana · Varianza campionaria
Skills: Calcolare · Stimare