A statistician's toolbox

Vocabulary

• census: recensement
• average: moyenne
• mean: moyenne
• median: médiane
• standard deviation: écart-type
• variance: variance
• box plot: boîte à moustaches
• scatter plot: nuage de points

Formulas

Let $x_1, x_2, ... , x_{n-1}, x_n\in\mathbb{R}$ be real numbers.

• The mean is often denoted by $\overline x$ and is given by

• The variance measures the dispersion of the values $x_k$. It is denoted by the the letter $V$ and is given by

• The standard deviation is denoted by the Greek letter $\sigma$ and is given by

Let $y_1, y_2, ... , y_{n-1}, y_n\in\mathbb{R}$ be real numbers. We now have two samples of data $x$ and $y$.

• We can compute their covariance

If the respective standard deviations of the values $x$ and $y$ are denoted by $\sigma_x$ and $\sigma_y$, we then obtain their

• Pearson correlation coefficient, denoted by $r$, and given by

We always have $-1\leq r \leq 1$. If the values $x$ and $y$ are mostly independant, we obtain $r\approx0$, whereas a value $r\approx\pm1$ means that the values $x$ and $y$ are correlated.

Application

Three students, Alice, Beatriz, and Charles, are in the same class. Out of guilt, they admitted that they cheated during their year together. By analyzing their marks, can you understand who cheated?

Correlations

Sometimes two things can be correlated (have $r\approx1$), just by coincidence.