Descriptive statistics: complete lesson (mean, median, quartiles, standard deviation)

Basic vocabulary

• Statistical series: list of values $x_1, x_2, \ldots, x_n$.
• Frequency count $n_i$: number of times value $x_i$ appears.
• Total count $N = \sum n_i$.
• Relative frequency $f_i = n_i / N$ (often as %).

Mean

Simple mean: $\bar{x} = \dfrac{x_1 + x_2 + \ldots + x_n}{n}$

Weighted mean (with counts): $\bar{x} = \dfrac{\sum n_i x_i}{N} = \sum f_i x_i$

Median

The median $M$ splits the sorted series into two equal halves.

• If $N$ odd: median = middle value.
• If $N$ even: median = mean of the 2 central values.

Robust to outliers, unlike the mean.

Quartiles

• $Q_1$: 25 % of values are $\leq Q_1$.
• $Q_2$: the median.
• $Q_3$: 75 % of values are $\leq Q_3$.

Interquartile range: $Q_3 - Q_1$. Measures dispersion around the median.

Mode

Most frequent value (or class).

Range

$\text{range} = \max - \min$. Measures the span (sensitive to outliers).

Variance and standard deviation

Variance: mean of squared deviations from the mean.

$V = \dfrac{1}{N} \sum n_i (x_i - \bar{x})^2$

Alternative formula (often faster):

$V = \bar{x^2} - \bar{x}^2 \quad \text{where } \bar{x^2} = \dfrac{1}{N} \sum n_i x_i^2$

Standard deviation: $\sigma = \sqrt{V}$.

Standard deviation is in the same unit as the data, so more “readable”.

Mean vs median

• Mean: good when data are symmetric and without outliers.
• Median: preferred when there are outliers (e.g., salaries, real-estate prices).

Common charts

• Bar chart: counts per value.
• Box plot: min, $Q_1$, median, $Q_3$, max.
• Histogram: continuous classes.

Common mistakes

1. Forgetting to sort before computing the median.
2. Confusing count and relative frequency.
3. Forgetting weighting with counts.
4. Confusing standard deviation and variance (order 2).