Median and mode

The median is the middle value of the sorted data (robust to outliers); the mode is the most frequent value (the only centre for nominal data).

\text{median} = \begin{cases} x_{((n+1)/2)} & n \text{ odd} \\[4pt] \tfrac{1}{2}\left(x_{(n/2)} + x_{(n/2+1)}\right) & n \text{ even} \end{cases}

Median of the sorted values.

$x_{(k)}$: The $k$ -th smallest value (sorted order).
$n$: Number of values.

When to use

Median: skewed quantitative data or data with outliers (incomes, house prices, delays); ordinal data.
Mode: categorical data ("most common payment method"), or to describe peaks of a distribution (unimodal, bimodal).

How to interpret

Half the values are at or below the median and half at or above it. Mean > median suggests right skew; mean < median suggests left skew (a rule of thumb, not a guarantee). The engine reports all tied modes, and no mode when every value occurs once.

Common mistakes

Taking the middle of the unsorted list.
For even $n$ , picking one of the two middle values instead of averaging them.
Reporting a mode for continuous measurements where every value is distinct — use a histogram peak instead.

Example

Values 3, 8, 5, 12, 5, 9. Sorted: 3, 5, 5, 8, 9, 12. With $n = 6$ the median is $(5 + 8)/2 = 6.5$ ; the mode is 5 (it appears twice).