Complete course on the median (statistics)

Definition, calculation method (even/odd n), weighted median, grouped data (classes), robustness to extreme values, interpretation, traps, and quiz.

7chapters0%progression

Chapters

1) Understanding the median

The value 'in the middle': 50% of the data below, 50% above.

definition

robust

central tendency

What is the purpose of the median?

The median summarizes a series by a central value: it divides the data into two halves.
It is particularly useful when the mean is misleading (skewed distribution, extreme values).
In real life, it is often used for salaries, real estate prices, income, waiting times...
It is said to be 'robust': a few outlier values change the median little.

Simple intuition

We sort the values: the median is the one in the middle (or the average of the two in the middle).
The goal is not to 'average', but to find a central position (the 50th percentile).
The median depends on the order, not on the distances between values.

Median vs mean: key idea

The mean uses all values and their deviations: it can be pulled up/down by extremes.
The median looks only at 'who is in the middle' after sorting: it is more resistant to extremes.
In a symmetric distribution, the mean and median are often close.

Mini-quiz

The median of a series is:

Use cases

Review a statistics chapterUnderstand the median, the calculation method, and common mistakes (sorting, rank, even/odd n).
Choose between mean and medianKnow why the median is often better when the distribution is skewed or there are outliers (salaries, prices…).
Handle frequency tablesCalculate a median with frequencies using cumulative totals, without reconstructing the entire list of values.
Understand the weighted medianWhen each value does not have the same "weight" (frequencies, importance), learn the logic of cumulative weights up to 50%.
Estimate a median on classesIdentify the median class (N/2) and use interpolation when only intervals are available (histograms).
Self-assess with quizzesMCQs and numerical exercises to check that you know how to apply the right method according to the data format.

What you get

A structured and progressive course: definition, intuition (50% below / 50% above), then calculation method with mandatory sorting.

Complete cases: odd n (central value), even n (average of the two central values), with step-by-step examples and corrected quizzes.

Useful extensions: weighted median (weights/frequencies) and median on grouped data (median class + interpolation).

An interpretation section: median as the 50th percentile (Q2), link with quartiles/IQR, and why the median is robust against outliers.

Limits and points of attention

The median depends on sorting: without a sorted series (or without cumulative frequencies), an immediate error is likely.
The median describes a position, not dispersion: two very different series can have the same median. To complement, quartiles and IQR are often looked at.
On grouped data in classes, the median is often an estimate: it depends on the assumption of uniform distribution in the median class.
The median alone is not always sufficient to decide: depending on the objective, one might prefer mean, median, or a median + IQR pair (or even a robust mean).

Frequently asked questions

Why is it necessary to sort the series to calculate the median?

Because the median is defined by the position "in the middle" in the ordered series. Without sorting, we do not know which values are central.

What is the median when there is an even number of values?

We take the average of the two central values (after sorting). The median can then be a number that does not appear in the series, and that is normal.

Median or mean: which one to choose?

If the series contains outliers or strong skewness (salaries, prices), the median is often more representative. If the data is "homogeneous", the mean may be more informative.

What is the median on a frequency table or classes?

With frequencies, we use the cumulative totals to find where 50% of the total is located. With classes, we identify the median class via N/2 and then can estimate the median by interpolation in that class.

Tip

Error-avoidance reflex: after sorting, first identify the two "central" values (even if n is odd). If n is even, the median is between the two; if n is odd, it is exactly one of them.