Complete course on means (statistics)

Arithmetic mean, weighted mean, geometric mean, harmonic mean, quadratic mean (RMS), moving average, truncated mean, grouped data… definitions, methods, examples, pitfalls, and quizzes.

10chapters0%progression

Chapters

1) Understand what a 'mean' is

A central value to summarize a series… but not always 'the arithmetic mean'.

definition

central tendency

method

Why use a mean?

A mean summarizes a list of values with a single number (grades, prices, measurements, durations…).
It is used to compare groups (two classes, two periods), track a trend (monthly average), or measure overall performance.
It facilitates decision-making: a simple indicator is often more readable than a long list of values.
But a mean can be misleading if the distribution is very skewed or if there are extreme values.

Three words to know: mean, median, mode

Mean (in a broad sense): a family of 'central' indicators (arithmetic, weighted, geometric, harmonic…).
Median: the value that divides the series in two (50% below, 50% above).
Mode: the most frequent value (can be multiple).
In practice, we often look at mean + median + dispersion to understand a series.

The right reflex: choose the right type of mean

Arithmetic: when adding comparable values (grades, sizes, quantities).
Weighted: when some values count more (coefficients, quantities, durations).
Geometric: when effects multiply (growth, successive returns).
Harmonic: for 'per unit' ratios (speed, flow) when the base unit is the same (same distance, same quantity).
RMS / quadratic: when working with squares (signals, effective amplitude, error).
Truncated / winsorized: when wanting to limit the impact of extreme values.
Moving: when wanting to smooth a series over time.

Mini-quiz

A mean primarily serves to:

Use cases

Reviewing for a test or examReview definitions, formulas, examples, and typical questions (arithmetic, weighted, geometric…).
Choosing the right mean in real situationsGrades with coefficients, average price paid, average speed, compound growth: each context calls for a different mean.
Understanding common pitfallsAvoid the "mean of means", errors on speeds, or the mean of percentages when there is composition.
Interpreting a mean without being misledKnow when a mean is misleading (extreme values, skewed distribution) and when to prefer a more robust approach.
Learning with step-by-step examplesEach concept is explained with series of values, detailed calculations, and formulas (LaTeX) for clear visualization.
Self-assessing with quizzesMultiple choice and numerical questions to check that you know how to apply the right methods at the right time.

What you get

A structured course by types of means: arithmetic, weighted, geometric, harmonic, RMS (quadratic), robust (trimmed / winsorized), moving, and mean with frequencies or grouped data.

Clear formulas and guided examples to move from definition to application (grades, prices, speeds, yields, time series…).

A quick choice method: recognize the context (coefficients, ratios, composition, outliers, time) to select the relevant mean.

Quizzes and corrected explanations to reinforce habits and identify your typical mistakes.

Limits and points of attention

A mean does not summarize an entire distribution: without information on dispersion (standard deviation, quartiles...), it can mask significant differences.
The right type of mean depends on the context: addition of values (arithmetic), coefficients/quantities (weighted), composition (geometric), ratios on the same basis (harmonic), amplitude/error (RMS).
Be careful with extreme values: the arithmetic mean is sensitive to outliers, hence the interest in robust means (trimmed/winsorized) or the median depending on the need.
Data grouped by classes often gives an approximate mean (class centers): the wider the classes, the coarser the approximation can be.

Frequently Asked Questions

Does this course replace a mean calculator?

No: the main goal is to explain and help choose the right mean. For "quick calculations", then use dedicated tools (arithmetic mean, weighted, moving...).

Why not take the arithmetic mean in all cases?

Because some phenomena are composed (returns), others are ratios (speed), and others have weights (coefficients). In these cases, arithmetic can be false or misleading.

What mean for successive returns (+10% then -10%)?

We reason in factors and use the geometric mean (compound effect). This is precisely a trap explained in the course.

What mean for average speed?

Over equal distances, it’s the harmonic mean; over equal durations, it’s the arithmetic mean. The course shows how to recognize the right case.

Tip

Before calculating, ask yourself one question: "Am I adding values, or am I combining weights/ratios/multipliers?" This step avoids most mean errors.