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Geometric mean

Geometric mean is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Geometric mean rather than just read about it. In short: In mathematics, the geometric mean (also known as the mean proportional) is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean, which uses their sum). The geometric mean of ⁠ n {\displaystyle n} ⁠ numbers is the nth root of their product, i.e., for a collection of numbers a1, a2, ..., an, the geo…

Geometric mean — main illustration
Geometric mean — illustration

Key takeaways

  • Geometric mean belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Geometric mean to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Geometric mean from memory before moving on to harder problems.

Reference excerpt

In mathematics, the geometric mean (also known as the mean proportional) is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean, which uses their sum). The geometric mean of ⁠ n {\displaystyle n} ⁠ numbers is the nth root of their product, i.e., for a collection of numbers a1, a2, ..., an, the geometric mean is defined as

a 1 a 2 ⋯ a n t n . {\displaystyle {\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}{\vphantom {t}}}}.}

Hence, the geometric mean of two numbers is the square root of their product, for example with numbers ⁠ 2 {\displaystyle 2} ⁠ and ⁠ 8 {\displaystyle 8} ⁠ the geometric mean is 2 ⋅ 8 =

{\displaystyle \textstyle {\sqrt {2\cdot 8}}={}}

16 = 4 {\displaystyle \textstyle {\sqrt {16}}=4} . The geometric mean of the three numbers is the cube root of their product, for example with numbers ⁠ 1 {\displaystyle 1} ⁠, ⁠ 12 {\displaystyle 12} ⁠, and ⁠ 18 {\displaystyle 18} ⁠, the geometric mean is 1 ⋅ 12 ⋅ 18 3 =

{\displaystyle \textstyle {\sqrt[{3}]{1\cdot 12\cdot 18}}={}}

216 3 = 6 {\displaystyle \textstyle {\sqrt[{3}]{216}}=6} . When the collection of numbers and their geometric mean are plotted in logarithmic scale, the geometric mean is transformed into an arithmetic mean, so the geometric mean can equivalently be calculated by taking the natural logarithm ⁠ ln {\displaystyle \ln } ⁠ of each number, finding the arithmetic mean of the logarithms, and then returning the result to linear scale by using the exponential function ⁠ exp {\displaystyle \exp } ⁠,

a 1 a 2 ⋯ a n t n = exp ⁡ ( ln ⁡ a 1 + ln ⁡ a 2 + ⋯ + ln ⁡ a n n ) . {\displaystyle {\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}{\vphantom {t}}}}=\exp \left({\frac {\ln a_{1}+\ln a_{2}+\cdots +\ln a_{n}}{n}}\right).}

… excerpt ends here. Continue reading the full article.

Illustrations

Geometric mean: Example of the geometric mean: l g {\displaystyle l_{g}} (red) is the geometric mean of l 1 {\displaystyle l_{1}} and l 2 {\displaystyle l_{2}} ,[1][2] is an example in which the line segment l 2 ( B C ¯ ) {\displaystyle l_{2}\;({\overline {BC}})} is given as a perpendicular to A B ¯ {\displaystyle {\overline {AB}}} . A C ′ ¯ {\displaystyle {\overline {AC'}}} is the diameter of a circle and B C ¯ ≅ B C ′ ¯ {\displaystyle {\overline {BC}}\cong {\overline {BC'}}} .
Example of the geometric mean: l g {\displaystyle l_{g}} (red) is the geometric mean of l 1 {\displaystyle l_{1}} and l 2 {\displaystyle l_{2}} ,[1][2] is an example in which the line segment l 2 ( B C ¯ ) {\displaystyle l_{2}\;({\overline {BC}})} is given as a perpendicular to A B ¯ {\displaystyle {\overline {AB}}} . A C ′ ¯ {\displaystyle {\overline {AC'}}} is the diameter of a circle and B C ¯ ≅ B C ′ ¯ {\displaystyle {\overline {BC}}\cong {\overline {BC'}}} .
Geometric mean: Proof without words of the AM–GM inequality:PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Triangle PGR is a right triangle from Thales's theorem, enabling use of the geometric mean theorem to show that its altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ.
Proof without words of the AM–GM inequality:PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Triangle PGR is a right triangle from Thales's theorem, enabling use of the geometric mean theorem to show that its altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ.
Geometric mean: Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two distinct positive numbers a and b[note 1]
Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two distinct positive numbers a and b[note 1]
Geometric mean: The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides (p + q, r, s ), (r, p, h ) and (s, h, q ), ( p + q ) 2 = r 2 + s 2 p 2 + 2 p q + q 2 = p 2 + h 2 ⏞ + h 2 + q 2 ⏞ 2 p q = 2 h 2 ∴ h = p q {\displaystyle {\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}}
The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides (p + q, r, s ), (r, p, h ) and (s, h, q ), ( p + q ) 2 = r 2 + s 2 p 2 + 2 p q + q 2 = p 2 + h 2 ⏞ + h 2 + q 2 ⏞ 2 p q = 2 h 2 ∴ h = p q {\displaystyle {\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}}
Geometric mean: Equal area comparison of the aspect ratios used by Kerns Powers to derive the SMPTE 16:9 standard.[13] .mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}  TV 4:3/1.33 in red,   1.66 in orange,   16:9/1.77 in blue,   1.85 in yellow,   Panavision/2.2 in mauve and   CinemaScope/2.35 in purple.
Equal area comparison of the aspect ratios used by Kerns Powers to derive the SMPTE 16:9 standard.[13] .mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}  TV 4:3/1.33 in red,   1.66 in orange,   16:9/1.77 in blue,   1.85 in yellow,   Panavision/2.2 in mauve and   CinemaScope/2.35 in purple.

Worked examples

Example 1 — a first encounter with Geometric mean

Start with the simplest possible case. Write down what Geometric mean claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Geometric mean before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Geometric mean ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Geometric mean

In research
Geometric mean appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Geometric mean in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Geometric mean is common in secondary-school and first-year university syllabi. It links to neighbouring topics Means, so understanding it makes those chapters shorter.
In everyday life
Look for Geometric mean outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Geometric mean in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Geometric mean means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Geometric mean out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Geometric mean in simple terms?

In mathematics, the geometric mean (also known as the mean proportional) is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean, which uses their sum). The geometric mean of ⁠ n…

Why does Geometric mean matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Geometric mean?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Geometric mean.

Tags

  • Means

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