2435: Geothmetic Meandian - Revision history

82.132.236.174: /* Explanation */ Reclarified that it's actually the maxima/minima, being formatted, not just the second of each thing they alternate between. Realigned the table data, by fixed-width spacing, purely for source-viewing aesthetics!

2025-10-22T07:20:01Z

‎Explanation: Reclarified that it's actually the maxima/minima, being formatted, not just the second of each thing they alternate between. Realigned the table data, by fixed-width spacing, purely for source-viewing aesthetics!

2607:F140:400:6F:9832:655B:9E48:6AC2: /* Explanation */ Added italics for minima

2025-10-22T01:52:00Z

‎Explanation: Added italics for minima

172.71.241.145: /* Explanation */ Rationalising wikilinks to site's templated format.

2025-02-10T23:21:54Z

‎Explanation: Rationalising wikilinks to site's templated format.

Jukamala: /* Explanation */

2025-02-10T20:25:08Z

‎Explanation

Jukamala: Fixed the table, added information on convergence proofs, and dis-ambiguation of the definition.

2025-02-10T20:19:58Z

Fixed the table, added information on convergence proofs, and dis-ambiguation of the definition.

Jukamala: /* Trivia */

2025-02-10T18:33:23Z

‎Trivia

GreatWyrmGold: /* Explanation */ Added explanations of arithmetic mean and median. If anyone dislikes how I explained median, please feel free to write a more technical definition.

2024-06-25T17:39:44Z

‎Explanation: Added explanations of arithmetic mean and median. If anyone dislikes how I explained median, please feel free to write a more technical definition.

Kynde: /* Explanation */

2024-01-18T12:31:22Z

‎Explanation

162.158.38.240: /* Explanation */ sorry, thought the Arithmetic–geometric mean was a pythagorean mean

2023-10-29T10:41:54Z

‎Explanation: sorry, thought the Arithmetic–geometric mean was a pythagorean mean

162.158.230.56: /* Explanation */

2023-10-29T10:35:30Z

‎Explanation

@@ Line 26: / Line 26: @@
   |-
   ! F1
-  | 2.4 || ''1.974350486'' || 2
+  | '''2.4'''         || ''1.974350486'' ||   2
   |-
   ! F2
-  | '''2.124783495''' ||	2.116192461 || ''2''
+  | '''2.124783495''' ||   2.116192461   ||  ''2''
   |-
   ! F3
-  | 2.080325319 || ''2.079536819'' || '''2.116192461'''
+  |    2.080325319    || ''2.079536819'' || '''2.116192461'''
   |-
   ! F4
-  | '''2.0920182''' || 2.091948605 || ''2.080325319''
+  | '''2.0920182'''   ||   2.091948605   ||  ''2.080325319''
   |-
   ! F5
-  | 2.088097374 || ''2.088090133'' || '''2.091948605'''
+  |    2.088097374    || ''2.088090133'' || '''2.091948605'''
   |-
   ! F6
-  | '''2.089378704''' ||	2.089377914 || ''2.088097374''
+  | '''2.089378704''' ||   2.089377914   ||  ''2.088097374''
   |-
   ! F7
-  | 2.088951331 ||	''2.088951244'' || '''2.089377914'''
+  |    2.088951331    ||	''2.088951244'' || '''2.089377914'''
   |-
   ! F8
-  | '''2.089093496''' || 2.089093487 || ''2.088951331''
+  | '''2.089093496''' ||   2.089093487   ||  ''2.088951331''
   |-
   ! F9
-  | 2.089046105 || ''2.089046103'' || '''2.089093487'''
+  |    2.089046105    || ''2.089046103'' || '''2.089093487'''
   |-
   ! F10
-  | '''2.089061898''' || 2.089061898 || ''2.089046105''
+  | '''2.089061898''' ||   2.089061898   ||  ''2.089046105''
   |}
-Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value alternates between the average and the median (bolded in the table), while the minimum value alternates between the geometric mean and the median (italiticzed in the table). This observation holds for many inputs.
+Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value (bolded in the table) alternates between the arithmetic average and the median, while the minimum value (italiticzed in the table) alternates between the geometric mean and the median. This observation holds for many inputs.
 To not distract from the comedic effect, the definition of the GMDN in the comic is left as a simplified sketch. To make the definition mathematically rigorous the implied infinite limit in the second line can be made precise, for example, as the result of a {{w|fixed-point iteration}} via <code>G = lim_{k -> infinity} m_k </code> where <code>m_0 = F(x_1, x_2, ..., x_n)</code> and <code>m_{k+1} = F(m_k) for k > 0</code>. This definition is well-defined only if we can proof convergence to a fixpoint of F for a set of inputs. Indeed, convergence holds if all numbers are non-negative (see discussions for proof and more cases). Note that the above definition yields a three-dimensional fixpoint G. Because all fixpoints of F are of the form <code>G=(g, g, g)</code>, with elements that are all equal, we can define <code>GMDN(x_1, x_2, ..., x_n) = G_1</code>, as the first element of G. This formal definition avoids the inconsistency present in the comic's definition sketch where the function GMDN as defined in the second line has the same three-dimensional output as F, but GMDN in the last line is shown to produce a single real number rather than a vector and is thus missing a final operation of returning a single component. Note also that the comic's definition of the median as the (n+1)/2-th {{w|order statistic}}, i.e. the (n+1)/2-th smallest value, coincides with the more regularly used sample median only on lists of odd length. For lists of even length the sample median is usually defined as the (arithmetic) mean of the two middle values <code>X_{n/2}</code> and <code>X_{(n+2)/2}</code> instead. Indeed, for lists of even lengths <code>X_{(n+1)/2}</code> is not well-defined without adding a flooring operation as (n+1)/2 is not an integer.

@@ Line 26: / Line 26: @@
   |-
   ! F1
-  | 2.4 || 1.974350486 || 2
+  | 2.4 || ''1.974350486'' || 2
   |-
   ! F2
-  | '''2.124783495''' ||	2.116192461 || 2
+  | '''2.124783495''' ||	2.116192461 || ''2''
   |-
   ! F3
-  | 2.080325319 || 2.079536819 || '''2.116192461'''
+  | 2.080325319 || ''2.079536819'' || '''2.116192461'''
   |-
   ! F4
-  | '''2.0920182''' || 2.091948605 || 2.080325319
+  | '''2.0920182''' || 2.091948605 || ''2.080325319''
   |-
   ! F5
-  | 2.088097374 || 2.088090133 || '''2.091948605'''
+  | 2.088097374 || ''2.088090133'' || '''2.091948605'''
   |-
   ! F6
-  | '''2.089378704''' ||	2.089377914 || 2.088097374
+  | '''2.089378704''' ||	2.089377914 || ''2.088097374''
   |-
   ! F7
-  | 2.088951331 ||	2.088951244 || '''2.089377914'''
+  | 2.088951331 ||	''2.088951244'' || '''2.089377914'''
   |-
   ! F8
-  | '''2.089093496''' || 2.089093487 || 2.088951331
+  | '''2.089093496''' || 2.089093487 || ''2.088951331''
   |-
   ! F9
-  | 2.089046105 || 2.089046103 || '''2.089093487'''
+  | 2.089046105 || ''2.089046103'' || '''2.089093487'''
   |-
   ! F10
-  | '''2.089061898''' || 2.089061898 || 2.089046105
+  | '''2.089061898''' || 2.089061898 || ''2.089046105''
   |}
-Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value alternates between the average and the median (highlighted in bold in the table), while the minimum value alternates between the geomean and the median. This observation holds for many inputs.
+Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value alternates between the average and the median (bolded in the table), while the minimum value alternates between the geometric mean and the median (italiticzed in the table). This observation holds for many inputs.
 To not distract from the comedic effect, the definition of the GMDN in the comic is left as a simplified sketch. To make the definition mathematically rigorous the implied infinite limit in the second line can be made precise, for example, as the result of a {{w|fixed-point iteration}} via <code>G = lim_{k -> infinity} m_k </code> where <code>m_0 = F(x_1, x_2, ..., x_n)</code> and <code>m_{k+1} = F(m_k) for k > 0</code>. This definition is well-defined only if we can proof convergence to a fixpoint of F for a set of inputs. Indeed, convergence holds if all numbers are non-negative (see discussions for proof and more cases). Note that the above definition yields a three-dimensional fixpoint G. Because all fixpoints of F are of the form <code>G=(g, g, g)</code>, with elements that are all equal, we can define <code>GMDN(x_1, x_2, ..., x_n) = G_1</code>, as the first element of G. This formal definition avoids the inconsistency present in the comic's definition sketch where the function GMDN as defined in the second line has the same three-dimensional output as F, but GMDN in the last line is shown to produce a single real number rather than a vector and is thus missing a final operation of returning a single component. Note also that the comic's definition of the median as the (n+1)/2-th {{w|order statistic}}, i.e. the (n+1)/2-th smallest value, coincides with the more regularly used sample median only on lists of odd length. For lists of even length the sample median is usually defined as the (arithmetic) mean of the two middle values <code>X_{n/2}</code> and <code>X_{(n+2)/2}</code> instead. Indeed, for lists of even lengths <code>X_{(n+1)/2}</code> is not well-defined without adding a flooring operation as (n+1)/2 is not an integer.

@@ Line 58: / Line 58: @@
 Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value alternates between the average and the median (highlighted in bold in the table), while the minimum value alternates between the geomean and the median. This observation holds for many inputs.
-To not distract from the comedic effect, the definition of the GMDN in the comic is left as a simplified sketch. To make the definition mathematically rigorous the implied infinite limit in the second line can be made precise, for example, as the result of a [https://en.wikipedia.org/wiki/Fixed-point_iteration fixed-point iteration] via <code>G = lim_{k -> infinity} m_k </code> where <code>m_0 = F(x_1, x_2, ..., x_n)</code> and <code>m_{k+1} = F(m_k) for k > 0</code>. This definition is well-defined only if we can proof convergence to a fixpoint of F for a set of inputs. Indeed, convergence holds if all numbers are non-negative (see discussions for proof and more cases). Note that the above definition yields a three-dimensional fixpoint G. Because all fixpoints of F are of the form <code>G=(g, g, g)</code>, with elements that are all equal, we can define <code>GMDN(x_1, x_2, ..., x_n) = G_1</code>, as the first element of G. This formal definition avoids the inconsistency present in the comic's definition sketch where the function GMDN as defined in the second line has the same three-dimensional output as F, but GMDN in the last line is shown to produce a single real number rather than a vector and is thus missing a final operation of returning a single component. Note also that the comic's definition of the median as the (n+1)/2-th [https://en.wikipedia.org/wiki/Order_statistic order statistic], i.e. the (n+1)/2-th smallest value, coincides with the more regularly used sample median only on lists of odd length. For lists of even length the sample median is usually defined as the (arithmetic) mean of the two middle values <code>X_{n/2}</code> and <code>X_{(n+2)/2}</code> instead. Indeed, for lists of even lengths <code>X_{(n+1)/2}</code> is not well-defined without adding a flooring operation as (n+1)/2 is not an integer.
+To not distract from the comedic effect, the definition of the GMDN in the comic is left as a simplified sketch. To make the definition mathematically rigorous the implied infinite limit in the second line can be made precise, for example, as the result of a {{w|fixed-point iteration}} via <code>G = lim_{k -> infinity} m_k </code> where <code>m_0 = F(x_1, x_2, ..., x_n)</code> and <code>m_{k+1} = F(m_k) for k > 0</code>. This definition is well-defined only if we can proof convergence to a fixpoint of F for a set of inputs. Indeed, convergence holds if all numbers are non-negative (see discussions for proof and more cases). Note that the above definition yields a three-dimensional fixpoint G. Because all fixpoints of F are of the form <code>G=(g, g, g)</code>, with elements that are all equal, we can define <code>GMDN(x_1, x_2, ..., x_n) = G_1</code>, as the first element of G. This formal definition avoids the inconsistency present in the comic's definition sketch where the function GMDN as defined in the second line has the same three-dimensional output as F, but GMDN in the last line is shown to produce a single real number rather than a vector and is thus missing a final operation of returning a single component. Note also that the comic's definition of the median as the (n+1)/2-th {{w|order statistic}}, i.e. the (n+1)/2-th smallest value, coincides with the more regularly used sample median only on lists of odd length. For lists of even length the sample median is usually defined as the (arithmetic) mean of the two middle values <code>X_{n/2}</code> and <code>X_{(n+2)/2}</code> instead. Indeed, for lists of even lengths <code>X_{(n+1)/2}</code> is not well-defined without adding a flooring operation as (n+1)/2 is not an integer.
 The comment in the title text about suggests that this will save you the trouble of committing to the 'wrong' analysis as it gradually shaves down any 'outlier average' that is unduly affected by anomalies in the original inputs. It is a method without any danger of divergence of values, since all three averaging methods stay within the interval covering the input values (and two of them will stay strictly within that interval).
@@ Line 76: / Line 76: @@
 There does exist an {{w|arithmetic-geometric mean}}, which is defined identically to this except with the arithmetic and geometric means, and sees some use in calculus.  In some ways it's also philosophically similar to the {{w|truncated mean}} (extremities of the value range, e.g. the highest and lowest 10%s, are ignored as not acceptable and not counted) or {{w|Winsorized mean}} (instead of ignored, the values are readjusted to be the chosen floor/ceiling values that they lie beyond, to still effectively be counted as "edge" conditions), only with a strange dilution-and-compromise method rather than one where quantities can be culled or neutered just for being unexpectedly different from most of the other data.
-The input sequence of numbers (1, 1, 2, 3, 5) chosen by Randall is also the opening of the {{w|Fibonacci sequence}}.  This may have been selected because the Fibonacci sequence also has a convergent property: the ratio of two adjacent numbers in the sequence approaches the [https://en.wikipedia.org/wiki/Golden_ratio#Relationship_to_Fibonacci_sequence golden ratio] as the length of the sequence approaches infinity.
+The input sequence of numbers (1, 1, 2, 3, 5) chosen by Randall is also the opening of the {{w|Fibonacci sequence}}.  This may have been selected because the Fibonacci sequence also has a convergent property: the ratio of two adjacent numbers in the sequence approaches the {{w|Golden ratio#Relationship to Fibonacci sequence|golden ratio}} as the length of the sequence approaches infinity.
 Here is a table of averages classified by the various methods referenced:

@@ Line 58: / Line 58: @@
 Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value alternates between the average and the median (highlighted in bold in the table), while the minimum value alternates between the geomean and the median. This observation holds for many inputs.
-To not distract from the comedic effect, the definition of the GMDN in the comic is left as a simplified sketch. To make the definition mathematically rigorous the implied infinite limit in the second line can be made precise, for example, as the result of a [https://en.wikipedia.org/wiki/Fixed-point_iteration fixed-point iteration] via <code>G = lim_{k -> infinity} m_k </code> where <code>m_0 = F(x_1, x_2, ..., x_n)</code> and <code>m_{k+1} = F(m_k) for k > 0</code>. This definition is well-defined only if we can proof convergence to a fixpoint of F for a set of inputs. Indeed, convergence holds if all numbers are non-negative (see discussions for proof and more cases). Note that the above definition yields a three-dimensional fixpoint G. Because all fixpoints of F are of the form <code>G=(g, g, g)</code>, with elements that are all equal, we can define <code>GMDN(x_1, x_2, ..., x_n) = G_1</code>, as the first element of G. This formal definition avoids the inconsistency present in the comic's definition sketch where the function GMDN as defined in the second line has the same three-dimensional output as F, but GMDN in the last line is shown to produce a single real number rather than a vector and is thus missing a final operation of returning a single component. Note also that the comic's definition of the median as the (n+1)/2-th [https://en.wikipedia.org/wiki/Order_statistic order statistic], i.e. the (n+1)/2-th smallest value, coincides with the more regularly used sample median only on lists of odd length. For lists of even length the sample median is usually defined as the (arithmetic) mean of the two middle values <code>X_{n/2}</code> and <code>X_{(n+2)/2}</code> instead. Indeed, for lists of even lengths <code>X_{(n+1)/2}</code> is not well-defined as (n+1)/2 is not an integer.
+To not distract from the comedic effect, the definition of the GMDN in the comic is left as a simplified sketch. To make the definition mathematically rigorous the implied infinite limit in the second line can be made precise, for example, as the result of a [https://en.wikipedia.org/wiki/Fixed-point_iteration fixed-point iteration] via <code>G = lim_{k -> infinity} m_k </code> where <code>m_0 = F(x_1, x_2, ..., x_n)</code> and <code>m_{k+1} = F(m_k) for k > 0</code>. This definition is well-defined only if we can proof convergence to a fixpoint of F for a set of inputs. Indeed, convergence holds if all numbers are non-negative (see discussions for proof and more cases). Note that the above definition yields a three-dimensional fixpoint G. Because all fixpoints of F are of the form <code>G=(g, g, g)</code>, with elements that are all equal, we can define <code>GMDN(x_1, x_2, ..., x_n) = G_1</code>, as the first element of G. This formal definition avoids the inconsistency present in the comic's definition sketch where the function GMDN as defined in the second line has the same three-dimensional output as F, but GMDN in the last line is shown to produce a single real number rather than a vector and is thus missing a final operation of returning a single component. Note also that the comic's definition of the median as the (n+1)/2-th [https://en.wikipedia.org/wiki/Order_statistic order statistic], i.e. the (n+1)/2-th smallest value, coincides with the more regularly used sample median only on lists of odd length. For lists of even length the sample median is usually defined as the (arithmetic) mean of the two middle values <code>X_{n/2}</code> and <code>X_{(n+2)/2}</code> instead. Indeed, for lists of even lengths <code>X_{(n+1)/2}</code> is not well-defined without adding a flooring operation as (n+1)/2 is not an integer.
 The comment in the title text about suggests that this will save you the trouble of committing to the 'wrong' analysis as it gradually shaves down any 'outlier average' that is unduly affected by anomalies in the original inputs. It is a method without any danger of divergence of values, since all three averaging methods stay within the interval covering the input values (and two of them will stay strictly within that interval).

@@ Line 29: / Line 29: @@
   |-
   ! F2
-  | 2.124783495 ||	2.116192461 || 2
+  | '''2.124783495''' ||	2.116192461 || 2
   |-
   ! F3
-  | '''2.080325319''' || 2.079536819 || 2.116192461
+  | 2.080325319 || 2.079536819 || '''2.116192461'''
   |-
   ! F4
-  | 2.0920182 || 2.091948605 || '''2.080325319'''
+  | '''2.0920182''' || 2.091948605 || 2.080325319
   |-
   ! F5
-  | '''2.088097374''' || 2.088090133 || 2.091948605
+  | 2.088097374 || 2.088090133 || '''2.091948605'''
   |-
   ! F6
-  | 2.089378704 ||	2.089377914 || '''2.088097374'''
+  | '''2.089378704''' ||	2.089377914 || 2.088097374
   |-
   ! F7
-  | '''2.088951331''' ||	2.088951244 || 2.089377914
+  | 2.088951331 ||	2.088951244 || '''2.089377914'''
   |-
   ! F8
-  | 2.089093496 || 2.089093487 || '''2.088951331'''
+  | '''2.089093496''' || 2.089093487 || 2.088951331
   |-
   ! F9
-  | '''2.089046105''' || 2.089046103 || 2.089093487
+  | 2.089046105 || 2.089046103 || '''2.089093487'''
   |-
   ! F10
-  | 2.089061898 || 2.089061898 || '''2.089046105'''
+  | '''2.089061898''' || 2.089061898 || 2.089046105
   |}
-The function GMDN in the comic is properly defined in the second row since F acts on a vector to produce another three vector, however GMDN in the last line is shown to produce a single real number rather than a vector and is thus missing a final operation of returning a single component. Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value alternates between the average and the median (highlighted in bold in the table), while the minimum value alternates between the geomean and the median. This holds for many inputs thus providing the basis for a possible proof-by-induction of convergence on the range (see discussions).
+Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value alternates between the average and the median (highlighted in bold in the table), while the minimum value alternates between the geomean and the median. This observation holds for many inputs.
 The comment in the title text about suggests that this will save you the trouble of committing to the 'wrong' analysis as it gradually shaves down any 'outlier average' that is unduly affected by anomalies in the original inputs. It is a method without any danger of divergence of values, since all three averaging methods stay within the interval covering the input values (and two of them will stay strictly within that interval).

@@ Line 114: / Line 114: @@
 Geothm means "counting earths" (From Ancient Greek γεω- (geō-), combining form of γῆ (gê, “earth”) and ἀριθμός arithmos, 'counting').  Geothmetic means "art of Geothming" based on the etymology of Arithmetic (from Ancient Greek ἀριθμητική (τέχνη) (arithmētikḗ (tékhnē), “(art of) counting”).  This is an exciting new terminology that is eminently suitable for modern cosmology & high energy physics - particularly when doing math on the multiverse.  However, it is unlikely this etymology is related to the term "geothmetic meandian" as coined by Randall, as it can be more simply explained as a portmanteau of the three averages in its construction: '''geo'''metric mean, ari'''thmetic mean''', and me'''dian'''.
-The following Python code (inefficiently) implements the above algorithm:
+The following Python code (inefficiently) implements the above algorithm for a list of non-negative numbers:
 <pre>

@@ Line 85: / Line 85: @@
 |-
 ! Arithmetic
-| 2.4 ||
+| 2.4
 |-
 ! Geometric
 | 1.9743504858348
-| Multiply all numbers, then take it to the nth root, where n is the number of terms.
+| Multiply all numbers, then take the product's nth root, where n is the number of terms.
 |-
 ! Median
-| 2 ||
+| 2
 |-
 ! GMDN
-| 2.089 ||
+| 2.089
 |}

@@ Line 7: / Line 7: @@
 }}
 ==Explanation==
 There are a number of different ways to identify the "{{w|average}}" value of a series of values, the most common unweighted methods being the {{w|median}} (take the central value from the ordered list of values if there are an odd number - or the value half-way between the two that straddle the divide between two halves if there are an even number) and the {{w|arithmetic mean}} (add all the numbers up, divide by the number of numbers). The {{w|geometric mean}} is less well-known but works similarly to the arithmetic mean. The geometric mean of ''n'' positive numbers is the ''n''th root of the product of those numbers. If all of the numbers in a sequence are identical, then its arithmetic mean, geometric mean and median will be identical, since they would all be equal to the common value of the terms of the sequence. However, if the sequence is not constant, then {{w|Inequality_of_arithmetic_and_geometric_means#Geometric_interpretation|the arithmetic mean will be greater than the geometric mean}}, and the median may be different than either of those means.

@@ Line 10: / Line 10: @@
 There are a number of different ways to identify the "{{w|average}}" value of a series of values, the most common unweighted methods being the {{w|median}} (take the central value from the ordered list of values if there are an odd number - or the value half-way between the two that straddle the divide between two halves if there are an even number) and the {{w|arithmetic mean}} (add all the numbers up, divide by the number of numbers). The {{w|geometric mean}} is less well-known but works similarly to the arithmetic mean. The geometric mean of ''n'' positive numbers is the ''n''th root of the product of those numbers. If all of the numbers in a sequence are identical, then its arithmetic mean, geometric mean and median will be identical, since they would all be equal to the common value of the terms of the sequence. However, if the sequence is not constant, then {{w|Inequality_of_arithmetic_and_geometric_means#Geometric_interpretation|the arithmetic mean will be greater than the geometric mean}}, and the median may be different than either of those means.
-The geometric mean, arithmetic mean, {{w|harmonic mean}} (not shown), and the {{w|arithmetic-geometric mean}} (not shown) are collectively known as the {{w|Pythagorean means}}, as specific modes of a greater and more generalized mean formula that extends arbitrarily to various other possible nuances of mean-value rationisations (cubic, etc.).
+The geometric mean, arithmetic mean, and the {{w|harmonic mean}} (not shown) are collectively known as the {{w|Pythagorean means}}, as specific modes of a greater and more generalized mean formula that extends arbitrarily to various other possible nuances of mean-value rationisations (cubic, etc.).
 {{w|Outlier}}s and internal biases within the original sample can make boiling down a set of values into a single 'average' sometimes overly biased by flaws in the data, with your choice of which method to use perhaps resulting in a value that is misleading, exaggerating or suppressing the significance of any blips.