How to calculate a geometric mean: 6 steps (with pictures)

How to calculate a geometric mean: 6 steps (with pictures)
How to calculate a geometric mean: 6 steps (with pictures)
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The geometric mean is another type of mean, but instead of adding up your numbers and dividing them by the number of the series, as is the case with an arithmetic mean, here you have to multiply them before calculating a root of the result. This geometric mean is, for example, used to realize the performance of a portfolio of stocks over several periods. So, for the calculation of a geometric mean, you will multiply the values, then take the n-th root of the result, n being the number of values ​​in the series. There is another method of calculation that uses decimal logarithms.

Steps

Method 1 of 2: Calculate the geometric mean of a series of values

Calculate the Geometric Mean Step 1

Step 1. Multiply all the values ​​in the series

Depending on the case, you will use a calculator, or you will do the calculations by hand or by head. Do not forget any value otherwise your calculation will be wrong. Write the result of the product on a separate sheet, it will be used soon.

  • Take as an example, the numerical series composed of the values ​​3, 5 and 12. You are going to calculate the following product: 3 × 5 × 12 = 180 { displaystyle 3 \ times 5 \ times 12 = 180}

    {displaystyle 3\times 5\times 12=180}
  • Si votre série ne comprend que deux valeurs, le principe reste le même, à l'image de la série comprenant 2 et 18, le produit est le suivant: 2×18=36{displaystyle 2\times 18=36}
  • {displaystyle 2\times 18=36}
Calculate the Geometric Mean Step 2

Step 2. Calculate the n-th root of this product

The date of the root corresponds to the number of values ​​in the series. After the product of the values ​​made in the previous step, determine the size of the series by counting the number of values. It is this number which will be the date of the root to use. This is how you will take the square root of the product if you only have two values, the cubic root for three values ​​etc. For this root calculation, you need a calculator.

  • Let's take the series composed of 3, 5 and 12. The root here is cubic (3 values), so do the following calculation: 1803≈5, 65 { displaystyle { sqrt [{3}] {180}} approx 5, 65}

    {displaystyle {sqrt[{3}]{180}}\approx 5, 65}
  • Reprenons aussi la série composée des seules valeurs 2 et 18. La racine est ici carrée (2 valeurs), aussi faites le calcul suivant:: 36=6{displaystyle {sqrt {36}}=6}
  • {displaystyle {sqrt {36}}=6}

Variante:

la racine n-ième d'une valeur peut se calculer différemment, à savoir en élevant cette valeur à la puissance 1n{displaystyle {frac {1}{n}}}

{frac {1}{n}}

. Si votre calculatrice n'a pas la fonction xn{displaystyle {sqrt[{n}]{x}}}

{displaystyle {sqrt[{n}]{x}}}

, c'est une solution. Pour la série composée de 3, 5 et 12, la notation 1803{displaystyle {sqrt[{3}]{180}}}

{displaystyle {sqrt[{3}]{180}}}

est équivalente à 18013{displaystyle 180^{frac {1}{3}}}

{displaystyle 180^{frac {1}{3}}} Calculate the Geometric Mean Step 3

Step 3. Convert the percentages to decimal values

If your series is made up of percentages, you have to operate differently, because they are not values ​​like numeric values. If you operate directly as we have seen, you will get a false result. Transform each percent increase by dividing it 100 and add 1 and each percent decrease by dividing it 100 and subtracting that result from 1.

  • Say you have to calculate the geometric mean of the price of an object, which first increases by 10%, then decreases by 3%.
  • Convert 10% to a decimal number (10100 = 0.10 { displaystyle { frac {10} {100}} = 0.10}

    {displaystyle {frac {10}{100}}=0, 10}

    ) et ajoutez 1, ce qui vous donne 1, 10.

  • Convertissez ensuite 3 % en un chiffre décimal (3100=0, 03{displaystyle {frac {3}{100}}=0, 03}
  • {displaystyle {frac {3}{100}}=0, 03}

    ), puis soustrayez-le de 1, soit 0, 97.

  • Servez-vous de ces 2 valeurs pour la moyenne géométrique:

    1, 10×0, 97≈1, 03{displaystyle {sqrt {1, 10\times 0, 97}}\approx 1, 03}

    {displaystyle {sqrt {1, 10\times 0, 97}}\approx 1, 03}
  • Convertissez ce résultat en pourcentage. Soustrayez 1 du résultat obtenu précédemment, puis multipliez ce nouveau résultat par 100, ce qui donne ici:

    1, 03−1=0, 03{displaystyle 1, 03-1=0, 03}

    {displaystyle 1, 03-1=0, 03}

    , soit 3 % (0, 03×100=3{displaystyle 0, 03\times 100=3}

    {displaystyle 0, 03\times 100=3}

    ).

Méthode 2 sur 2: Calculer une moyenne géométrique à l'aide des logarithmes

Calculate the Geometric Mean Step 4

Step 1. Sum the logarithms of each of the values ​​in the series

It is a question of using here the decimal logarithm (of base 10). This calculation must be carried out with a scientific calculator. Locate the log key, type in the value you want to log, then simply press log. Press the + key, then the second value, then press log, etc. Don't forget to type the + sign after each log, it's important.

  • Consider a series made up of three values: 7, 9 and 12. You will type the following sum on your calculator: log (7) + log (9) + log (12) { displaystyle log (7) + log (9) + log (12)}

    {displaystyle log(7)+log(9)+log(12)}

    avant d'appuyer sur =. Dans ce cas très précis, vous allez avoir comme résultat 2, 878521796.

  • Vous pouvez aussi calculer chacun des logarithmes, noter les résultats et faire la somme après.
Calculate the Geometric Mean Step 5

Step 2. Divide the sum of the logarithmic values ​​by the series size

Count the number of values ​​(count) in your series, then divide the sum of the logarithms by the count. What you get is the logarithm of the geometric mean, not the geometric mean itself.

  • The series 7, 9 and 12 is made up of 3 values, so the calculation looks like this: 2, 8785217963≈0, 959507265 { displaystyle { frac {2, 878521796} {3}} approx 0, 959507265}

    {displaystyle {frac {2, 878521796}{3}}\approx 0, 959507265}
Calculate the Geometric Mean Step 6

Step 3. Calculate the geometric mean

To do this, you must use the inverse function of log (x), i.e. 10x. On your calculator, the two functions being linked, they are on the same key. The log function is marked on the key, 10x is above, in yellow and smaller. Press the 2nd key { displaystyle 2nd}

{displaystyle 2nd}

dans le coin supérieur gauche de la calculatrice, puis sur la touche log pour bénéficier de la fonction réciproque. tapez ensuite le résultat de la division précédente et vous aurez votre moyenne géométrique.

  • reprenons notre exemple. le calcul final se présente ainsi:

    100, 959507265≈9, 11{displaystyle 10^{0, 959507265}\approx 9, 11}

    {displaystyle 10^{0, 959507265}\approx 9, 11}

    . la moyenne géométrique est de 9, 11.

conseils

  • la moyenne géométrique des nombres négatifs n'existe tout simplement pas.
  • si vous avez un 0 dans votre série, inutile de faire tous ces calculs: la moyenne géométrique sera 0.

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