geometric mean

l*he idea of a proportional multiplier needs to be mtroduced at this stage in order to .

Consider increasing some value, £200 say, by 20%. The most efficient way to do this is to multiply the value by

120

120%. That is, resultant value = £300 x = £300 x 1.2 = £360. Notę that the 120% comes from (100+20)% or, equivalentły, the 1.2 is madę up of 1 + 0.2.

Thus, for example:

to add	35%	multiply by	135%	or 1.35
to add	50%	multiply by	150%	or 1.50
to add	87%	multiply by	187%	or 1.87

In the above, 1.35,1.50 and 1.87 are called proportional multipliers.. Using proportions rather than percentages simpiifies the arithmetic involved in całculations.

14. Formula for the geometrie mean

The geometrie mean is a speciaiised measure, used to average proportional inereases.

In calculafcing the geometrie mean, there are three steps to follow.

STEP 1 Express the proportional inereases (p, say) as proportional multipliers (1+p)

For example, suppose a smali firm had been growing over a four year period, with its average number of empłoyees per year given as

84, 97, 116 and 129.

The proportional inereases from each year to the next can be całculated as:

97-84

84

= 0.155;

116-97

97

= 0.196;

129-126

116

= 0.112

STEP 2

The proportional multipliers are therefore: 1.155, 1.196 and 1.112 Calculate the geometrie mean multiplier using:

Geometrie mean multiplier = ^(l + p₁)(l + p₂)-**(l + p_fi) For the previous example we have:

g.m. multipl;ier = \i(l. 155)(1.196X1.112)

= V1.5361 = 1.154 (3D)

STEP 3 Subtract 1 from the g.m. multiplier to obtain the average proportional inerease.

For the above example, average proportional inerease = 1.154 — 1

= 0.154 = 15.4%

That is, the (geometrie) mean rise in empłoyees per year is 15.4%

Notę that (using the original data) a 15.4% inerease appłied 3 times successively to 84 will give a result of 129. That is, 84 x 1.154³ = 129.

The geometrie mean can be used to average proportional inereases in wages or goods, such as percentages or index numbers. Because of the way it is defined, it takes little account of extremes and is occasionalły used as an altemative to the arithmetic mean.

The Financial Times (FT) Index is the most well known example of the practical use of the geometrie mean. It is całculated as the geometrie mean of a set of selected share values.

15. Example 5 (Calculation of the geometrie mean)

If it is known that the price of a commodity has risen by 6%, 13%, 11% and 15% in each of four successive years* then the geometrie mean rise can be całculated as follows.    -

STEP 1 The four proportional multipliers are 1.06, 1/1'3, 1.11 and 1.15

STEP 2 The geometrie mean is given by

g.m.

= V 1.06x1.13x1.11x1.15