Who in their right mind would use the geometric mean?

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

08 November 2009

A few weeks ago, this column looked at the proper way to determine the average value of a given set of data. Three methods were discussed, namely, the mean, the mode and the median.  It was explained that there are two types of mean – arithmetic and geometric.

The arithmetic mean is well known. It is found by adding all the data and dividing the sum by the number of values. The geometric mean on the other hand, is found by multiplying all the data and then calculating the “n-th” root of the product (where n is the number of values).

If, for example, there are ten values, then their geometric mean is the 10th root of their product. That is, the number which, when multiplied by itself ten times, gives the product of the ten values.

These two mean values are very different. Suppose the data comprises two equal quantities of 100 each. The arithmetic mean is 200 (100 + 100) divided by 2 which comes to 100. The geometric mean is the square root of 10,000 (100 x 100) which is also 100.

Now if the data is changed to 80 and 120, the arithmetic mean remains 100 but the geometric mean changes: 80 x 120 is 9,600; and the square root of 9,600 is approximately 98.

But the big question is: who in their right mind use the geometric mean? This average is quite common in financial circles. It is used in determining the average interest rate over a period of time.

Suppose you have invested some sh100,000 and you earn 5 percent in the first year and then 35 percent in the second year. What is the average annual earning rate? It is tempting to add 5 to 35 and the divide by two to get 20 percent but this would be wrong.

After the first year, you investment has grown by 5 percent to sh105,000. In the second year, you earn 35 percent of sh105,000 bringing you total to sh141,750. Now, if you used the arithmetic mean value of 20 percent, the result would come to sh120,000 in the first year and then sh144,000 in the second – a difference of sh2,250

The correct way to get the average is to appreciate that in the first year the investment grows by a factor of 1.05 (5 percent) and in the second year by 1.35 (35 percent). The combined result in the two-year period is 1.05 x 1.35 = 1.4175. Now the question to consider is: what number would be multiplied by itself to give 1.4175? In other words, what is the square root of 1.4175?

The answer is 1.191. Therefore, the investment earned and average of 19.1 percent per annum. The difference may seem small but it translates to sh2,250 out of sh100,000.