“Tell Me About Statistics!” vol.36 What is the geometric average?

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Usually, the average value we use is that obtained by adding all the data such as “average age,” “average height,” “average weight,” and dividing the aggregate by the total number of data (number of people). This is referred to as the “arithmetic average.”

So, what kind of average is the “geometric average”? In this study, we explain the geometric average.

Geometric average

The geometric average is the “average value of the rate of change.” For example, let us say that the annual prescription volume for a drug doubles after one year and triples thereafter. In this case, the average annual growth rate is referred to as the geometric average.
(It is incorrect to use the arithmetic average calculation here and multiply by (2 + 3) ÷ 2 = 2.5.)

If there are n data points, this is obtained by multiplying n numbers and taking their n-squared roots.

[Formula]

When n data are X₁ , X₂ , X_3・・・X_n ,

Data in Table 1 were obtained by examining the prescription volume growth rate of drug X, which is useful for certain diseases, from the first to the fourth year of its launch. Let us find the geometric average prescription volume growth rate.

[Table 1]

We substitute the growth rate in the second year as X1, third year as X2, and fourth year as X3 into the formula.

We observe that the average annual growth rate of prescriptions is doubling.

Notes on geometric average

An average annual increase of two times means that the prescription amount in the second year will double from the year of launch: 2 × 2 = four times in the 3rd year, and 2 × 2 × 2 = eight times in the 4th year.

If the prescription amount is d1 in the first year, d2 in the second year, d3 in the third year, and d4 in the fourth year, the growth rate ( X2, X3 ) and the geometric average for each year can be determined using the following formula:

In other words, the geometric average of the growth rate can also be obtained by dividing the data in the final year (8,000) by that in the release year (1,000), and taking the cube root of the value (8).

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