Is it possible to calculate the arithmetic mean from the geometric mean?

Solution 1:

Unfortunately the AM-GM inequality is the best you can do. If your data is $\{x,\frac{1}{x}\}$ the geometric mean will be $1$, yet you can make your arithmetic mean any value in $[1,+\infty)$ by choosing $x$ large enough.

Solution 2:

You can use the A.M. - G.M. inequality which is as follows- $$\frac{x_1+x_2+\cdots+x_n}{n}\ge\sqrt[\large n]{x_1\cdot x_2\cdots x_n}$$

Solution 3:

Since the geometric mean for both $(2,2)$ and $(1,4)$ is $2$, while the arithmetic means are $2$ and $2.5$, the answer is a clear no. The only thing you can say is that the geometric mean is smaller or equal to the arithmetic.

Solution 4:

No it is not. Arithmetic mean gives you one equation. And there are two numbers to solve for. So there are infinite possibilities. For example, $$a=1,b=100$$ $$ a=0,b=101$$ They both have same A.M but widely different G.M

Solution 5:

Of course the answer is "no," not without more data. There's not a direct calculation.

In finance, what's interesting is that the arithmetic mean will always be a bit higher than the geometric for year on year stock returns.

From 1929 to 2013, the average was 11.41% (S&P return) yet the geometric mean was 9.43% nearly 2% lower per year. An understanding of the math behind this difference is helpful when someone mentions the market's return over a particular period.