Is there anything significant about a geometric mean and arithmetic mean that fall very close to one another, say ~0.1%? What conjectures can be made about such a data set?
I've been working on analyzing a data set, and I notice that ironically the values are very, very close. Not exact, but close. Also, a quick sanity check of the arithmetic mean-geometric mean inequality as well as a review of data acquisition reveal that there is nothing fishy about the integrity of my data set in terms of how I came up with the values.
Best Answer
The arithmetic mean is related to the geometric mean through the Arithmetic-Mean-Geometric-Mean (AMGM) inequality which states that:
$$\frac{x_1+x_2+\cdots+x_n} n \geq \sqrt[n]{x_1 x_2\cdots x_n},$$
where equality is achieved iff $x_1=x_2=\cdots=x_n$. So probably your data points are all very close to each other.