We have seen earlier that the geometric mean is the antilogarithm of the arithmetic mean of the logarithms, and the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.
Likewise, the quadratic mean (Q) is the square root of the arithmetic mean of the squares.
Symbolically,
Instead of using original values, the quadratic mean can be used while averaging deviations when the standard deviation is to be calculated. This will be used in the next chapter on dispersion.
Relative Position of Different Means
The relative position of different means will always be:
Q> x >G>H provided that all the individual observations in a series are positive and all of them are not the same.
Composite Average or Average of Means
Sometimes, we may have to calculate an average of several averages. In such cases, we should use the same method of averaging that was employed in calculating the original averages.
Thus, we should calculate the arithmetic mean of several values of x, the geometric mean of several values of GM, and the harmonic mean of several values of HM. It will be wrong if we use some other average in averaging of means.