How to Find Two Observations Given Arithmetic Mean and Geometric Mean

When dealing with statistical problems, it's often necessary to find the values of two observations given their arithmetic mean and geometric mean. This article will guide you through the process of finding two such observations given that their arithmetic mean is 25 and their geometric mean is 15.

Understanding the Problem

Let x and y be the two observations. We are given that the arithmetic mean of these two observations is 25, and their geometric mean is 15.

Arithmetic Mean (AM)

The arithmetic mean is calculated as follows:

$$ AM frac{x y}{2} $$

From the problem, we know:

$$ frac{x y}{2} 25 $$

Therefore:

$$ x y 50 $$ Equation 1

Geometric Mean (GM)

The geometric mean is calculated as follows:

$$ GM sqrt{xy} $$

From the problem, we know:

$$ sqrt{xy} 15 $$

Therefore:

$$ xy 15^2 225 $$ Equation 2

We now have a system of two equations:

1. ( x y 50 ) 2. ( xy 225 )

Let's solve this system step-by-step.

Formulating the Equations and Solving

From Equation 1, we can express y in terms of x as:

$$ y 50 - x $$

Substituting this into Equation 2:

$$ x(50 - x) 225 $$

This simplifies to:

$$ 5 - x^2 225 $$

Rearranging gives us a quadratic equation:

$$ x^2 - 5 225 0 $$

We can solve this quadratic equation using the quadratic formula:

$$ x frac{-b pm sqrt{b^2 - 4ac}}{2a} $$

where ( a 1 ), ( b -50 ), and ( c 225 ).

Calculating the discriminant:

$$ b^2 - 4ac (-50)^2 - 4 cdot 1 cdot 225 2500 - 900 1600 $$

Now applying the quadratic formula:

$$ x frac{50 pm sqrt{1600}}{2} $$

$$ x frac{50 pm 40}{2} $$

Which gives us two possible values for x:

1. ( x frac{90}{2} 45 ) 2. ( x frac{10}{2} 5 )

We can now find y using Equation 1:

1. If ( x 45 ), then ( y 50 - 45 5 ) 2. If ( x 5 ), then ( y 50 - 5 45 )

Thus, the two observations are 45 and 5.