Question

How do you find the geometric mean and harmonic mean $?$

Answer 1

Finding geometric mean and harmonic mean :

In Mathematics, geometric mean and harmonic mean are the two different types of mean to calculate the average of the given values. The procedure to calculate the geometric and harmonic mean are given below:

Step 1:Steps to calculate the geometric mean:

1. Multiply the given numbers together.

2. Take $nth$ root for the result obtained from step 1, if $“n”$values are given.

If two values are given, take square root for the result obtained from step 1, or take cube root, if three values are given.

Geometric mean (GM)$=\mathbf{}\sqrt[\mathbf{n}]{{\mathbf{a}}_{\mathbf{1}}\mathbf{\times }{\mathbf{a}}_{\mathbf{2}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\mathbf{a}}_{\mathbf{n}}}$

Where,

$a_{1}x{a}_{2}x\dots a_n$ are the given values.

For example, $4,6$ are the two given values, the geometric mean is calculated as follows:

1. Multiply $4,6$ we get $24.$

2. Take the square root of $24$, because the number of values given is $2.$

Now,$\sqrt{24}=4.89$

Hence, the geometric mean for $4,6$is $4.89.$

Step 2: Steps to calculate the harmonic mean:

1. Write down the reciprocal of each given value.

2. Now, calculate the average value of the reciprocal obtained from step 1.

3. Finally, take the reciprocal of the average value.

The formula to calculate the harmonic mean is:

Harmonic mean (HM) $=\frac{\mathbf{n}}{{\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+...}}$

Where,

$a,b,c,\dots$ are the given data values

$n$ is the number of values.

Now, consider the same example. $4$and $6$are the two values, then the harmonic mean is calculated as follows:

Step 1: The reciprocal of $4,6$ are$\frac{1}{4},\frac{1}{6}$, respectively.

Step 2: The average value of $\frac{1}{4},\frac{1}{6}$ is:

Average$=\frac{0.25+0.16}{2}$

$=\frac{0.41}{2}$

Step 3: Now, take the reciprocal of the average value obtained in step 2.

Harmonic mean $=\frac{2}{0.81}=4.88$

Hence, in this way we can find the geometric mean and harmonic mean.