Question

What is the geometric mean of 14 and 56? What is the simplest form?

Answer 1

Hint: In statistics, the geometric mean (G.M) is the average or mean that signifies the central tendency of the given collection of data by finding the product of their values.
Generally, We have to multiply the given number altogether and then take out the ${n^{th}}$ root of the multiplied numbers, where $n$ is the total number of values.
Formula to be used:
The formula to find the geometric mean for the given collection of data is as follows:
     $G.M = \sqrt[n]{{{x_1} \times {x_2} \times ...... \times {x_n}}}$
               Or
     $G.M = {({x_1} \times {x_2} \times ........ \times {x_3})^{\dfrac{1}{n}}}$
Where,${x_i}(1 \leqslant i \leqslant n)$ denotes each value in the given set and $n$ is the number of observations.

Complete step-by-step solution:
The geometric mean refers to the \[{n^{th}}\] root of the product of $n$ numbers. We can note that the geometric mean is totally different from the arithmetic mean .Because, in the arithmetic mean; we need to add the given values and then divide it by the total number of observations. But in geometric means, we need to multiply the given values and then take the root with respect to the number of values.
Examples: if we have three data values, then product the three values and takes the cube root.
Now, let us get into our question. It is given that the set contains two values 14 and 56.
Let ${x_1} = 14$ and ${x_2} = 56$
Then, the number of observation,$n = 2$ we know that the formula to calculate the geometric mean for two values is as follows:
$GM = \sqrt {{x_1} \times {x_2}} $
Hence,$G.M,\overline x = \sqrt {14 \times 56} $
$ = \sqrt {784} $
$ = \sqrt {28 \times 28} $ $ = 28$ .

Note: Generally, the mean is the most commonly used measures of central tendency. We know that the central tendency is a single value which acts as a representative for the whole collection of data and the different types of mean are arithmetic mean (A.M), geometric mean (G.M) and harmonic mean (H.M).
Also, the value of the geometric mean is always less than that of the arithmetic mean for the given collection of data.