Recall that in the proportion a b = c d , b and c are called the means , and a and d are called the extremes . When the means of a proportion are the same number, that number is called the geometric mean of the extremes. So if p x = x q , then cross-multiplying gives x 2 = p q . Taking the square root of both sides, we get x = p q as the geometric mean of p and q . More generally, the geometric mean of a set of n numbers is the n th root of their product.
Example 1: Find the geometric mean of 25 and 9 There are two numbers. So, the geometric mean of the two numbers is the square root of their product. Geometric mean = 25 ⋅ 9 = 225 = 15 The geometric mean of 25 and 9 is 15 .
Example 2: Find the geometric mean of 4 , 10 and 25 . There are three numbers. So, the geometric mean of the three numbers is the cube root of their product. Geometric mean = 4 ⋅ 10 ⋅ 25 3 = 1000 3 = 10 The geometric mean of 4 , 10 and 25 is 10 . For geometric mean calculation, please enter numerical data separated with comma (or space, tab, semicolon, or newline). For example: 853.4 709.0 457.7 980.3 -670.0 404.0 809.6 283.1 383.9 579.6 539.3 914.6 976.1 The geometric mean of the set of positive numbers is nth root of the product of the values (n=count of values). Similarity to the arithmetic average is after replace operators: sum by product and dividing by the n-th root. Simple. First-type data elements (separated by spaces or commas, etc.), then type f: and further write frequency of each data item. Each element must have a defined frequency that count of numbers before and after symbol f: must be equal. For example:1.1 2.5 3.99
10-20 20-30 30-40 f: 5 10 15Similar to a frequency table, but instead f: type cf: in the second line. For example: 10 20 30 40 50 60 70 80
more math problems »
The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.
In one line: Geometric Mean of 2 and 18 = √(2 × 18) = 6 It is like the area is the same!
In one line: Geometric Mean = 3√(10 × 51.2 × 8) = 16 It is like the volume is the same:
In one line: Geometric Mean = 5√(1 × 3 × 9 × 27 × 81) = 9 I can't show you a nice picture of this, but it is still true that: 1 × 3 × 9 × 27 × 81 = 9 × 9 × 9 × 9 × 9
Using scientific notation:
Geometric Mean= √(0.275 × 10-9 × 8.8 × 103) = √(2.42 × 10-6) ≈ 0.0016 m Which is 1.6 millimeters, or about the thickness of a coin. We could say, in a rough kind of way, "a millimeter is half-way between a molecule and a mountain!" Another cool one:
Geometric Mean= √(3 × 10-8 × 1.3 × 107) = √(3.9 × 10-1) = √0.39 ≈ 0.6 m A child is about 0.6 m tall! So we could say, in a rough kind of way, "A child is half-way between a cell and the Earth" So the geometric mean gives us a way of finding a value in between widely different values. DefinitionFor n numbers: multiply them all together and then take the nth root (written n√ ) More formally, the geometric mean of n numbers a1 to an is: n√(a1 × a2 × ... × an) UsefulThe Geometric Mean is useful when we want to compare things with very different properties.
Example: you want to buy a new camera.
Comparing using the usual arithmetic mean gives (200+8)/2 = 104 vs (250+6)/2 = 128. The zoom is such a big number that the user rating gets lost. But the geometric means of the two cameras are:
So, even though the zoom is 50 bigger, the lower user rating of 6 is still important. 9379, 9380, 9381, 9382, 9383, 9384, 9385, 9386, 9387, 9388 Copyright © 2022 Rod Pierce |