1、Geometric Mean and Harmonic Mean In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. In the context of mathematics and statistics, the word “average” can be the mean, median and mode, which
2、are all known as measures of central tendency or central location. In colloquial usage, any of these might be called an average value. The mean, in statistics, unless otherwise specified, usually refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. Wh
3、ile except the arithmetic mean, there are also two types of special mean, the geometric mean and harmonic mean. Both are the average but only used in special situations. Since we have learned the arithmetic mean, median and mode in our lectures, here we mainly discuss geometric mean and harmonic mea
4、n. Geometric Mean Definition The Geometric Mean is a special type of average where we multiply the numbers together for n numbers and then take the nth root (written ). More formally, the geometric mean of n numbers x1 to xn is: GM=12 Useful The Geometric Mean is useful when we want to compare thing
5、s with very different properties. For example: you want to buy a new camera. One camera has a zoom of 200 and gets an 8 in reviews. The other has a zoom of 250 and gets a 6 in reviews. Comparing using the usual arithmetic mean gives (200+8)/2 = 104 vs (250+6)/2 = 128. The zoom is such a big number t
6、hat the user rating gets lost. But the geometric means of the two cameras are: GM1=2008=40 GM2=2506=38.7 So, even though the zoom of the second camera is 50 bigger, the lower user rating of 6 is still important. A typical of an example of a geometric mean is finding the average percentage return in
7、finance. When the returns are annual, it is called the Compound Annual Growth Rate (CAGR). For example, if we are considering a period of two years, and the investment return in the first year is 10% and the return in the second year is +60%, then the average percentage return or CAGR, R, can be obt
8、ained by solving the equation: GM=(110%)(1+60%)1 = 0.2 = 20% The value of R that makes this equation true is 0.2, or 20%. This means that the total return over the 2-year period is the same as if there had been 20% growth each year. The order of the years makes no difference-the average percentage r
9、eturns of +60% and 10% is the same result as that for 10% and +60%. Harmonic Mean Definition The harmonic mean for a non-empty collection of numbers x1, x2, ., xn, all different from 0, is defined as the reciprocal of the arithmetic mean of the reciprocals of the xis. Yes, there is a lot of reciproc
10、als! The reciprocal of a number n is simply 1/n. The formula of harmonic mean is: HM= 11+12+1 Useful In some rate type questions the harmonic mean gives the true answer! One example where the harmonic mean is useful is when examining the speed for a number of fixed-distance trips. For example, if th
11、e speed for going from point A to B was 60 km/h, and the speed for returning from B to A was 40 km/h, then the harmonic mean speed is given by 2/(1/60 + 1/ 40) = 48 km/h. The harmonic mean is also good at handling large outliers. For example: for data 2, 4, 6 and 100 The arithmetic mean is (2+4+6+10
12、0)/4=28 The harmonic mean is 4/(1/2+1/4+1/6+1/100) = 4.32 (to 2 places) But small outliers will make things worse! A well known inequality concerning arithmetic, geometric, and harmonic means for any set of positive numbers is: AM = GM = HM. The alphabetical order of the letters A, G, and H is preserved in the inequality. Cited by https:/ https:/ https:/ https:/
