Understanding Types of Mean

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Introduction

The average value of a set of data points, observations, or values is known as the Mean of the data. It is the measure of the central tendency.

Mathematically, the mean is obtained by dividing the sum of values by the number of values or observations. It is also called the expected value. The mean itself is not restricted to this simple form but has different types such as the arithmetic mean, geometric mean, harmonic mean, and weighted average.

It is given mathematically as,

$$\mathrm{Mean=\frac{\sum x}{N}}$$

where,

x = set of observations

N = number of observations

Different types of Mean

Arithmetic Mean

It is given the arithmetic average of the data/observations. It is represented as the sum of the observations divided by the total number of such observations.

For example, let us consider x_1,x_2,x_3,x_4,....,x_n to be n observations, then Arithmetic mean μ,

$$\mathrm{\mu=\frac{x_1 +x_2+x_3+...+x_n}{n}=\frac{\sum x}{n}}$$

If 3, 4, 5,8,10 is a set of 5 observations, then the arithmetic mean will be,

$$\mathrm{\mu=\frac{3+4+5+8+10}{5}=6}$$

Geometric Mean

Geometric mean is represented as the nth root of the product of all the observations/values in the dataset. It is generally lesser than the arithmetic mean and commonly used where there is fluctuation in the data such as in investment use cases.

For example, let us consider x_1,x_2,x_3,x_4,....,x_n to be n observations, then Geometric mean G is given as

$$\mathrm{G=\sqrt[n]{x_1,x_2,x_3,\ldots,xn}}$$

The GM of three observations 2, 4, and 8 will be

$$\mathrm{G=\sqrt[n]{2\times 4\times 8}=\sqrt[n]{64}=4}$$

However, taking the log of the observations to compute Geometric Mean makes it much easier when the roots are larger. This is evident from the below representation

$$\mathrm{\log G = \frac{\log x_1+log x_2+log x_3+...+log x_n}{n}=\frac{log \sum x}{n}}$$

In the above example of observation 2,4,8 applying log,

$$\mathrm{\log G = \frac{\log 2+\log 4+\log 8}{n}=\frac{0.3010+0.6020+0.9030}{3}=0.602}$$

G = antilog 0.602 = 4

Harmonic Mean

Harmonic Mean is defined as the reciprocal of the arithmetic mean of the reciprocal of the values. In other words, the number of observations is divided by the sum of the reciprocal of the values to obtain the Harmonic Mean. Mathematically, for observations x_1,x_2,x_3,x_4,....,x_n

$$\mathrm{H=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4}+\dotso+\frac{1}{x_n}}}$$

For example, the harmonic mean of 2,4 and 8 will be

$$\mathrm{H=\frac{3}{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}}=\frac{3}{0.5 + 0.25+ 0.125}=3.428}$$

Weighted Mean/Average

Weighted average is used when each of the observations has different importance in the mean calculation. Each observation is weighted by a factor of w. The Weighted mean is calculated by summing the weighted value of each observation divided by the sum of the weights of observations. Mathematical, Mathematically, for observations x_1,x_2,x_3,...,x_n with weights w_1,w_2,w_3,...w_n,

$$\mathrm{\mu_w=\frac{w_1x_1+w_2x_2+w_3x_3\dotso+w_nx_n}{w_1+w_2+w_3+\dotso+w_n}=\frac{\sum wx}{\sum w}}$$

For example, let us consider the marks of a student in three subjects to be 90, 80, and 70 and weights assigned to the subjects accordingly be 5, 7, and 9 then the weighted mean will be

$$\mathrm{\mu_w = \frac{5\times 90+7\times 80+9\times 70}{5+7+9}=78.09}$$

Conclusion

The mean is one of the most popular and important central tendencies widely used in statistics and Machine Learning. It gives a summary of the entire data. Knowing the mean helps derives important insights about the datasets. There are different types of means including Arithmetic mean, Geometric Mean, Harmonic Mean, and Weighted Mean.