Averages of a series

Arithmetic mean

The arithmetic mean of a series of nunmbers is calculated by summing all the numbers and then dividing them by the number of elements in the series.

Example: Calculate the arithmetic mean of this series of numbers: 14, 12, 16, 13, 14, 15, 12, 10, 16.

The sum of the numbers is 14 + 12 + 16 + 13 + 14 + 15 + 12 + 10 + 16 = 122
The number of the series elements is 9
The arithmetic mean of the series is therefore 122/9 = 13.56

Median

In statistics, the median of a series is computed by ordering these elements and then splitting the ordered series into two equal parts. The median is the value that separates the two series.

Two cases are distinguished depending on whether the number of elements is even or odd.

Odd case
We consider the above series : 14, 12, 16, 13, 14, 15, 12, 10, 16.
What is the median of this series?

We order the numbers : 10, 12, 12, 13, 14, 14, 15, 16, 16.
The number of elements is 9 which is an odd number, the series can be divided into two parts as follows :
10, 12, 12, 13, 14 , 14, 15, 16, 16

14 is the central value that divides the series into two parts so it is the median.

Even case
What is the median of this series : 4, 12, 7, 13, 4, 5, 2, 0, 6, 17.

We order the sample : 0, 2, 4, 4, 5, 6, 7, 12, 13, 17.
The number of values is 10 which is an even number, the series can be divided into two parts as follows:
0, 2, 4, 4, 5   |   6 , 7, 12, 13, 17

Unlike the odd case, there is not a central value. In this case, the two (underlined) values are averaged that is, (5+6) /2 = 5.5. So the median is 5.5.

Mode

The Mode is the most frequent value in a statistical series.

Example 1: What is the mode of the series 1, 1, 16, 13, 1, 14, 1, 5 ?

The mode is 1 because its frequency is 4 while the other values appear only once in the dataset.

Example 2: what is the mode of the series 10, 1, 10, 12, 15, 1.

The mode is equal to [1, 10] because these two values appear twice; all other values appear only once.

Geometric Mean

For a dataset of n numbers, the geometric mean is equal to the n-th root of the product of the n series values.

Let X be a dataset defined by,

`X = {x_1, x_2, ..., x_n}`

The geometric mean is then equal to,

`bar X_g = root(n)(\prod_{i=1}^{i=n}x_i)`

Example: `X = {1, 2, 5, 3,8}`

`bar X_g = root(5)(1.2.5.3.8) approx 2.99`

Harmonic Mean

For a series of n numbers, the harmonic mean is equal to the inverse of the arithmetic mean of the inverses of the series values.

If X is a series defined by,

`X = {x_1, x_2, ..., x_n}`

The harmonic mean is then written,

`bar X_h = n / (sum_{i=1}^{i=n}\frac{1}{x_i})`

Example: `X = {1, 2, 5, 3,8}`

`bar X_h = 5 / (\frac{1}{1} + \frac{1}{2} + \frac{1}{5} + \frac{1}{3} + \frac{1}{8}) approx 2.32`

Quadratic mean

For a series of n elements, the quadratic mean is equal to the square root of the mean of the squares of the series values.

Let X be a series defined by,

`X = {x_1, x_2, ..., x_n}`

The quadratic mean is then equal to,

`bar X_q = sqrt(1/n.sum_{i=1}^{i=n}x_i^2)`

Example: `X = {1, 2, 5, 3,8}`

`bar X_q = sqrt(1/5.(1^2+2^2+5^2+3^2+8^2)) approx 4.54`

Interquartile Mean

The interquartile mean is a statistical measure based on a truncated mean of interquartile values. This mean is calculated by discarding the 25% lowest values and the 25% highest values in the series and calculating the average of the remaining values.
X is a dataset defined by,

`X = {x_1, x_2, ..., x_n}`

We assume that the values has been ordered.
The interquartile mean is then calculated as follows,

`IQM = (2/n) * sum_{i=n/4+1}^{i=3*n/4}x_i`

Cubic Mean

For a series of size n, the cubic mean is equal to the cubic root of the cubic mean of the values in the series.

If X is the dataset defined by,

`X = {x_1, x_2, ..., x_n}`

The cubic mean is defined by,

`bar X_c = root(3)((1/n.sum_{i=1}^{i=n}x_i^3))`

Example: `X = {1, 2, 5, 3,8}`

`bar X_c = root(3)((1/5.(1^3+2^3+5^3+3^3+8^3))) approx 5.12`