Arithmetic Mean

Arithmetic mean of two or more quantities is the sum of all the quantities divided by the number of values added in the sum.The general symbol used to denoted arithmetic mean is x. The same notation is used to represent the sample mean while the population mean is denoted by the Greek letter μ.

The arithmetic mean of n observations x₁, x₂,......x_n is given by the formula
x = $\frac{x_{1}+x_{2}+....+x_{n}}{n}$ and written using Sigma notation as $\frac{\sum_{i=1}^{n}x_{i}}{n}$

If these n observations have corresponding frequencies, the arithmetic mean is computed using the formula
x = $\frac{x_{1}f_{1}+x_{2}f_{2}+......+x_{n}f_{n}}{N}$ and using Sigma notation = $\frac{\sum_{i=1}^{n}x_{i}f_{i}}{N}$ where N = f₁ + f₂ + ..........+ f_n.

The above formula can also be used to find the weighted arithmetic mean by taking f₁, f₂.....f_n as the weights of x₁, x₂...... x_n. When the frequencies divided by N are replaced by probabilities p₁, p₂, ......p_n we get the formula for the expected value of a discrete random variable.
X = x₁p₁ + x₂p₂ +.......+ x_np_n. or using Sigma notation = $\sum_{i=1}^{n}x_{i}p_{i}$

1. When each item in the data set is increased by k, then the arithmetic mean is also increased by k.

2. The algebraic sum of deviations of all the items from the arithmetic mean is zero. That means if x is the arithmetic mean of n observations x₁, x₂,......x_n then (x₁ - x) + (x₂ - x) +.......(x_n - x) = 0. This property can be expressed using Sigma Notation as $\sum_{i=1}^{n}(x_{i}-\overline{x})$ = 0

3. If each of the observations is multiplied by a constant k, then the AM also gets multiplied by k.