Arithmetic Mean

The arithmetic mean is the sum of all values, x₁, x₂, ..., x_n, divided by the total number of values n. $$ \mu = \frac{x_1+x_2+...+x_n}{n} = \frac{1}{n} \cdot \sum_{i=1}^n x_i $$

The arithmetic mean is the most well-known type of average and also the easiest to calculate.

When people mention the "average" without further context, they are usually referring to the arithmetic mean.

What is it used for?

The arithmetic mean is one of the summary measures of central tendency.

It gives an indication of the central value of a dataset.

Limitations of the Arithmetic Mean. The arithmetic mean doesn’t provide any insight into how the data is distributed or the overall trend. The mean value can sometimes be far from the actual values in the dataset. Additionally, the mean is only a good measure of central tendency if it falls within the range where the majority of data points are concentrated, which isn’t always the case.

A Practical Example

Consider the following set of numbers:

$$ 1 \ , \ 5 \ , \ 7 \ , \ 3 \ , \ 6 \ , \ 8 $$

Here, n=6 because there are 6 values in the set.

The arithmetic mean of this set is 5:

$$ \mu = \frac{1+5+7+3+6+8}{6 }$$

$$ \mu = \frac{30}{6 }$$

$$ \mu = 5 $$

Example 2

Now, consider this different set of numbers:

$$ 1 \ , \ 2 \ , \ 3 \ , \ 27 \ , \ 28 \ , \ 29 $$

The arithmetic mean of this set is 15:

$$ \mu = \frac{1+2+3+27+28+29}{6 }$$

$$ \mu = \frac{90}{6 }$$

$$ \mu = 15 $$

Note. In this case, the mean is quite distant from most of the actual values, as the numbers are clustered at the extremes.

Example 3

Given a set of n=5 numbers:

$$ 7 \:, \: 10 \:, \: 6 \:, \: 2 \:, \: 5$$

The arithmetic mean is 6:

$$ x = \frac{1}{5} \sum_{i=1}^5 x_i $$

$$ x = \frac{1}{5} \cdot (7+10+6+2+5) $$

$$ x = \frac{1}{5} \cdot (30) $$

$$ x = 6 $$

Further Observations

Here are a few additional notes about the arithmetic mean:

• Weighted Arithmetic Mean
  The weighted arithmetic mean is a variation where each value is assigned a specific weight (w_i). $$ \mu = \frac{ \sum_{i=1}^n x_i \cdot w_i}{ \sum_{i=1}^n w_i } $$

And so on.